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Rajendra Acharya4, 5, 6 +1 Department of Computer Science, Faculty of Computer Science and Telecommunications, Cracow University of +Technology, Krakow, Poland +2 Department of Computer Engineering, Ferdowsi University of Mashhad +3 Institute of Theoretical and Applied Informatics, Polish Academy of Sciences, Gliwice, Poland +4 Department of ECE, Ngee Ann Polytechnic, 535 Clementi Road, Singapore 599 489, Singapore +5 Department of Biomedical Engineering, School of Science and Technology, SUSS University, Singapore +6 Department of Biomedical Informatics and Medical Engineering, Asia University, Taichung, Taiwan + +Objectives: To extract datasets containing useful information from two drug databases and +recommend a list of drugs to physicians and patients with high accuracy while considering +the most important features of patients and drugs. The history and review of the target +patient and similar patients, and drug information, are used as a reference to recommend +drugs. +Methods: A comprehensive pharmaceutical recommendation system was designed based +on the patients’ and drugs’ features extracted from Drugs.com and Druglib.com. First, data +from these databases were combined, and a dataset of patients and drug information was +built. Secondly, the patients and drugs were clustered, and then the recommendation was +performed using different ratings provided by patients, and importantly by the knowledge +obtained from patients and drug specifications, and considering drug interactions. To the +best of our knowledge, we are the first group to consider patients’ conditions and history +in the proposed approach for selecting a specific medicine appropriate for that particular +user. Our approach applies artificial intelligence (AI) models for the implementation. +Sentiment analysis using natural language processing approaches is employed in pre- + +processing along with neural network-based methods and recommender system algorithms +for modeling the system. In our work, patients’ conditions and drugs’ features are used for +making two models based on matrix factorization. Then we used drug interaction to filter +drugs with severe or mild interactions with other drugs. +We developed a deep learning model for recommending drugs by using data from 2304 +patients as a training set, and then we used data from 660 patients as our validation set. +After that, we used knowledge from critical information about drugs and combined the +outcome of the model into a knowledge-based system with the rules obtained from +constraints on taking medicine. +Results: The results show that our recommendation system is able to recommend the best +combination of medicines according to the existing real-life prescriptions available. It also +has the best accuracy and other metrics in recommending a specific drug compared to other +existing approaches, which are generally based only on patient ratings or comments. Our +proposed model improves the accuracy, sensitivity, and hit rate by 26%, 34%, and 40%, +respectively, compared with conventional matrix factorization. In addition, it improves the +accuracy, sensitivity, and hit rate by an average of 31%, 29%, and 28% compared to other +machine learning methods. We have also open-sourced our implementation in Python. +Conclusions: Our proposed RECOMMED system extracts all vital information from the +drug, patient databases and considers all necessary factors for recommending accurate +medicine which can be trusted more by doctors and patients. We have shown the efficacy +of our proposed model in real test cases. +Keywords: recommendation system; drug recommendation system; drug information +extraction; hybrid recommendation method + +1- INTRODUCTION +Recommendation systems (RS) are knowledge extraction systems that use information retrieval +approaches to help people make better decisions and discover items through a complex information +space [1], [2]. They have been around for many years, and with the advancement in machine + +learning approaches, their use has been widened, and it helps people to make more appropriate +decisions in using different products. The popularity of using RS in different fields has increased +since the announcement of the Netflix Prize competition that aimed to predict movie rates [3]. The +application of recommender systems is very extensive: from entertainment to e-commerce, the +tourism industry, and medical recommender systems. Also, with rapid progress in artificial +intelligence, there has been a greater acceleration in the application of recommender systems and +their development. +Medical recommender systems are a particular type of recommender system, and they have some +distinct features that make them special: They have to be used very carefully, and because they +affect people’s health, there are many concerns about using RS for them. On the other hand, many +people die every year because of medication errors. It has been reported as the third leading cause +of death in the world [4]. This makes the use of intelligent systems in medical science valuable +and necessary. Drug prescription is also vital for physicians, and it involves considering different +aspects. Patient history of using drugs, the specification of drugs for diseases related to the +recommendation in question, and the drug's effectiveness for that specific case are among such +concerns. +Having as many different medicines as 24000 [5] in just one database, they can benefit from a +recommendation system to perform a set of suggestions for a particular patient with a specific +disease to help physicians in prescribing the most appropriate medicines and also help patients to +have a better choice in using drugs. +Recommender systems can be distinguished by the degree of risk imposed when a user accepts a +recommendation [6]. In this regard, the medical domain can be seen as high risk, mostly due to the +recommendation given to the user. +On the other hand, while having a comprehensive drug recommender system is important, +designing a complete system requires a dataset of drugs with patient ratings, reviews, and also +information about drugs. +We gathered this information from two different and well-known databases Drugs.com [5] and +Druglib.com [7] and we built three datasets which train the system and construct the final model. +Finally, putting all of them together, we proposed a novel drug recommender system called + +RECOMMED that learns the patient and drug features and their previous drugs taken, and also the +user reviews for different drugs to recommend a new drug to a patient. +The novelty of this work lies in the following parts: +1- Propose a pharmaceutical recommender system by considering the features of patients and +drugs, including patients’ conditions, age, gender, drug side effects, and drug categories. +2- Performing pre-processing steps on databases Druglib.com and Drugs.com websites to +gather the appropriate data for our recommendation system, leading to comprehensive +datasets for drug information. +3- Our system considers sentiment analysis of reviews, the prescriptions of doctors, and +different similarity measures for recommending a medicine and its dose and other +recommendations, including side effects and warnings for their usage. +4- Our system consists of a knowledge-based component to exclude drugs with serious side +effects for a specific patient. +5- We proposed a model to predict the efficiency of medicine for patients. + +In the next section, we provide some background in the general and general recommender systems +field; later in Section 3 we particularly introduce the drug recommender systems and their +challenges. Then in Section 4 we provide the current state-of-the-art of recommender systems +methods, specifically drug recommender systems. Section 5 is an elaborate explanation of our +proposed comprehensive drug recommender system in detail. Section 6 provides the results of this +work, plus a discussion about them. Finally, in Section 7 we conclude the paper and present the +future directions of this research. + +2- RECOMMENDATION SYSTEMS BACKGROUND +Recommender systems are decision-making systems that extract information from different kinds +of knowledge. For many years, many recommender systems have been in various domains with +different purposes [8]. In this section, we review the basic concepts of various types of +recommender systems and the way they are categorized. + +According to the type of data that recommender systems use to make decisions, the algorithms +utilized in recommender systems have two major categories: - collaborative filtering (CF) and - +content-based filtering (CB). CF approaches are further divided into user-based and item-based +approaches. CF and CB approaches have shown acceptable results when they are used in +recommending different kinds of products like movies, books, and music. +Also, the recommendation system can combine these two major techniques, usually called hybrid +recommendations. In addition, there are some specific types of recommender systems that have +their strength in various domains. One of these types is a knowledge-based recommender system. +Here, we briefly introduce the major recommender system techniques considered in this work. +2-1 Collaborative recommender systems +The idea behind this group of recommendation methods is to use a measure of similarity between +users or items to recommend something to a given user. It states that if two users share some +interest in the past, they will likely have similar interests in the future. A collaborative approach is +based on the rating a user gives to items; in its basic form, it doesn’t need any other information +about users and items. CF approaches can be divided into two basic types, neighborhood methods +(also known as memory-based) and latent factor models. Neighborhood methods are divided into +one of the following two basic methods: +2-1-1 User-based neighborhood recommender system +This approach aims at suggesting the products based on the similarity between users. In this regard, +several similarity measures can be used. +We denote 𝒰 as the set of users, ℐ as the set of items, and 𝑅 as the set of existing ratings, +Pearson Correlation (PC) is one of the popular ones, which is computed as equation (1) for users +𝑢 and 𝑣 [9]: +𝑃𝑒𝑎𝑟𝑠𝑜𝑛_𝐶𝑜𝑟𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛(𝑢, 𝑣) = +∑ +(𝑟𝑢𝑖−𝑟̅𝑢)(𝑟𝑣𝑖−𝑟̅𝑣) +𝑖∈ℐ𝑢𝑣 +√∑ +(𝑟𝑢𝑖−𝑟̅𝑢)2 +𝑖∈ℐ𝑢𝑣 +√∑ +(𝑟𝑣𝑖−𝑟̅𝑣)2 +𝑖∈ℐ𝑢𝑣 + +(1) +In this equation, ℐ𝑢 is the set of items rated by user 𝑢 and ℐ𝑢𝑣 is the items rated by both 𝑢 and 𝑣. +Also, 𝑟𝑢𝑖 is the rating of the user 𝑢 for a new item 𝑖 and 𝑟̅𝑢 is the average of the ratings given by 𝑢. + +And the prediction for the rating of user 𝑢 for item 𝑖 is calculated as equation (2): +𝑝𝑟𝑒𝑑(𝑢, 𝑖) = +∑ +𝑠𝑖𝑚(𝑢,𝑗)∗(𝑟𝑖,𝑗−𝑟̅𝑗) +𝑗 ∈𝑢𝑠𝑒𝑟𝑠 +∑ +𝑠𝑖𝑚(𝑢,𝑗) +𝑗 ∈𝑢𝑠𝑒𝑟𝑠 ++ 𝑟̅𝑢 +(2) +Where 𝑠𝑖𝑚 is the measure of similarity between user 𝑢 and item 𝑖 and 𝑢𝑠𝑒𝑟𝑠 is the set of users +most similar to user 𝑢. Therefore, the ratings are weighted by the similarity measure in this +prediction. +2-1-2 Item-based neighborhood recommender system +In contrast to user-based recommender systems, item-based recommender systems use item +similarity to suggest a product to a specific user. Similar to the user-based approach, here the +prediction for user 𝑢 for item 𝑖 is also calculated as equation (3): +𝑝𝑟𝑒𝑑(𝑢, 𝑖) = +∑ +𝑠𝑖𝑚(𝑖,𝑗)∗(𝑟𝑢,𝑗−𝑟̅𝑗) +𝑗 ∈𝑁 +∑ +𝑠𝑖𝑚(𝑖,𝑗) +𝑗 ∈𝑁 ++ 𝑟̅𝑖 +(3) +Where 𝑠𝑖𝑚 is the measure of similarity between items 𝑖 and 𝑗 and 𝑁 is the set of items similar to +item 𝑖 rated by 𝑢. +2-1-3 Matrix factorization +One of the biggest challenges for standard methods in CF is the sparsity of the rating matrix (or +user-item matrix); model-based CF can help overcome this challenge. There are many ways to +build models based on which we can make recommendations. Matrix factorization is one the +popular methods with the idea of decomposition of a matrix into the product of two or maybe three +matrices. +Having a dataset of the ratings of various users for different items, this model transforms the rating +matrix to the individual user and item matrices. +So the model is defined as follows: +Assume there is a set of users 𝑈 and items 𝐷, with rating matrix 𝑅(𝑀 × 𝑁), which is the ratings +given by users on items. 𝑀 and 𝑁 are the total numbers of users and items, respectively. Matrix +factorization in recommender systems aims to find 𝑘 total latent features/factors by decomposing +𝑅 according to equation (4) to user matrix 𝑈 and item matrix 𝐼. + +𝑅 ≈ 𝑈 × 𝐼𝑇 = 𝑅̂ +(4) +U is a 𝑀 × 𝑘 embedding matrix and, +I is a 𝑁 × 𝑘 embedding matrix. +2-2 Content-based recommender systems +Content-based recommendation uses the attributes of the users or user profile and the attributes of +items to recommend an item to a user [10]. Providing this information requires extra work and +effort to represent items properly and build a user profile appropriate for the recommendation +process [10]. +This kind of recommender system learns the user preferences and tries to recommend items similar +to the user's preferences. +Having 𝐷 rated items by user 𝑈, content-based RS aims to find the rating for item 𝑖, which is not +seen by user 𝑈. +In this method, items’ features are extracted, then used to find similarities between items. Then, in +a simple nearest neighbor approach top-𝑛 nearest neighbors of item 𝑖 in 𝐷 are selected. This +selection is based on a similarity measure like cosine similarity which is calculated as equation +(5): +cos(𝜃) = +𝑿 .𝒀 +‖𝑿‖×‖𝒀‖ = +∑ +𝑋𝑖𝑌𝑖 +𝑛 +𝑖=1 +√∑ +𝑋𝑖 +2 +𝑛 +𝑖=1 +√∑ +𝑌𝑖 +2 +𝑛 +𝑖=1 + +(5) +The ratings of these 𝑛 items are used to predict the rating for item 𝑖 by user 𝑈. +In most content-based recommender systems, item features are textual descriptions and don’t have +well-defined values. So, natural language processing approaches like TF-IDF or the bag-of-words +are used to assign numerical values to the textual features. +2-3 Hybrid recommender systems +Hybrid approaches combine different recommender system algorithms to make a more accurate +system that considers the benefits of different approaches for recommending an item to the users. + +A combination of content-based and collaborating filtering is the most common type of +hybridization method [11]. +2-4 Knowledge-based recommender systems +This RS aims to produce recommendations based on existing rules that satisfy a user’s needs. In +the context of drug recommendation, this knowledge involves many different conditions. For +example, death reports for a specific drug and drug interactions are two important information that +a drug recommender system has to consider before recommending a list of medicine to a patient. +2-5 AI-based recommendation systems +Over time many different artificial intelligent approaches have been applied to recommendation +systems. However, the tendency to use AI methods in recommender systems is mostly because of +the big data availability and diversity of recommendation systems approaches, which can benefit +from AI, particularly machine learning algorithms. +Deep learning as a subfield of machine learning has attracted many researchers from a broad +variety of disciplines due to its learning capabilities from data. Recently there have been many +researches on deep learning-based recommendation systems [12], and Multilayer Perceptron +(MLP), Autoencoder (AE), Convolutional Neural Networks (CNN), Recurrent Neural Networks +(RNN), are among the mostly used deep learning models in RS [13]. Many of these deep learning- +based approaches have contributed to the works on CB, CF, and other types of RS [14]. Also, some +works utilize hybrid deep networks, like the combination of RNN and CNN [15]. Moreover, to +integrate the advantages of memorization and generalization for recommender systems, a wide & +deep neural network has been used [16], and the model shows better results with increased +acquisitions on the Google Play app. +2-6 Other types of recommendation systems +Although these techniques are the basic and mostly used recommender system approaches, several +more types of recommender systems are suggested in the literature, and authors in [17] give a +detailed classification. + + +Medical and drug recommendation is one of the important applications of recommender systems +which uses techniques in recommender systems to recommend medicine, predict the usefulness of +drugs, etc. In the following sections, the position of recommender systems in medical science, +particularly in the pharmaceutical sector and the state-of-the-art in this field, is discussed. + +3- RECOMMENDER SYSTEMS IN MEDICAL SCIENCE + +One of the attractive and important applications of recommendation systems is medical +recommendations and drug products. +Here are the major differences between medical recommender systems and other recommender +systems: +- +Medical recommender systems care more about the health of patients than to make a profit. +- +Security is the primary goal in drug recommender systems. +- +Many existing recommender system techniques cannot be used, and others must be used +with caution because of safety issues. +- +In the long term, time is considered an important factor in recommending a drug. There are +many situations where some drugs' negative effects are discovered over time. One example +is the drug zimeldine [18]. So, a comprehensive medical recommender system should +consider ratings in different time stamps. +In the drug recommender system, the domain is medicine, and the exact contents to be +recommended are one or more of the following lists: +1. A list of drugs, at least one. +2. The dose, is the amount of drug taken at one time. +3. The frequency at which the drug doses are taken over time. +4. Duration, which is how long the drug is taken. +Numbers 2 to 4 in the above list are referred to as the dosage. Therefore, we can define a drug +recommender system as a smart system that is able to recommend a list of drugs plus their dosages + +with high accuracy in terms of a real prescription of a physician and also to have a positive effect +on a patient, which available data can partially verify. +It should be noted that, of course, there is no medicine recommender system that we can trust +thoroughly, and like other artificial intelligence systems applied in healthcare, their use and ethical +issues must be addressed appropriately [19], [20]. + +4- LITERATURE REVIEW + +Medical recommender systems have been around for many years, even before the emergence of +recommender systems as a new field in computer science. According to [21], medicine +recommender systems fall into two broad categories named “ontology and rule-based +approaches” and “data mining and machine learning-based” approaches. Ontology-based +recommender systems use the hierarchical organization of users and items to improve the +recommendation [22]. +Data mining and machine learning algorithms in the medical field are used to predict and +recommend things like drug usefulness, having a disease [23], [24], the condition of the user, or +ratings [25], [26]. For example, SVM, backpropagation neural network, and ID3 decision tree have +been used in [27] for recommending drugs. The performance of these approaches has been +compared in the above work, and the authors have shown that SVM has better accuracy and +efficiency compared to the other algorithms. Their data set contains patients’ features age, sex, +blood pressure, cholesterol, Na and K levels, and drug. +Some other researchers, while described as medical or medicine recommender systems, consider +a detection and classification task where the dataset which is trained has some patient attributes, +and based on that, the objective of the work is the detection or prediction of a disease and then for +each disease a set of medicines is recommended [27]. +Sentiment analysis of drug reviews is one of the basic approaches for drug recommendations [28], +[29], [30], [31], [32]. The sentiment analysis in these works mainly aims to recommend a drug or +extract useful information like adverse drug reactions. + +In [31], different deep learning approaches, such as CNN, LSTM, and BERT, have been +investigated for sentiment analysis of patients’ drug reviews. In another work, the combination of +CNN-RNN has been applied +In addition to recommendation systems, sentiment analysis and opinion mining of drug reviews is +an active research area in drug review processing [33]. This analysis can be used for automatic +opinion mining and recommending drugs. +A hybrid knowledge-based medical prescription approach has been presented in [34]. The authors +use historical medical prescriptions to recommend a list of medicines to physicians. The approach uses the +similarity between cases where a case is medical information like demography, treatment, age, sex, +symptoms, and diagnosis. Based on the degree of similarity, a drug list is produced. The list is +complemented by Bayesian reasoning, where a model of the conditional probability of drugs is built. This +approach has been applied in Humphrey & Partners Medical Services Limited medical center. +Some works in medical recommendation have focused on particular drugs like diabetes [35]. Their +model is based on the ontology of medical knowledge and a decision decision-making approach +for multiple criteria and computes the medication. Then by using the entropy, the information +about patient’ history has been computed, and finally, the most appropriate medications have been +recommended to the physicians. +Many recommender system approaches have not been well considered in the medical and +pharmaceutical recommendations. However, using polarity in sentiment analysis of user +comments is one of the important parts of using NLP in recommendation systems. It can be viewed +as determining whether a word or phrase in the document or even a whole document is positive, +negative, or neutral in general. +Figure 1 shows the broad classification of different recommendation system approaches in +pharmaceutical research. We can see a growing tendency to use machine learning approaches in +this field. + + + +Figure 1: Broad classification of recommendation system techniques. + +5- Material and methods +In this section, we formulate the medicine recommender system problem and present our approach +for the general medicine recommender system. +Many recommendation systems, like collaborative filtering and content-based approaches, mostly +rely on past information to make decisions for the current situation. It is not always the case in the +domain of drug recommendation. The patient condition is different compared to the other patients +and compared to the same patient over time. So, in addition to the history information like general +rates, reviews, and the effective rate of the drug, it is necessary to use the patient's current condition +to make a more accurate decision. We also cannot rely on diversity-based recommendations as it +is used in some recommender systems, like the one used in Netflix, even if the drug is not rated +high, it can be suitable for some patients. +On the other hand, many recommender systems rely on knowledge from users; when there is a +lack of users’ knowledge, we cannot personalize them. While we have an adequate dataset for our +recommendation task, the problem emerges when new inputs enter the system. In our medicine +Drug +Recommendation +Systems +Collaborative Filtering +Memory Based +User-Based Filtering +WaveLet [36] +Item-Based Filtering +Model Based +Matrix Factorization +SVD +SVD++ +Content Based +Machine Learning +T-Recs [37] +Knowledge-Based +Machine Learning +Data mining Framework +[38] +Ontology-based +MCDM & Entropy [35] +SWRL [39] , [40], [41] +GalenOWL [42] +Panacea [43] +SemMed [44] +Hybrid +CB & Knowledge Based +Machine Learning +LOD cloud mining [45] +User CF & Knowledge +Based +Machine Learning +DiaTrack [46] +Personalized Clinical +Prescription [47] +CADRE [48] +Item Based CF & +Knowledge Based +Machine Learning +Data Driven[49] + +recommender system, these inputs can be new patients or new drugs. Cold start problem is a term +used for this problem, and it is a challenging issue in designing any recommender system. We +reduced this effect by applying a clustering-based approach. Because drugs are clustered into a +specific category, we can put a new drug in the category which belongs to it, so we use the same +rating for the new drugs as those in that category. This is effective in solving the cold start problem +in our recommender system. +Proposed method +Since every recommendation technique has its own benefits, a universal recommender system +should be able to take advantage of all of these techniques to improve the outcome of a +recommender system. The drug recommendation system in our work has the benefits of different +recommendation categories and combines their advantages by using several steps. First, natural +language processing and machine learning algorithms are applied in the context of basic +recommender system techniques. +This section discusses all phases of our model for building a comprehensive drug recommender +system. +This paper presents a novel hybrid drug recommender system (RS) with features of several +recommender systems. It uses natural language processing (NLP) and other machine learning +techniques to implement the system. The proposed RS approach is a new recommendation system +method for pharmaceutical recommendation, which can be considered a hybrid of CB, CF model- +based, knowledge-based, and AI-based methods. Here in this section, we elaborate on each step +toward the final drug recommendation for each patient. After a very intensive web crawling +through two well-known pharmaceutical websites, Drugs.com and Druglib.com, and building +three different datasets, feature extraction and modeling are performed. Then in the next step, +recommendations for proper drugs are performed. At the final stage, the list of drugs is refined +based on defined rules in addition to the ratings and drug features which is an important aspect of +our medicine recommender system. + + + +Figure 2: Components of RECOMMED drug recommendation system in the training stage. +Figure 2 presents the whole RECOMMED model in the training stage of our work, consists of +four components, and we elaborate on each phase of our approach in more detail in the following +parts: +5-1 Dataset extraction +In this work, any recommendation for drugs and their dosage is based on the patients’ features like +age, gender, previous illness, and other drugs they consume, and drugs’ features like drug +classification, side effects, and drug interactions. So, in this phase, the extraction of user features, +drug features, and drug interaction datasets from Drugs.com and Druglib.com databases is +accomplished. In the second step of this phase, the dataset is prepared for clustering and modelling +the recommendation system. The review field in the drug recommendation database contains users' +and caregivers opinions about drugs' effectiveness. According to our knowledge, none of the +existing datasets have complete and comprehensive patient and drug information. + + +Start + + +Remove HTML Frames, + +Tags and advertisement + +End + + +Crawl Review Webpages + +Dataset +Extraction +Modeling +User and Drug +Feature Set + +Keyword Search + +Drug +Interation Set +Pre- +processing + +Feature Extraction + +Normalized +User and Drug +Feature Set + +Combining User Comment + +Rates And Effectivness + + +Normalize Feature Sets + +User Clusters +Feature Set +User Rating +Matrix +Drug Clusters +Feature Set +Drug Rating +Matrix + +Clustering Users + + +Clustering Drugs + + +Generating + +User Rating + + +Generating + +Drug Rating + +Knowledge- +based +User Weights +& Biases Sets +Drug Weights +& Biases Sets + +Initialize Weights & Biases + + +Forward Propagation + + +Backward Propagation + + +Update Weights & Biases + +Error < Threshold + + +New User Registration + + +Compute 10 High Rated +Drug for Recommendation + +Drug[i] + + Interaction? + + i=0 + +i<10 + + i=i+1 + +User Features +Filter Set +Drug[i] + +Allowed? + + + Recommendation + +append(Drug[i]) + + +We built three different datasets named users, drugs, and interactions. +5.1.1 Drugs and users datasets +In this work, Druglib.com and Drugs.com were employed to extract information about patients +and drugs and build two datasets named drugs and patients. We should mention that there are also +other databases for drug information and recommendations, like SIDER [50], for drug side effects. +We will include them in future works to build a complete dataset for drugs. Three features +consisting of side effects, benefits, and membership in a given drug category were considered for +drugs. +First, different drug categories and side effects were extracted in tables 1 and 2. There are 150 +different drug categories, and 128 different side effects were extracted from the Druglib.com +database. +TABLE 1 - DRUG CATEGORIES LIST +Category +Index +Acetylcholine-Agonists +1 +Adrenergic-Alpha-Agonists +2 + +… +Vasodilators +150 + +TABLE 2 -DRUGS SIDE EFFECTS +Side Effects +Index +Completed suicide +1 +Confusional state +2 + +… +Wrong drug administered +128 + +Then drug benefits were also extracted and combined with the information in the above tables, and +finally, the drugs dataset was prepared, as is partially shown in Table 3. +TABLE 3- DRUGS DATASET +Benefits +Side Effects +Drug Category +Drug Name +Index +88 +… +2 +1 +128 +… +2 +1 +150 +… +2 +1 +0 +… +1 +1 +0 +… +0 +0 +0 +… +0 +1 +Hytrin Terazosin +1 +0 +… +0 +0 +0 +… +1 +1 +0 +… +0 +1 +Mirtazapine +2 +… +… +… +… +… +… +… +… +… +… +… +… +… +… +0 +… +0 +0 +0 +… +0 +0 +1 +… +0 +0 +Proscar Finasteride +480 + +We also extracted the users dataset of patient features and comments on different drugs. Six +features are considered for users datasets: age, gender, current disease (condition), other +conditions, other drugs are taken, and user level, which is patient or caregiver. +Table 4 represents the structure of this dataset. +TABLE3- USERS DATASET +Comment +Side +Effects +Effective +ness +Overall +Rating +Drug +Name +Other +Drug +Other +Conditio +n +Conditio +n +Genus +Age +Level +index +… +Severe Side +Effects +Ineffective + +1 +Mirtazapine + +None + +Sleeplessness + +Depression + +Male +22 +Patient + +1 +… +Moderate +Side Effects +Ineffective + +2 +Mirtazapine + +None + +None + +Depression + +Male +38 +Patient + +2 + ... + ... + ... + ... + ... + ... + ... + ... + ... + ... + ... +… +… +Mild Side +Effect +Moderately +Effective + +4 +Proscar +Finasteride + +None + +None + +Hair loss + +Male + + +28 +Patient +3294 + +5.1.2 Interactions dataset +The last dataset prepared in this work is the interactions dataset. This information is important for +recommending the appropriate medicine list to the patients. We extracted drug interaction +information from Drugs.com, and after mapping drugs’ names with their counterparts in +Druglib.com, the interaction dataset, partially presented in Table 5, was created with 180 drug +interaction information. +TABLE 4 -DRUG INTERACTION DATASET + +Abilifish + +… + +Cimbaita + +… + +Syntroid + +… + +zyban + +Abilifish + +- +- +Moderate + +… + +- +… + +- +Accupril + +- +- +- +… + +Moderate + +… + +- +Aciphex + +- +- +Major + +… + +- +… + +- +… + +… + +… + +… + +… + +… + +… + +… + +Zyban + +- +- +- +… + +- +… + +- + +5.2 Dataset preparation + +In this phase, our dataset is prepared for creating the recommendation model in the next step. First, +using Natural Language Processing (NLP) techniques, user and drug features are extracted, and +then normalization and clustering are accomplished to prepare the datasets for modeling the +recommendation system. Here, we elaborate on each of these steps: + +5.2.1 Feature extraction +The first pre-processing step is feature extraction from user feature and drug features datasets. +Bag-Of-Words (BOW) method is used for this purpose. +NLP for extracting drug and user features +The feature extraction was mainly performed using natural language processing (NLP) techniques. +Two well-known methods to extract text features by NLP are Bag-of-Words (BOW) and term +frequency-inverse term frequency (TF-IDF). +Our proposed pharmaceutical recommendation system uses the BOW feature extraction method +to perform feature extraction from database texts. This method consists of four steps: +• Text-pre-processing pre-processing +• Vocabulary creation +• Building feature matrix +• Polarity of user comments +Here, every part of this process has been described: +Text-pre-processing pre-processing +In the text- pre-processing step, all punctuations and symbols are removed, and abbreviations are +converted into their full names or phrases. Some of these conversions are presented in Table 6. +Moreover, spelling mistakes were corrected using the TexBlob library of Python, and stop words +were removed using a predefined list of stop words. + + +TABLE 6- EXAMPLES OF ABBREVIATIONS TO FULL NAME CONVERSIONS +Original Form + +Abbreviations + +high blood pressure + +HBP + +chronic obstructive pulmonary disease + +COPD + +premenstrual syndrome + +PMS + +obsessive-compulsive disorder + +OCD + + +Vocabulary creation +Using NLP techniques, a vocabulary of words is created in the second step of feature extraction. +For this purpose, an array of words is created by checking all registered words in the dataset. This +array is constructed from unique words of the dataset and their frequency. To deal with the random +filling of the feature matrix, words are rearranged according to their frequency. Moreover, to deal +with the sparseness of the feature matrix, words with low frequency are removed. Some of the +most frequent words extracted from the datasets created and discussed in the previous section can +be seen in Table 7. + +TABLE 7. EXAMPLES OF MOST FREQUENT WORDS IN DATASETS. +Frequency + +Term + +33 + +Pain + +22 + +Infection + +15 + +Surgery + +13 + +Chronic + + +Building feature matrix +The feature matrix is created in the third step of extracting the features. For this purpose, a unique +word is assigned to each matrix column, and a new row is considered for each user review. Each +cell of this matrix represents the existence of the word in the user’s review, which is essentially +zero or one. + +Polarity of user comments (PUC) +We used NLP and opinion mining to extract PUC. This approach aims at extracting the opinion of +users as a positive or negative comment. The output of this component is used in the users’ rating +matrix. + + +ALGORITHM1. COMMENT POLARITY ACQUISITION +Input:𝑈𝑠𝑒𝑟𝐶𝑜𝑚𝑚𝑒𝑛𝑡𝑠, 𝑆𝑡𝑜𝑝𝑊𝑜𝑟𝑑𝑠 +Output: 𝑃𝑜𝑙𝑎𝑟𝑖𝑡𝑦𝑂𝑓𝑈𝑠𝑒𝑟𝐶𝑜𝑚𝑚𝑒𝑛𝑡𝑠 +1. 𝑅𝑒𝑚𝑜𝑣𝑖𝑛𝑔 𝐶𝑎𝑝𝑖𝑡𝑎𝑙 𝐿𝑒𝑡𝑡𝑒𝑟𝑠 𝑎𝑛𝑑 𝐸𝑚𝑜𝑗𝑖𝑠 from 𝑈𝑠𝑒𝑟𝐶𝑜𝑚𝑚𝑒𝑛𝑡𝑠 +2. 𝑅𝑒𝑚𝑜𝑣𝑖𝑛𝑔 𝑆𝑡𝑜𝑝𝑊𝑜𝑟𝑑𝑠 𝑓𝑟𝑜𝑚 𝑈𝑠𝑒𝑟𝐶𝑜𝑚𝑚𝑒𝑛𝑡𝑠 +3. 𝑊𝑜𝑟𝑑 𝑇𝑜𝑘𝑒𝑛𝑖𝑧𝑒 (𝑈𝑠𝑒𝑟𝐶𝑜𝑚𝑚𝑒𝑛𝑡𝑠) +4. 𝑊𝑜𝑟𝑑𝐿𝑒𝑚𝑎𝑡𝑖𝑎𝑡𝑖𝑜𝑛(𝑈𝑠𝑒𝑟𝐶𝑜𝑚𝑚𝑒𝑛𝑡𝑠) +5. 𝐹𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 𝑜𝑓 𝑊𝑜𝑟𝑑𝑠(𝑈𝑠𝑒𝑟𝐶𝑜𝑚𝑚𝑒𝑛𝑡𝑠) +6. 𝑇𝑒𝑥𝑡𝐵𝑙𝑜𝑏(𝑈𝑠𝑒𝑟𝐶𝑜𝑚𝑚𝑒𝑛𝑡𝑠) + +Combined User Rating Acquisition +To have a more accurate rating for drugs, we considered the combined user comments and ratings +from different sources. This overall rating is called Combined User Rating Acquisition (CUR) +parameter and is obtained from analyzing user comments and ratings as follows: +1. 𝑂𝑣𝑒𝑟𝑎𝑙𝑙 𝑅𝑎𝑡𝑖𝑛𝑔 ∈ 𝑍, 0 ≤ 𝑂𝑣𝑒𝑟𝑎𝑙𝑙 𝑅𝑎𝑡𝑖𝑛𝑔 ≤ 10 +2. 𝐸𝑓𝑓𝑒𝑐𝑡𝑖𝑣𝑛𝑒𝑠𝑠 ∈ 𝐸, 𝐸 = {Ineffective, Marginally Effective, Moderately Effective, +Considerably Effective, Highly Effective} +3. 𝑆𝑖𝑑𝑒 𝐸𝑓𝑓𝑒𝑐𝑡 ∈ 𝑆, 𝑆 = {𝑁𝑜 𝑆𝑖𝑑𝑒 𝐸𝑓𝑓𝑒𝑐𝑡, 𝑀𝑖𝑙𝑑 𝑆𝑖𝑑𝑒 𝐸𝑓𝑓𝑒𝑐𝑡, 𝑀𝑜𝑑𝑒𝑟𝑎𝑡𝑒 𝑆𝑖𝑑𝑒 𝐸𝑓𝑓𝑒𝑐𝑡, +𝑆𝑒𝑣𝑒𝑟𝑒 𝑆𝑖𝑑𝑒 𝐸𝑓𝑓𝑒𝑐𝑡, 𝐸𝑥𝑡𝑒𝑟𝑒𝑚𝑙 𝑆𝑒𝑣𝑒𝑟𝑒 𝑆𝑖𝑑𝑒 𝐸𝑓𝑓𝑒𝑐𝑡} +4. User Comment + +CUR parameter is calculated as equation (6), and the above parameters are replaced by CUR in +the user feature matrix: +CUR = +(𝑂𝑣𝑒𝑟𝑎𝑙𝑙𝑅𝑎𝑡𝑖𝑛𝑔 +10 ++𝐷𝑂𝐸 +4 +) +2 +− 𝐷𝑂𝑆 +4 +𝑃𝑈𝐶 +2 + +(6) +In equation (6), DOE (Degree of Effectiveness) represents the degree of drug effectiveness. The +user selects the effectiveness of a drug from a list of five different options: Ineffective, Marginally +Effective, Moderately Effective, Considerably Effective, and Highly Effective, and it takes a number +in the range [0-4]. Similarly, DOS (Degree of Side Effects) is the degree that a drug has a side +effect (range [0-4]), and the numbers applied in the denominator are for normalization purpose. +PUC (Polarity of User Comments) is calculated using Natural Language Processing (NLP), and +opinion mining techniques and the nltk library in Python are used in this regard. Algorithm 1 shows +the steps of the work for calculating PUC. + +Normalization- After extracting features from drug and user datasets, these features should also +be normalized to perform better in training the model. + + +Combined User +Rating (CUR) +Drug Name +Other Drug +Other Condition +Condition +Genus +Age +Level +index + +Comment +Side Effects (DOS) +Effectiveness (DOE) +Overall Rating +Drug Name +Other Drug +Other Condition +Condition +Genus +Age +Level +Index +0.05 +Mirtazapine +None +Sleeplesness +depression +male +22 +patient +1 + +0.8 +(Obtained +from +PUC) +0.75 +0 +0.1 +Mirtazapine +None +Sleeplesness +depression +male +22 +patient +1 +… +… +… +… +… +… +… +… +… + +… +… +… +… +… +… +… +… +… +… +… +… +0.1 +Proscar +None +None +Hair loss +male +28 +patient +3294 + +0.2 +0.25 +0.5 +0.4 +Proscar +None +None +Hair loss +male +28 +patient +3294 + +Figure 3- Combination of different user ratings for a given drug + +Figure 3 is the final user rating dataset after applying the combined user rating acquisition stage. +This stage converts the dataset on the left side into the right side dataset. Each column in both +datasets has a given user’s features along with the drug name they rate. In the left side dataset, we +can see different ratings of the user, and then in the right side dataset, these ratings are combined +into CUR using equation (6). + +5.3 Clustering +Clustering is considered one of the main steps in a recommender system for improving the +diversity, consistency, and reliability [51], which has been considered in many works in +recommender systems, particularly for reducing the sparsity of data [52], [53]. Due to the +sparseness of the rating matrix, we consider a clustering-based approach, and patients are clustered + +before performing the matrix factorization, which is elaborated in the next part. This clustering is +mostly required because users usually review only one drug corresponding to a specific disease, +so the rating matrix is highly sparse. Clustering can help group the users and drugs with similar +features and significantly resolve the sparsity problem. Users are clustered based on their gender, +age, comments, and being patient or caregiver. It is clear that after clustering, each class of users +reviews several drugs, which can improve the matrix factorization process. +We used a modified K-means algorithm in [54] to perform this clustering. While the original K- +means algorithm is unsupervised, which is used for clustering, the number of clusters is pre- +determined, and so it couldn't be utilized in the same way in our proposed drug recommendation +system. Therefore, in this paper, we employed the U-Kmeans method [54]. This method performs +the unsupervised K-means and determines the best cluster numbers that lead to better classification +performance. +If each row of the dataset and the center of each cluster are represented by F= {𝑓1, … , 𝑓𝑛} and A= +{𝑎1, … , 𝑎𝑘} respectively, the K-means objective function is defined as (7). + +𝐽(𝑀, 𝐴) = ∑ +∑ +𝑀𝑖𝑗‖𝑓𝑖 − 𝑎𝑗‖ +𝑘 +𝑗=1 +𝑛 +𝑖=1 + +(7) + + Where in (7), 𝑘 is the number of clusters, 𝑛 is the number of dataset features, and 𝑀𝑖𝑗 indicates +the membership of 𝐹𝑖 to the 𝑗𝑡ℎ cluster. In the K-means algorithm, this objective function must be +minimized. In [55], an entropy-based method is proposed to improve K-means. In this method, to +determine the centers of the clusters, Equation (8) is added to the objective function. +𝐵𝑛 ∑ +𝑎𝑗 +𝑘 +𝑗=1 +ln 𝑎𝑗 +(8) + In (8), the effect of the cluster imbalance is added to the objective function. As can be seen in +(9), when the 𝐵𝑛 coefficient of the improved objective function is zero. The following K-means +objective function is obtained. + +𝐽(𝑀, 𝐴) = ∑ +∑ +𝑀𝑖𝑗‖𝑥𝑖 − 𝑎𝑗‖ +𝑘 +𝑗=1 +𝑛 +𝑖=1 +− 𝐵 ∑ +𝜂𝑗 +𝑘 +𝑗=1 +ln 𝜂𝑗 + +(9) + + + Where in this equation, 𝜂𝑗 represents the number of members of a cluster, which is determined +by (10). +𝜂𝑗 = +∑ +𝑀𝑖𝑗 +𝑛 +𝑖=1 +𝑥𝑖 +∑ +𝑀𝑖𝑗 +𝑛 +𝑖=1 + +(10) + + In [54], equation (11) is considered to determine the optimized number of clusters. By adding +this term to equation (10), the final objective function is obtained as (12). +L ∑ +∑ +𝑀𝑖𝑗 +𝑘 +𝑗=1 +ln 𝑎𝑗 +𝑛 +𝑖=1 + +(11) +𝐽(𝑀, 𝐴, 𝑎) = ∑ +∑ +𝑀𝑖𝑗‖𝑥𝑖 − 𝑎𝑗‖ +𝑘 +𝑗=1 +𝑛 +𝑖=1 +− 𝐵 ∑ +𝑎𝑗 +𝑘 +𝑗=1 +ln 𝑎𝑗 − L ∑ +∑ +𝑀𝑖𝑗 +𝑘 +𝑗=1 +ln 𝑎𝑗 +𝑛 +𝑖=1 + +(12) + The pseudocode of the U-K-means classification method based on the approach in [54] is +presented in Algorithm 2. +Algorithm . +2 Our modified Pseudo code of U-Kmeans based on [54]. +1. 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑐(0) = 𝑛, 𝛼𝑘 +(0) = +1 +𝑛 , 𝑎𝑘 +(0) = 𝑥𝑖 +2. 𝐼𝑛𝑖𝑡𝑖𝑎𝑙 𝑙𝑒𝑎𝑟𝑛𝑖𝑛𝑔 𝑟𝑎𝑡𝑒𝑠 𝐿(0) = 𝐵(0) = 1 +3. 𝑆𝑒𝑡 𝑡 = 0 , 𝜀 > 0 +4. 𝑤ℎ𝑖𝑙𝑒 𝑚𝑎𝑥‖𝑎𝑘 +𝑡+1 − 𝑎𝑘 +𝑡 ‖ < 𝜀 +5. If ‖𝑥𝑖 − 𝛼𝑘‖2 − 𝐿𝑙𝑛𝛼𝑘 = min +1≤𝑘≤𝑐‖𝑥𝑖 − 𝑎𝑘‖2 − 𝐿𝑙𝑛𝛼𝑘 +6. 𝑀𝑖𝑘 +(𝑡+1) = 1 +7. Else +8. 𝑀𝑖𝑘 +(𝑡+1) = 0 +9. 𝐿(𝑡+1) = 𝑒−𝑐(𝑡+1)/250 +10. 𝛼𝑘 +(𝑡+1) = ∑ +𝑀𝑖𝑘 +𝑛 + ( +𝐵 +𝐿) 𝛼𝑘 +(𝑡) ln 𝑎𝑘 +𝑡 − ∑ +𝛼𝑠 +𝑡 +𝑐 +𝑠=1 +𝑛 +𝑖=1 +ln 𝑎𝑠 +𝑡 +11. 𝐵𝑡+1 = 𝑚𝑖𝑛 ( +∑ +exp (−𝜂𝑛|𝑎𝑘 +𝑡+1−𝑎𝑘 +𝑡 |) +𝑐 +𝑘=1 +𝑐 +, +1− max +1≤𝑘≤𝑐(1 +𝑛 ∑ +𝑀𝑖𝑘 +𝑛 +𝑖=1 +) +− max +1≤𝑘≤𝑐 𝑎𝑘 +𝑡 ∑ +ln 𝑎𝑘 +𝑡 +𝑐 +𝑘=1 +) +12. 𝑢𝑝𝑑𝑎𝑡𝑒 𝐶𝑡 𝑡𝑜 𝐶𝑡+1 𝑏𝑦 𝑑𝑖𝑠𝑐𝑎𝑟𝑑 𝑡ℎ𝑜𝑠𝑒 𝑐𝑙𝑢𝑠𝑡𝑒𝑟 𝑤𝑖𝑡ℎ 𝑎𝑘 +𝑡+1 ≤ +1 +𝑛 +13. 𝑎𝑘 +∗ = +𝑎𝑘 +∗ +∑ +𝑎𝑠∗ +𝑐(𝑡+1) +𝑠=1 + +14. 𝑀𝑖𝑘 +∗ = +𝑀𝑖𝑘 +∗ +∑ +𝑀𝑖𝑠 +∗ +𝑐(𝑡+1) +𝑠=1 + +15. 𝑎𝑘 = +∑ +𝑀𝑖𝑘𝑥𝑖𝑗 +𝑛 +𝑖=1 +∑ +𝑀𝑖𝑘 +𝑛 +𝑖=1 + +16. 𝑖𝑓 𝑡 ≥ 60 𝑎𝑛𝑑 𝑐(𝑡−60) − 𝑐𝑡 = 0 +17. 𝐵(𝑡+1) = 0 +18. t=t+1 + + + + +TABLE 8: RATE MATRIX WITHOUT CLASSIFICATION +619 + +618 + +617 + +616 + +615 + +… + +6 + +5 + +4 + +3 + +2 + +1 + +Drugs + + +Users + + + + + + + + + + + + +1 + +1 + +3 + + + + +3 + + + + + + + + +2 + + + + +5 + + + + + + + + + +3 + + + + + + + + + + + + + +… + + + + + + + + + + + +1 + + +979 + + +3 + + + + + + + + + + + +980 + + + + + + + +3 + + + + + + +981 + + +TABLE 9: RATE MATRIX AFTER CLASSIFICATION +619 + +618 + +617 + +616 + +615 + +… + +6 + +5 + +4 + +3 + +2 + +1 + +Drugs + + +Users + + + + + + + + + + + + +1 + +1 + +3 + + + + +3 + + + + + + + + +2 + + + + +5 + + + + + + + + + +3 + + + + + + + + + + + + + +… + + + + + + + + + + + +1 + + +38 + + +3 + + + + + + + + + + + +39 + + + + + + + +3 + + + + + + +40 + + + +5.4 Modeling +In the next step, the clustering outcome is used to build a recommender system model able to +recommend the best drugs. Later, we filter the model's output with a knowledge-based component +for safety reasons. + +Neural Network-based Matrix Factorization +Matrix factorization is a popular method for recommender systems aiming at finding two +rectangular matrices called user and item matrices with smaller sizes than the rating matrix [56]. +The dot product between these two matrices results in the rating matrix. + +To reduce the computational overhead, copeTo reduces the computational overhead, cope with the +sparsity of the ratings, and increase accuracy. We proposed a neural network-based matrix +factorization technique. The first two matrices, Rating and Effectiveness, are constructed by +extracting information from Druglib.com. +In our model, the rating matrix 𝑅𝑎𝑡𝑖𝑛𝑔 ∈ R𝑛∗𝑚 is estimated as the multiplication of two matrices +𝐶𝑙𝑢𝑠𝑡𝑒𝑟𝑠𝑛∗𝑘 and 𝐷𝑟𝑢𝑔𝑠𝑘∗𝑚 as (13): +𝑅𝑎𝑖𝑛𝑔 ≈ Clusters. 𝐷𝑟𝑢𝑔𝑠𝑇 (13) + +This model applies a neural network algorithm to estimate the users’ comments for each medicine. +Clustered users and drugs and users’ and drugs’ features are used in building this new model, as +illustrated in Figure 4. +Figure 4- Our proposed customized matrix factorization method. + + + + + + + + + + + + + +𝑏𝑚 + +𝑏2 +𝑏1 + + + + + + + + + + + + + + + + + + + + + + + +Drug +Cluster k + +… + +Drug +Cluster 2 + +Drug +Cluster 1 + + + + + + + + + + +Drug m + +… + +Drug 2 + +Drug 1 + + + + + +𝑐1,𝑚 + +𝑐1,2 +𝑐1,1 +Drug +Feature +1 + + + + + + + + + + +𝑑1,𝑚 + +𝑑1,2 +𝑑1,1 +Latent +Factor 1 + + + +𝑐2,𝑚 + +𝑐2,2 +𝑐2,1 +Drug +Feature +2 + + + + + + + + + + +𝑑2,𝑚 + +𝑑2,2 +𝑑2,1 +Latent +Factor 2 + + + + + +… + + + + + + + + + + + + + + +… + +𝑐𝑘,𝑚 + +𝑐𝑘,2 +𝑐𝑘,1 +Drug +Feature +k + + + + + + + + + + +𝑑𝑘,𝑚 + +𝑑𝑘,2 +𝑑𝑘,1 +Latent +Factor k + + + + + + + + +User +Feature k + +… + + +User +Feature 2 + +User +Feature 1 + + + + + + + +User +Feature k + +… + +User +Feature 2 + +User +Feature 1 + + +𝑟1,𝑚 + +𝑟1,2 +𝑟1,1 + +𝑢1,𝑘 + + +𝑢1,2 +𝑢1,1 +User +1 + + +𝑏1 + +𝑟1,𝑚 + +𝑟1,2 +𝑟1,1 + +𝑐1,𝑘 + +𝑐1,2 +𝑐1,1 +User +Cluster +1 + +𝑟1,𝑚 + +𝑟2,2 +𝑟2,1 + +𝑢2,𝑘 + + +𝑢2,2 +𝑢2,1 +User +2 + + +𝑏1 + +𝑟1,𝑚 + +𝑟2,2 +𝑟2,1 +𝑐2,𝑘 + +𝑐2,2 +𝑐2,1 +User +Cluster +2 + + + + + + + + + + + +… + + + + + + + + + + + + +… + +𝑟𝑛,𝑚 + +𝑟𝑛,2 +𝑟𝑛,1 + +𝑢𝑛,𝑘 + + +𝑢𝑛,2 +𝑢𝑛,1 +User +n + + +𝑏𝑛 + +𝑟𝑛,𝑚 + +𝑟𝑛,2 +𝑟𝑛,1 +𝑐𝑛,𝑘 + +𝑐𝑛,2 +𝑐𝑛,1 +User +Cluster +n + +User Ratings + + + + + + +Drug Ratings + + + +User embedding +(User Weights) +Drug Cluster Features +sEmbedding +Drug Embedding +(Drug Weights) + + Drug Bias + +User Cluster Features + User Bias + + +The input to the neural network is user and drug-clustered features. Drug Embedding and User +Embedding matrices are the input to this network, and drug and user are the network's outputs. +With sparse rating matrices, the forward and backward pass calculations are accomplished just for +non-zero ratings to reduce the computation load. The neural network layer output is calculated as: +𝑎𝑧+1 = 𝑓𝑧+1(∑ +𝑤𝑖 +𝑧+1. 𝜓𝑧+1(𝑛, 𝑚). 𝑎𝑖 +𝑧 + 𝑏𝑗 +𝑧+1 +𝐾 +𝑖=1 +) 𝑖 ∈ (1, 𝐾), 𝑗 ∈ (1, 𝑀𝑁), 𝑧 ∈ (0, 𝑍 − 1), 𝑛 ∈ +(0, 𝑁), 𝑚 ∈ (0, 𝑀) +(14) +In this equation, 𝑓𝑧+1 is the activation function, 𝑤𝑖 +𝑧+1 are the weights, 𝑏𝑗 +𝑧+1 are the biases, 𝑍 is the +number of layers, 𝑀 is the number of drugs, 𝑁 is the number of users, 𝑎𝑍 is the output of the +network, 𝐾 is the number of the features for drugs or users, and 𝑀𝑁 represents the number of +drugs in the user network and represents the number of users in the drugs network. And finally +𝜓𝑧+1 is the rating existence function defined as: +{𝜓𝑧+1(𝑛, 𝑚) = 1 𝑖𝑓(𝑅𝑎𝑡𝑖𝑛𝑔 𝑛, 𝑚 𝑒𝑥𝑖𝑡 𝑜𝑟 𝑧 < 𝑍 − 1) +𝜓𝑧+1(𝑛, 𝑚) = 0 𝑖𝑓(𝑅𝑎𝑡𝑖𝑛𝑔 𝑛, 𝑚 𝑛𝑜𝑡 𝑒𝑥𝑖𝑠𝑡) +(15) +Also, the backward pass calculations are as equations (16) to (19) for the output and hidden layers +respectively: +For the output: +∆𝑜𝑢𝑡 = (𝑅(𝑛,𝑚) − 𝑎𝑍). 𝜓𝑍(𝑛, 𝑚). 𝑓𝑧+1′(𝑎𝑍) 𝑛 ∈ (0, 𝑁), 𝑚 ∈ (0, 𝑀) +(16) +∆𝑊𝑍 = ∆𝑜𝑢𝑡. 𝑎𝑍. 𝛾𝑍 +(17) +For the hidden layers: +∆𝐻𝑖𝑑𝑑𝑒𝑛𝑧 = 𝑓′(𝑎𝑧). ∑ ∆𝑜𝑢𝑡𝑖 +𝑖 +. 𝑤𝑖 +𝑧 𝑖 ∈ (1, 𝐾), 𝑧 ∈ (0, 𝑍 − 1) +(18) +∆𝑤𝑧 = ∆𝐻𝑖𝑑𝑑𝑒𝑛𝑧. 𝑎𝑧. 𝛾𝑧 𝑧 ∈ (0, 𝑍 − 1) +(19) +In these equations, 𝑓𝑧+1′ is the gradient of the activation function, 𝑅(𝑛,𝑚) is the rating +corresponding to the users or drugs, ∆𝑊𝑍 is the error correction for the output layer, ∆𝑤𝑧 are the +error corrections for the hidden layers, and 𝛾𝑧 is the learning rate. The weight updates are also +according to equation (20): +𝑤𝑛𝑒𝑤 +𝑧 += 𝑤𝑜𝑙𝑑 +𝑧 ++ ∆𝑤𝑧 𝑧 ∈ (0, 𝑍) (20) + + +5.5 Knowledge-based component +After modeling the recommendation system, several constraints on the model output are applied. +The final stage in the recommendation process is based on the knowledge-based technique. The +knowledge-based recommendation is a specific recommender system that can be used in +combination with other algorithms or alone. +The aim of using this module is its huge impact in increasing the safety of the recommendations. +We extracted and gathered rules in the drug recommendation domain as queries. These rules are +based on Drug Interactions and Adverse Events. Using these rules, we can prevent recommending +drugs that lead to events like death, hospitalization, disability, and life-threatening events. The +flowchart of this component has been extracted from Figure 2 and redrawn in Figure 5. +The set of these rules which our knowledge-based component considers falls into these two +categories: +- +Based on patients’ features: +o Gender is allowed to recommend a drug. +o The age is allowed for recommending a drug. +- +Based on drug interactions: +o The recommended drug has no interaction with other drugs taken by the user. + + +Figure 5- Knowledge-based component of our proposed approach based on the bottom left of +Figure 2. + +Table 10 presents knowledge-based rules based on patient’s features that have been considered in +this work. For example, according to this table, a drug can only be recommended if the patient's +age is in the allowed range and the gender is allowed for recommending the drug. +TABLE 10- USER FEATURE-BASED RULES +Zometa + +Actemra +Abilify +Drug Name +31 +… +29 +24 +Minimum +Not allowed age +ranges +67 +… +64 +45 +Maximum +0 +… +1 +0 +None +Allowed gender +0 +… +0 +0 +Female +0 +… +0 +0 +Male +1 +… +0 +1 +Both + + Start + + +New User Registration + +User Weights +& Biases Sets +Drug Weights +& Biases Sets + +Compute 10 High Rated +Drug Recommendation +Drug[i] Interaction? + + I=0 + +I<10 + + I=i+1 + +Drug +Interation Set + End + + + Recommendation.append(Drug[i] + +Knowledge- +based +Recommendations +based on the model + +User Features +Filter Set +Drug[i] Allowed? + +Drug Recommendation + + +For our proposed knowledge-based component, another adverse events dataset is generated from +Druglib.com. The structure of this dataset is presented in Table 11. Features in this dataset include +age, gender, the name of the drug taken by a given patient, its adverse event, reaction, and other +drugs used by the patient. +TABLE11-ADVERSE EVENTS DATASET +Other Drug +Adverse Event +Reaction +Genus +Age +Drug Name +Index +- +Death + +male +63 +Ability +(Airipiprazole) +1 +... +… +… +… +… +... +… +Insulin +Death; +Hospitalization + +female +47 +Acterma +12 +… +… +... +… +… +… +… +Fluticasone propiate; +Salmeterol; Carbemazepine +Hospitalization + +male +50 +Zyprexa +2486 + +We used Gaussian and Poisson distribution for patients' age and gender from the above dataset for +the adverse events of using a specific drug. These adverse events can be death, hospitalization, +disability, or other life-threatening events. +Since, in this case, we require the average and standard deviation, by using Poisson and Gaussian +distribution, it is possible to compute the allowed gender for recommending a drug to a patient +using much less memory than machine learning for this specific task. +Assume that on average, by recommending a drug 𝛾 for 𝜂 times to patients with 𝑔𝑒𝑛𝑑𝑒𝑟 = + 𝑓𝑒𝑚𝑎𝑙𝑒 they experience one of the adverse events mentioned in Table 11, then the probability +that by recommending drug 𝛾 to a female patient she experiences one of the adverse events is +calculated as equation (21): +𝑃𝑛(𝑥) = 𝑒−𝜆𝐹𝑒𝑚𝑎𝑙𝑒 +𝛶 +𝜆𝐹𝑒𝑚𝑎𝑙𝑒 +𝛶 + 𝜆𝐹𝑒𝑚𝑎𝑙𝑒 +𝛶 += +𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝐴𝑑𝑣𝑒𝑟𝑠𝑒 𝐸𝑣𝑒𝑛𝑡 +𝜂 + +(21) +And similarly, for a male patient, this probability is calculated as equation (22): +𝑃𝑛(𝑥) = 𝑒−𝜆𝑀𝑎𝑙𝑒 +𝛶 +𝜆𝑀𝑎𝑙𝑒 +𝛶 + 𝜆𝑀𝑎𝑙𝑒 +𝛶 += +𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝐴𝑑𝑣𝑒𝑟𝑠𝑒 𝐸𝑣𝑒𝑛𝑡 +𝜂 + +(22) + + +Using the above calculations, if the probability of an adverse event for each gender and each +medicine is more than a given threshold value, the medicine is removed from the list and is not +recommended to the patient. In this paper, we set the 𝑡ℎ𝑟𝑒𝑠ℎ𝑜𝑙𝑑 = 50%. +Also, normal distribution was used for setting the rules related to the patients ages. Suppose the +average and standard deviation of a patient’s age who have taken medicine 𝛾 and has an adverse +event is represented by 𝜇 and 𝜎, respectively. In that case, the normal distribution function related +to age is as equation (23): +𝑓(𝑥) = +1 +√2𝜋σ𝛶 𝑒 +−1 +2(𝑥−μ𝛶 +σ𝛶 ) +2 + +(23) +and so for patients who are taking medicine 𝛾, equation (17) for age range has to be met to +minimize the adverse event (24): +𝑋𝜖 (𝜇𝛶 − 1.96 ( +𝜎𝛶 +√𝑛) , 𝜇𝛶 + 1.96 ( +𝜎𝛶 +√𝑛)) , 1 − 𝑎 = 95%, 𝑍0.975 = 1.96 +(24) + +In this research, we used the rules related to the users’ features and the medicine rules and drug +interactions we are also considering. In this regard, the drug interactions dataset was used to +exclude recommendations for drugs having high interactions with other drugs. + +6- RESULTS AND DISCUSSION +This section discusses our proposed drug recommendation system implementation and the newly +generated datasets. First, we explain the extracted and newly generated datasets and then we will +demonstrate the results of our implemented system. + +The dataset +As discussed in the proposed method, we used the information from two databases of drugs +Druglib.com [7] and Drugs.com [5]. The first database Druglib.com is a comprehensive resource +for drug information. For each drug, a variety of information such as description, side-effects, drug +ratings & reviews by patients, and clinical pharmacology has been provided. Also, Drugs.com is + +another database for drug information, and many recommendation systems have been suggested +that use this database to build their models. Both the original and the revised version of Drugs.com +have been used in RS to evaluate the performance of the approaches. +We crawled these pharmaceutical websites to construct our intended datasets with the required +features in a structured way. As a result, we gathered much useful information about drugs and +patients’ conditions and collected them into three datasets as follows: +- +The first extracted dataset is the Rating dataset consists of patients’ features and their +ratings on drugs consisting of 3294 samples. +- +The second dataset consists of Drug features containing drug categories, side effects, and +benefits. +- +The last dataset is the Interaction dataset containing interactions between drugs. +To evaluate the performance of our system, we used the most popular existing machine learning +evaluation metrics. Accuracy, sensitivity (recall), specificity, and precision were the basic metrics +that we applied to our model. +We used 70 percent of the samples (2304 samples) in the dataset for training our model, 20 percent +(660 samples) for evaluation, and 10 percent (330 samples) for the test. +After obtaining the values for true positive (TP), false positive (FP), true negative (TN), and false +negative (FN), different metrics can be calculated. +We compared our results with the existing approaches in [27], [48], [57], [58], and [59]. We +implemented the algorithms in these papers with the datasets they have applied. +In [27], SVM and recurrent neural network (RNN) ve been used to recommend a drug to a patient. +In [48] the authors first considered the clustering of drugs according to the drug information, like +the algorithm proposed in this paper. Then collaborative filtering is used to recommend a drug. +But unlike our work, they haven’t considered the classification of users and their features. Finally, +in [57], an improved matrix factorization has been used, filters the results using NSGA-III to +improve the accuracy, diversity, novelty, and recall. +Table 12 represents the comparison results between our work and other drug recommendation +systems in terms of important machine learning metrics. + +TABLE 12: COMPARISON RESULT OF OUR PROPOSED RECOMMENDATION SYSTEM +WITH OTHER STATE-OF-THE-ART APPROACHES +F1-Measure + +Precision + +Specificity + +Sensitivity + +Accuracy + + +0.07 + +0.04 + +0.33 + +0.75 + +0.34 + +SVM[26] + +0.18 + +0.31 + +0.86 + +0.13 + +0.31 + +Neural Network [26] + +0.41 + +0.32 + +0.54 + +0.61 + +0.55 + +Kmeans User CF [48] + +0.39 + +0.41 + +0.66 + +0.39 + +0.63 + +NSGA III [57] + +0.38 + +0.33 + +0.49 + +0.45 + +0.48 + +Conventional MF [58] + +0.45 + +0.36 + +0.38 + +0.60 + +0.45 + +MLP [59] + +0.65 + +0.62 + +0.64 + +0.69 + +0.65 + +Proposed Method + + +Comparison results consist of the F2 measure, ROC, and confusion matrix of different approaches +depicted in Figures 6 to 8. + + + + + + + + +Figure 6- Comparison result of F2 measure metric + +0 +0.2 +0.4 +0.6 +F2 Measure +SVM [26] +Backpropagation [26] +Kmeans CF [48] +NSGA II [57] +ConvMF [58] +MLP [59] +Proposed Method + + +Figure 7- comparison result of ROC. + + +Figure 8- comparison result of the confusion matrix. + +ROC +10 +True Positive Rate +0.8 +0.6 +0.4 +ProposedMethod +Backpropagation +0.2 +KmeansUserCF +Conventional MF +0.0 +RandomClassifier +0.0 +0.2 +0.4 +0.6 +0.8 +10 +FalsePositiveRateProposed Method +110 +100 +S +proper +108 +47 +ActualLabels +90 +80 +70 +not proper +113 +60 + 50 +PredictedLabelsConventional MF +100 +I Labels +proper +53 +90 +Actual +80 +not proper +105 +108 +70 +60 +Predicted LabelsMLP +120 +110 +Labels +proper +73 +47 +100 +06 +Actual +80 +not_proper +128 +82 +70 +60 +F 50 +PredictedLabelsSVM +200 +175 +Labels +proper +10 +5 +150 +125 +Actual +100 +75 +not_proper +208 +107 +50 +25 +PredictedLabels +Figure 9 -Confusion matrix obtained for the proposed method. + +The construction of a confusion matrix for different ratings is also shown in Figure 9. The +predictions are compared with actual ratings of users, and drugs for the case when they are +considered separately and combined according to our proposed approach. +One of the important components of our recommender system is the final knowledge-based +approach. This component prevents death, hospitalization, and disability by considering drug +interactions and the user’s age. The adverse Events Dataset is used in this regard to our system's +performance for recognizing such cases and recommending the appropriate drugs. This dataset +contains 2486 samples, where 80% of them are used for rule extraction, and the remaining 20% +are for the test. + +JustUserRatings +160 +140 +Labels +proper +37 +19 +120 +100 +Actual +80 +not proper +112 +162 + 60 +40 +20 +PredictedLabelsustDrugRatings +180 +160 +S +proper +30 +14 +140 +Label +120 +Actual +100 +80 +not proper +98 +188 +60 +40 +F 20 +PredictedLabelsUsers&DrugRatings +110 +100 +S +proper +108 +47 +Label +90 +Actual +80 +70 +not proper +113 +60 + 50 +PredictedLabelsThe following parameters are considered for the evaluation: +𝐷𝑒𝑎𝑡ℎ 𝑅𝑎𝑡𝑖𝑜 = +𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝐷𝑒𝑎𝑡ℎ +𝑇𝑜𝑡𝑎𝑙 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑅𝑒𝑐𝑜𝑚𝑚𝑒𝑛𝑑𝑎𝑡𝑖𝑜𝑛 +𝐷𝑖𝑠𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑅𝑎𝑡𝑖𝑜 = +𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝐷𝑖𝑠𝑎𝑏𝑖𝑙𝑖𝑡𝑦 +𝑇𝑜𝑡𝑎𝑙 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑅𝑒𝑐𝑜𝑚𝑚𝑒𝑛𝑑𝑎𝑡𝑖𝑜𝑛 +𝐻𝑜𝑠𝑝𝑖𝑡𝑎𝑙𝑖𝑧𝑎𝑡𝑖𝑜𝑛 𝑅𝑎𝑡𝑖𝑜 = +𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝐻𝑜𝑠𝑝𝑖𝑡𝑎𝑙𝑖𝑧𝑎𝑡𝑖𝑜𝑛 +𝑇𝑜𝑡𝑎𝑙 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑅𝑒𝑐𝑜𝑚𝑚𝑒𝑛𝑑𝑎𝑡𝑖𝑜𝑛 +For comparison, the system's performance for different adverse events was calculated one time +without a knowledge-based component and the second time using this component. +Table 13 and Figure 10 represent the results of this comparison. +TABLE 13- COMPARISON RESULTS OF KNOWLEDGE-BASED COMPONENT +Adverse event +Without knowledge-based +component +With knowledge-based +component +Death rate +44% +6% +Hospitalization +15% +2% +Disability +4% +0.7% + +Knowledge-based component is an essential part of a drug recommendation system in reducing +adverse events and improving the quality of recommendations. + +Figure 10- Comparison result of adding knowledge-based in the recommendation system. +We also considered one more important metric for recommender systems evaluations: hit rate. +0 +10 +20 +30 +40 +50 +Death Rate +Disability Rate +Hospitalization Rate +With Knowledge Base +Without Knowledge base + +The data set's testing samples (330) are utilized in hit-rate evaluation. The hit-rate in evaluation is +calculated by the ratio of the total hits in the top 10 recommended drugs returned for all users and +the total testing samples. So if 𝜂 is the number of relevant predicted drugs for all users, and 𝑁 is +the total number of testing samples, according to [60], the hit-rate is calculated as equation (25): +ℎ𝑖𝑡_𝑟𝑎𝑡𝑒 = +𝜂 +𝑁 (25) +The result of hit rate evaluation is represented in Figure 11. As it can be seen from this figure, our +proposed approach has the ℎ𝑖𝑡 𝑟𝑎𝑡𝑒 = 0.49, which is better than all other approaches. + + +Figure 11- Top-10 hit-rate recommendation systems. + +The next evaluation metric is cumulative hit-rate, which represents the number of hits with ratings +above a given threshold and ignores the predicted ratings lower than the threshold. The result of +the cumulative hit-rate with the threshold set to 4 is shown in Figure 12. The cumulative hit-rate +is calculated as (26): +𝐶𝑢𝑚𝑢𝑙𝑎𝑡𝑖𝑣𝑒 𝐻𝑖𝑡 − 𝑅𝑎𝑡𝑒 = +𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 ℎ𝑖𝑡𝑠 𝑤𝑖𝑡ℎ 𝑟𝑎𝑡𝑖𝑛𝑔 𝑎𝑏𝑜𝑣𝑒 𝑡ℎ𝑟𝑒𝑠ℎ𝑜𝑙𝑑 +𝑁 + +(26) +The utilization of this threshold makes a better match with the user’s interest in the recommended +drug. +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +Top-10 Hit Rate +SVM [26] +NeuralNetwork [26] +Kmeans Used CF [48] +NSGA III [57] +Conventional MF [58] +MLP [59] +Proposed Method + + +Figure 12- Top-10 cumulative hit-rate of recommendation systems. + +Our results are encouraging in the field of drug recommendations. It has combined the benefits of +basic recommender approaches with less computational overhead through a novel modeling +approach and using statistical methods. It also classifies drugs and users in terms of their features, +leading to high accuracy compared to state-of-the-art algorithms. However, better results can be +achieved by considering the characteristics of diseases and recommending drugs based on disease +features in addition to the features of patients and drugs. +7- CONCLUSION +In this paper, we proposed a comprehensive drug recommender system that takes advantage of all +basic recommender system techniques and applies natural language processing, neural network- +based matrix factorization, and, more importantly, employing knowledge-based recommendations +to recommend the most accurate drugs to patients. Compared with conventional matrix +factorization, our proposed method improves the accuracy, sensitivity, and hit rate by 26%, 34%, +and 40%, respectively. In comparison with other machine learning approaches, we obtained an +accuracy, sensitivity, and hit rate by an average of 31%, 29%, and 28%, respectively. Our approach +can be used as an adjunct tool torecommend drugs to patients and improve the quality of +prescriptions and reduce the errors caused by medical practitioners. +0 +0.05 +0.1 +0.15 +0.2 +0.25 +0.3 +0.35 +Top-10 Cumulative Hit Rate +SVM [26] +NeuralNetwork [26] +Kmeans Used CF [48] +NSGA III [57] +Conventional MF [68] +MLP [59] +Proposed Method + +In the future, we will extend the knowledge and information extraction from drug databases and +include all existing patient features in the user features. Also, we are going to consider the features +of the disease in the recommendation. These features can be captured by general practitioners and +help improve the proposed drug recommender system performance and make more accurate +recommendations by having more relevant features. In the final output of the recommendation, we +also include the dosage and effectiveness of a drug in addition to the list of drugs. 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Dashpand Mukund, and George Karypis. “item-based top-n recommendation algorithms.” +ACM Transactions on Informatics Systems (TOIS), 22.1 2004: 143-177 + + diff --git a/4tAyT4oBgHgl3EQfcPfW/content/tmp_files/load_file.txt b/4tAyT4oBgHgl3EQfcPfW/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b8d9ec806a57a024a2244d58be0767253a58c388 --- /dev/null +++ b/4tAyT4oBgHgl3EQfcPfW/content/tmp_files/load_file.txt @@ -0,0 +1,1232 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf,len=1231 +page_content='RECOMMED: A Comprehensive Pharmaceutical Recommendation System Mariam Zomorodi1,*, Ismail Ghodsollahee2, Pawel Plawiak1,3, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Rajendra Acharya4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' 6 1 Department of Computer Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Faculty of Computer Science and Telecommunications,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Cracow University of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Krakow,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Poland 2 Department of Computer Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Ferdowsi University of Mashhad 3 Institute of Theoretical and Applied Informatics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Polish Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Gliwice,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Poland 4 Department of ECE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Ngee Ann Polytechnic,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' 535 Clementi Road,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Singapore 599 489,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Singapore 5 Department of Biomedical Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' School of Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' SUSS University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Singapore 6 Department of Biomedical Informatics and Medical Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Asia University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Taichung,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Taiwan Objectives: To extract datasets containing useful information from two drug databases and recommend a list of drugs to physicians and patients with high accuracy while considering the most important features of patients and drugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' The history and review of the target patient and similar patients, and drug information, are used as a reference to recommend drugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Methods: A comprehensive pharmaceutical recommendation system was designed based on the patients’ and drugs’ features extracted from Drugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='com and Druglib.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' First, data from these databases were combined, and a dataset of patients and drug information was built.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Secondly, the patients and drugs were clustered, and then the recommendation was performed using different ratings provided by patients, and importantly by the knowledge obtained from patients and drug specifications, and considering drug interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' To the best of our knowledge, we are the first group to consider patients’ conditions and history in the proposed approach for selecting a specific medicine appropriate for that particular user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Our approach applies artificial intelligence (AI) models for the implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Sentiment analysis using natural language processing approaches is employed in pre- processing along with neural network-based methods and recommender system algorithms for modeling the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' In our work, patients’ conditions and drugs’ features are used for making two models based on matrix factorization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Then we used drug interaction to filter drugs with severe or mild interactions with other drugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' We developed a deep learning model for recommending drugs by using data from 2304 patients as a training set, and then we used data from 660 patients as our validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' After that, we used knowledge from critical information about drugs and combined the outcome of the model into a knowledge-based system with the rules obtained from constraints on taking medicine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Results: The results show that our recommendation system is able to recommend the best combination of medicines according to the existing real-life prescriptions available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' It also has the best accuracy and other metrics in recommending a specific drug compared to other existing approaches, which are generally based only on patient ratings or comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Our proposed model improves the accuracy, sensitivity, and hit rate by 26%, 34%, and 40%, respectively, compared with conventional matrix factorization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' In addition, it improves the accuracy, sensitivity, and hit rate by an average of 31%, 29%, and 28% compared to other machine learning methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' We have also open-sourced our implementation in Python.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Conclusions: Our proposed RECOMMED system extracts all vital information from the drug, patient databases and considers all necessary factors for recommending accurate medicine which can be trusted more by doctors and patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' We have shown the efficacy of our proposed model in real test cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Keywords: recommendation system;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' drug recommendation system;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' drug information extraction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' hybrid recommendation method 1- INTRODUCTION Recommendation systems (RS) are knowledge extraction systems that use information retrieval approaches to help people make better decisions and discover items through a complex information space [1], [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' They have been around for many years, and with the advancement in machine learning approaches, their use has been widened, and it helps people to make more appropriate decisions in using different products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' The popularity of using RS in different fields has increased since the announcement of the Netflix Prize competition that aimed to predict movie rates [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' The application of recommender systems is very extensive: from entertainment to e-commerce, the tourism industry, and medical recommender systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Also, with rapid progress in artificial intelligence, there has been a greater acceleration in the application of recommender systems and their development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Medical recommender systems are a particular type of recommender system, and they have some distinct features that make them special: They have to be used very carefully, and because they affect people’s health, there are many concerns about using RS for them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' On the other hand, many people die every year because of medication errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' It has been reported as the third leading cause of death in the world [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' This makes the use of intelligent systems in medical science valuable and necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Drug prescription is also vital for physicians, and it involves considering different aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=" Patient history of using drugs, the specification of drugs for diseases related to the recommendation in question, and the drug's effectiveness for that specific case are among such concerns." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Having as many different medicines as 24000 [5] in just one database, they can benefit from a recommendation system to perform a set of suggestions for a particular patient with a specific disease to help physicians in prescribing the most appropriate medicines and also help patients to have a better choice in using drugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Recommender systems can be distinguished by the degree of risk imposed when a user accepts a recommendation [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' In this regard, the medical domain can be seen as high risk, mostly due to the recommendation given to the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' On the other hand, while having a comprehensive drug recommender system is important, designing a complete system requires a dataset of drugs with patient ratings, reviews, and also information about drugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' We gathered this information from two different and well-known databases Drugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='com [5] and Druglib.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='com [7] and we built three datasets which train the system and construct the final model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Finally, putting all of them together, we proposed a novel drug recommender system called RECOMMED that learns the patient and drug features and their previous drugs taken, and also the user reviews for different drugs to recommend a new drug to a patient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' The novelty of this work lies in the following parts: 1- Propose a pharmaceutical recommender system by considering the features of patients and drugs, including patients’ conditions, age, gender, drug side effects, and drug categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' 2- Performing pre-processing steps on databases Druglib.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='com and Drugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='com websites to gather the appropriate data for our recommendation system, leading to comprehensive datasets for drug information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' 3- Our system considers sentiment analysis of reviews, the prescriptions of doctors, and different similarity measures for recommending a medicine and its dose and other recommendations, including side effects and warnings for their usage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' 4- Our system consists of a knowledge-based component to exclude drugs with serious side effects for a specific patient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' 5- We proposed a model to predict the efficiency of medicine for patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' In the next section, we provide some background in the general and general recommender systems field;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' later in Section 3 we particularly introduce the drug recommender systems and their challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Then in Section 4 we provide the current state-of-the-art of recommender systems methods, specifically drug recommender systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Section 5 is an elaborate explanation of our proposed comprehensive drug recommender system in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Section 6 provides the results of this work, plus a discussion about them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Finally, in Section 7 we conclude the paper and present the future directions of this research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' 2- RECOMMENDATION SYSTEMS BACKGROUND Recommender systems are decision-making systems that extract information from different kinds of knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' For many years, many recommender systems have been in various domains with different purposes [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' In this section, we review the basic concepts of various types of recommender systems and the way they are categorized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' According to the type of data that recommender systems use to make decisions, the algorithms utilized in recommender systems have two major categories: - collaborative filtering (CF) and - content-based filtering (CB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' CF approaches are further divided into user-based and item-based approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' CF and CB approaches have shown acceptable results when they are used in recommending different kinds of products like movies, books, and music.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Also, the recommendation system can combine these two major techniques, usually called hybrid recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' In addition, there are some specific types of recommender systems that have their strength in various domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' One of these types is a knowledge-based recommender system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Here, we briefly introduce the major recommender system techniques considered in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' 2-1 Collaborative recommender systems The idea behind this group of recommendation methods is to use a measure of similarity between users or items to recommend something to a given user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' It states that if two users share some interest in the past, they will likely have similar interests in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' A collaborative approach is based on the rating a user gives to items;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' in its basic form, it doesn’t need any other information about users and items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' CF approaches can be divided into two basic types, neighborhood methods (also known as memory-based) and latent factor models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Neighborhood methods are divided into one of the following two basic methods: 2-1-1 User-based neighborhood recommender system This approach aims at suggesting the products based on the similarity between users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' In this regard, several similarity measures can be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' We denote 𝒰 as the set of users, ℐ as the set of items, and 𝑅 as the set of existing ratings, Pearson Correlation (PC) is one of the popular ones, which is computed as equation (1) for users 𝑢 and 𝑣 [9]: 𝑃𝑒𝑎𝑟𝑠𝑜𝑛_𝐶𝑜𝑟𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛(𝑢, 𝑣) = ∑ (𝑟𝑢𝑖−𝑟̅𝑢)(𝑟𝑣𝑖−𝑟̅𝑣) 𝑖∈ℐ𝑢𝑣 √∑ (𝑟𝑢𝑖−𝑟̅𝑢)2 𝑖∈ℐ𝑢𝑣 √∑ (𝑟𝑣𝑖−𝑟̅𝑣)2 𝑖∈ℐ𝑢𝑣 (1) In this equation, ℐ𝑢 is the set of items rated by user 𝑢 and ℐ𝑢𝑣 is the items rated by both 𝑢 and 𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Also, 𝑟𝑢𝑖 is the rating of the user 𝑢 for a new item 𝑖 and 𝑟̅𝑢 is the average of the ratings given by 𝑢.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' And the prediction for the rating of user 𝑢 for item 𝑖 is calculated as equation (2): 𝑝𝑟𝑒𝑑(𝑢, 𝑖) = ∑ 𝑠𝑖𝑚(𝑢,𝑗)∗(𝑟𝑖,𝑗−𝑟̅𝑗) 𝑗 ∈𝑢𝑠𝑒𝑟𝑠 ∑ 𝑠𝑖𝑚(𝑢,𝑗) 𝑗 ∈𝑢𝑠𝑒𝑟𝑠 + 𝑟̅𝑢 (2) Where 𝑠𝑖𝑚 is the measure of similarity between user 𝑢 and item 𝑖 and 𝑢𝑠𝑒𝑟𝑠 is the set of users most similar to user 𝑢.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Therefore, the ratings are weighted by the similarity measure in this prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' 2-1-2 Item-based neighborhood recommender system In contrast to user-based recommender systems, item-based recommender systems use item similarity to suggest a product to a specific user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Similar to the user-based approach, here the prediction for user 𝑢 for item 𝑖 is also calculated as equation (3): 𝑝𝑟𝑒𝑑(𝑢, 𝑖) = ∑ 𝑠𝑖𝑚(𝑖,𝑗)∗(𝑟𝑢,𝑗−𝑟̅𝑗) 𝑗 ∈𝑁 ∑ 𝑠𝑖𝑚(𝑖,𝑗) 𝑗 ∈𝑁 + 𝑟̅𝑖 (3) Where 𝑠𝑖𝑚 is the measure of similarity between items 𝑖 and 𝑗 and 𝑁 is the set of items similar to item 𝑖 rated by 𝑢.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' 2-1-3 Matrix factorization One of the biggest challenges for standard methods in CF is the sparsity of the rating matrix (or user-item matrix);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' model-based CF can help overcome this challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' There are many ways to build models based on which we can make recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Matrix factorization is one the popular methods with the idea of decomposition of a matrix into the product of two or maybe three matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Having a dataset of the ratings of various users for different items, this model transforms the rating matrix to the individual user and item matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' So the model is defined as follows: Assume there is a set of users 𝑈 and items 𝐷, with rating matrix 𝑅(𝑀 × 𝑁), which is the ratings given by users on items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' 𝑀 and 𝑁 are the total numbers of users and items, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Matrix factorization in recommender systems aims to find 𝑘 total latent features/factors by decomposing 𝑅 according to equation (4) to user matrix 𝑈 and item matrix 𝐼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' 𝑅 ≈ 𝑈 × 𝐼𝑇 = 𝑅̂ (4) U is a 𝑀 × 𝑘 embedding matrix and, I is a 𝑁 × 𝑘 embedding matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' 2-2 Content-based recommender systems Content-based recommendation uses the attributes of the users or user profile and the attributes of items to recommend an item to a user [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Providing this information requires extra work and effort to represent items properly and build a user profile appropriate for the recommendation process [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=" This kind of recommender system learns the user preferences and tries to recommend items similar to the user's preferences." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Having 𝐷 rated items by user 𝑈, content-based RS aims to find the rating for item 𝑖, which is not seen by user 𝑈.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' In this method, items’ features are extracted, then used to find similarities between items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Then, in a simple nearest neighbor approach top-𝑛 nearest neighbors of item 𝑖 in 𝐷 are selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' This selection is based on a similarity measure like cosine similarity which is calculated as equation (5): cos(𝜃) = 𝑿 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='𝒀 ‖𝑿‖×‖𝒀‖ = ∑ 𝑋𝑖𝑌𝑖 𝑛 𝑖=1 √∑ 𝑋𝑖 2 𝑛 𝑖=1 √∑ 𝑌𝑖 2 𝑛 𝑖=1 (5) The ratings of these 𝑛 items are used to predict the rating for item 𝑖 by user 𝑈.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' In most content-based recommender systems, item features are textual descriptions and don’t have well-defined values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' So, natural language processing approaches like TF-IDF or the bag-of-words are used to assign numerical values to the textual features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' 2-3 Hybrid recommender systems Hybrid approaches combine different recommender system algorithms to make a more accurate system that considers the benefits of different approaches for recommending an item to the users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' A combination of content-based and collaborating filtering is the most common type of hybridization method [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' 2-4 Knowledge-based recommender systems This RS aims to produce recommendations based on existing rules that satisfy a user’s needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' In the context of drug recommendation, this knowledge involves many different conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' For example, death reports for a specific drug and drug interactions are two important information that a drug recommender system has to consider before recommending a list of medicine to a patient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' 2-5 AI-based recommendation systems Over time many different artificial intelligent approaches have been applied to recommendation systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' However, the tendency to use AI methods in recommender systems is mostly because of the big data availability and diversity of recommendation systems approaches, which can benefit from AI, particularly machine learning algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Deep learning as a subfield of machine learning has attracted many researchers from a broad variety of disciplines due to its learning capabilities from data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Recently there have been many researches on deep learning-based recommendation systems [12], and Multilayer Perceptron (MLP), Autoencoder (AE), Convolutional Neural Networks (CNN), Recurrent Neural Networks (RNN), are among the mostly used deep learning models in RS [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Many of these deep learning- based approaches have contributed to the works on CB, CF, and other types of RS [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Also, some works utilize hybrid deep networks, like the combination of RNN and CNN [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Moreover, to integrate the advantages of memorization and generalization for recommender systems, a wide & deep neural network has been used [16], and the model shows better results with increased acquisitions on the Google Play app.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' 2-6 Other types of recommendation systems Although these techniques are the basic and mostly used recommender system approaches, several more types of recommender systems are suggested in the literature, and authors in [17] give a detailed classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Medical and drug recommendation is one of the important applications of recommender systems which uses techniques in recommender systems to recommend medicine, predict the usefulness of drugs, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' In the following sections, the position of recommender systems in medical science, particularly in the pharmaceutical sector and the state-of-the-art in this field, is discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' 3- RECOMMENDER SYSTEMS IN MEDICAL SCIENCE One of the attractive and important applications of recommendation systems is medical recommendations and drug products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Here are the major differences between medical recommender systems and other recommender systems: - Medical recommender systems care more about the health of patients than to make a profit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' - Security is the primary goal in drug recommender systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' - Many existing recommender system techniques cannot be used, and others must be used with caution because of safety issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' - In the long term, time is considered an important factor in recommending a drug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=" There are many situations where some drugs' negative effects are discovered over time." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' One example is the drug zimeldine [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' So, a comprehensive medical recommender system should consider ratings in different time stamps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' In the drug recommender system, the domain is medicine, and the exact contents to be recommended are one or more of the following lists: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' A list of drugs, at least one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' The dose, is the amount of drug taken at one time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' The frequency at which the drug doses are taken over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Duration, which is how long the drug is taken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Numbers 2 to 4 in the above list are referred to as the dosage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Therefore, we can define a drug recommender system as a smart system that is able to recommend a list of drugs plus their dosages with high accuracy in terms of a real prescription of a physician and also to have a positive effect on a patient, which available data can partially verify.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' It should be noted that, of course, there is no medicine recommender system that we can trust thoroughly, and like other artificial intelligence systems applied in healthcare, their use and ethical issues must be addressed appropriately [19], [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' 4 LITERATURE REVIEW Medical recommender systems have been around for many years, even before the emergence of recommender systems as a new field in computer science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' According to [21], medicine recommender systems fall into two broad categories named “ontology and rule-based approaches” and “data mining and machine learning-based” approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Ontology-based recommender systems use the hierarchical organization of users and items to improve the recommendation [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Data mining and machine learning algorithms in the medical field are used to predict and recommend things like drug usefulness, having a disease [23], [24], the condition of the user, or ratings [25], [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' For example, SVM, backpropagation neural network, and ID3 decision tree have been used in [27] for recommending drugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' The performance of these approaches has been compared in the above work, and the authors have shown that SVM has better accuracy and efficiency compared to the other algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Their data set contains patients’ features age, sex, blood pressure, cholesterol, Na and K levels, and drug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Some other researchers, while described as medical or medicine recommender systems, consider a detection and classification task where the dataset which is trained has some patient attributes, and based on that, the objective of the work is the detection or prediction of a disease and then for each disease a set of medicines is recommended [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Sentiment analysis of drug reviews is one of the basic approaches for drug recommendations [28], [29], [30], [31], [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' The sentiment analysis in these works mainly aims to recommend a drug or extract useful information like adverse drug reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' In [31], different deep learning approaches, such as CNN, LSTM, and BERT, have been investigated for sentiment analysis of patients’ drug reviews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' In another work, the combination of CNN-RNN has been applied In addition to recommendation systems, sentiment analysis and opinion mining of drug reviews is an active research area in drug review processing [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' This analysis can be used for automatic opinion mining and recommending drugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' A hybrid knowledge-based medical prescription approach has been presented in [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' The authors use historical medical prescriptions to recommend a list of medicines to physicians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' The approach uses the similarity between cases where a case is medical information like demography, treatment, age, sex, symptoms, and diagnosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Based on the degree of similarity, a drug list is produced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' The list is complemented by Bayesian reasoning, where a model of the conditional probability of drugs is built.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' This approach has been applied in Humphrey & Partners Medical Services Limited medical center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Some works in medical recommendation have focused on particular drugs like diabetes [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Their model is based on the ontology of medical knowledge and a decision decision-making approach for multiple criteria and computes the medication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Then by using the entropy, the information about patient’ history has been computed, and finally, the most appropriate medications have been recommended to the physicians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Many recommender system approaches have not been well considered in the medical and pharmaceutical recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' However, using polarity in sentiment analysis of user comments is one of the important parts of using NLP in recommendation systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' It can be viewed as determining whether a word or phrase in the document or even a whole document is positive, negative, or neutral in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Figure 1 shows the broad classification of different recommendation system approaches in pharmaceutical research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' We can see a growing tendency to use machine learning approaches in this field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Figure 1: Broad classification of recommendation system techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' 5- Material and methods In this section, we formulate the medicine recommender system problem and present our approach for the general medicine recommender system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Many recommendation systems, like collaborative filtering and content-based approaches, mostly rely on past information to make decisions for the current situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' It is not always the case in the domain of drug recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' The patient condition is different compared to the other patients and compared to the same patient over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=" So, in addition to the history information like general rates, reviews, and the effective rate of the drug, it is necessary to use the patient's current condition to make a more accurate decision." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' We also cannot rely on diversity-based recommendations as it is used in some recommender systems, like the one used in Netflix, even if the drug is not rated high, it can be suitable for some patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' On the other hand, many recommender systems rely on knowledge from users;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' when there is a lack of users’ knowledge, we cannot personalize them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' While we have an adequate dataset for our recommendation task, the problem emerges when new inputs enter the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' In our medicine Drug Recommendation Systems Collaborative Filtering Memory Based User-Based Filtering WaveLet [36] Item-Based Filtering Model Based Matrix Factorization SVD SVD++ Content Based Machine Learning T-Recs [37] Knowledge-Based Machine Learning Data mining Framework [38] Ontology-based MCDM & Entropy [35] SWRL [39] ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' [40],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' [41] GalenOWL [42] Panacea [43] SemMed [44] Hybrid CB & Knowledge Based Machine Learning LOD cloud mining [45] User CF & Knowledge Based Machine Learning DiaTrack [46] Personalized Clinical Prescription [47] CADRE [48] Item Based CF & Knowledge Based Machine Learning Data Driven[49] recommender system,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' these inputs can be new patients or new drugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Cold start problem is a term used for this problem, and it is a challenging issue in designing any recommender system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' We reduced this effect by applying a clustering-based approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Because drugs are clustered into a specific category, we can put a new drug in the category which belongs to it, so we use the same rating for the new drugs as those in that category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' This is effective in solving the cold start problem in our recommender system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Proposed method Since every recommendation technique has its own benefits, a universal recommender system should be able to take advantage of all of these techniques to improve the outcome of a recommender system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' The drug recommendation system in our work has the benefits of different recommendation categories and combines their advantages by using several steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' First, natural language processing and machine learning algorithms are applied in the context of basic recommender system techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' This section discusses all phases of our model for building a comprehensive drug recommender system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' This paper presents a novel hybrid drug recommender system (RS) with features of several recommender systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' It uses natural language processing (NLP) and other machine learning techniques to implement the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' The proposed RS approach is a new recommendation system method for pharmaceutical recommendation, which can be considered a hybrid of CB, CF model- based, knowledge-based, and AI-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Here in this section, we elaborate on each step toward the final drug recommendation for each patient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' After a very intensive web crawling through two well-known pharmaceutical websites, Drugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='com and Druglib.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='com, and building three different datasets, feature extraction and modeling are performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Then in the next step, recommendations for proper drugs are performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' At the final stage, the list of drugs is refined based on defined rules in addition to the ratings and drug features which is an important aspect of our medicine recommender system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Figure 2: Components of RECOMMED drug recommendation system in the training stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Figure 2 presents the whole RECOMMED model in the training stage of our work, consists of four components, and we elaborate on each phase of our approach in more detail in the following parts: 5-1 Dataset extraction In this work, any recommendation for drugs and their dosage is based on the patients’ features like age, gender, previous illness, and other drugs they consume, and drugs’ features like drug classification, side effects, and drug interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' So, in this phase, the extraction of user features, drug features, and drug interaction datasets from Drugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='com and Druglib.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='com databases is accomplished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' In the second step of this phase, the dataset is prepared for clustering and modelling the recommendation system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=" The review field in the drug recommendation database contains users' and caregivers opinions about drugs' effectiveness." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' According to our knowledge, none of the existing datasets have complete and comprehensive patient and drug information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Start Remove HTML Frames,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='Tags and advertisement ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='End ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='Crawl Review Webpages ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='Dataset ' 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Weights ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='& Biases Sets ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='Initialize Weights & Biases ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='Forward Propagation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='Backward Propagation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='Update Weights & Biases ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='Error < Threshold ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='New User Registration ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='Compute 10 High Rated ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='Drug for Recommendation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='Drug[i] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='Interaction?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' i=0 i<10 i=i+1 User Features Filter Set Drug[i] Allowed?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Recommendation append(Drug[i]) We built three different datasets named users, drugs, and interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='1 Drugs and users datasets In this work, Druglib.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='com and Drugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='com were employed to extract information about patients and drugs and build two datasets named drugs and patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' We should mention that there are also other databases for drug information and recommendations, like SIDER [50], for drug side effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' We will include them in future works to build a complete dataset for drugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Three features consisting of side effects, benefits, and membership in a given drug category were considered for drugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' First, different drug categories and side effects were extracted in tables 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' There are 150 different drug categories, and 128 different side effects were extracted from the Druglib.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='com database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' TABLE 1 - DRUG CATEGORIES LIST Category Index Acetylcholine-Agonists 1 Adrenergic-Alpha-Agonists 2 … Vasodilators 150 TABLE 2 DRUGS SIDE EFFECTS Side Effects Index Completed suicide 1 Confusional state 2 … Wrong drug administered 128 Then drug benefits were also extracted and combined with the information in the above tables, and finally, the drugs dataset was prepared, as is partially shown in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' TABLE 3- DRUGS DATASET Benefits Side Effects Drug Category Drug Name Index 88 … 2 1 128 … 2 1 150 … 2 1 0 … 1 1 0 … 0 0 0 … 0 1 Hytrin Terazosin 1 0 … 0 0 0 … 1 1 0 … 0 1 Mirtazapine 2 … … … … … … … … … … … … … … 0 … 0 0 0 … 0 0 1 … 0 0 Proscar Finasteride 480 We also extracted the users dataset of patient features and comments on different drugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Six features are considered for users datasets: age, gender, current disease (condition), other conditions, other drugs are taken, and user level, which is patient or caregiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Table 4 represents the structure of this dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' TABLE3- USERS DATASET Comment Side Effects Effective ness Overall Rating Drug Name Other Drug Other Conditio n Conditio n Genus Age Level index … Severe Side Effects Ineffective 1 Mirtazapine None Sleeplessness Depression Male 22 Patient 1 … Moderate Side Effects Ineffective 2 Mirtazapine None None Depression Male 38 Patient 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' … … Mild Side Effect Moderately Effective 4 Proscar Finasteride None None Hair loss Male 28 Patient 3294 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='2 Interactions dataset The last dataset prepared in this work is the interactions dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' This information is important for recommending the appropriate medicine list to the patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' We extracted drug interaction information from Drugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='com, and after mapping drugs’ names with their counterparts in Druglib.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='com, the interaction dataset, partially presented in Table 5, was created with 180 drug interaction information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' TABLE 4 -DRUG INTERACTION DATASET Abilifish … Cimbaita … Syntroid … zyban Abilifish Moderate … … Accupril … Moderate … Aciphex Major … … … … … … … … … … Zyban … … 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='2 Dataset preparation In this phase, our dataset is prepared for creating the recommendation model in the next step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' First, using Natural Language Processing (NLP) techniques, user and drug features are extracted, and then normalization and clustering are accomplished to prepare the datasets for modeling the recommendation system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Here, we elaborate on each of these steps: 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='1 Feature extraction The first pre-processing step is feature extraction from user feature and drug features datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Bag-Of-Words (BOW) method is used for this purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' NLP for extracting drug and user features The feature extraction was mainly performed using natural language processing (NLP) techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Two well-known methods to extract text features by NLP are Bag-of-Words (BOW) and term frequency-inverse term frequency (TF-IDF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Our proposed pharmaceutical recommendation system uses the BOW feature extraction method to perform feature extraction from database texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' This method consists of four steps: • Text-pre-processing pre-processing • Vocabulary creation • Building feature matrix • Polarity of user comments Here, every part of this process has been described: Text-pre-processing pre-processing In the text- pre-processing step, all punctuations and symbols are removed, and abbreviations are converted into their full names or phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Some of these conversions are presented in Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Moreover, spelling mistakes were corrected using the TexBlob library of Python, and stop words were removed using a predefined list of stop words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' TABLE 6- EXAMPLES OF ABBREVIATIONS TO FULL NAME CONVERSIONS Original Form Abbreviations high blood pressure HBP chronic obstructive pulmonary disease COPD premenstrual syndrome PMS obsessive compulsive disorder OCD Vocabulary creation Using NLP techniques, a vocabulary of words is created in the second step of feature extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' For this purpose, an array of words is created by checking all registered words in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' This array is constructed from unique words of the dataset and their frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' To deal with the random filling of the feature matrix, words are rearranged according to their frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Moreover, to deal with the sparseness of the feature matrix, words with low frequency are removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Some of the most frequent words extracted from the datasets created and discussed in the previous section can be seen in Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' TABLE 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' EXAMPLES OF MOST FREQUENT WORDS IN DATASETS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Frequency Term 33 Pain 22 Infection 15 Surgery 13 Chronic Building feature matrix The feature matrix is created in the third step of extracting the features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' For this purpose, a unique word is assigned to each matrix column, and a new row is considered for each user review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Each cell of this matrix represents the existence of the word in the user’s review, which is essentially zero or one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Polarity of user comments (PUC) We used NLP and opinion mining to extract PUC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' This approach aims at extracting the opinion of users as a positive or negative comment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' The output of this component is used in the users’ rating matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' ALGORITHM1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' COMMENT POLARITY ACQUISITION Input:𝑈𝑠𝑒𝑟𝐶𝑜𝑚𝑚𝑒𝑛𝑡𝑠, 𝑆𝑡𝑜𝑝𝑊𝑜𝑟𝑑𝑠 Output: 𝑃𝑜𝑙𝑎𝑟𝑖𝑡𝑦𝑂𝑓𝑈𝑠𝑒𝑟𝐶𝑜𝑚𝑚𝑒𝑛𝑡𝑠 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' 𝑅𝑒𝑚𝑜𝑣𝑖𝑛𝑔 𝐶𝑎𝑝𝑖𝑡𝑎𝑙 𝐿𝑒𝑡𝑡𝑒𝑟𝑠 𝑎𝑛𝑑 𝐸𝑚𝑜𝑗𝑖𝑠 from 𝑈𝑠𝑒𝑟𝐶𝑜𝑚𝑚𝑒𝑛𝑡𝑠 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' 𝑅𝑒𝑚𝑜𝑣𝑖𝑛𝑔 𝑆𝑡𝑜𝑝𝑊𝑜𝑟𝑑𝑠 𝑓𝑟𝑜𝑚 𝑈𝑠𝑒𝑟𝐶𝑜𝑚𝑚𝑒𝑛𝑡𝑠 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' 𝑊𝑜𝑟𝑑 𝑇𝑜𝑘𝑒𝑛𝑖𝑧𝑒 (𝑈𝑠𝑒𝑟𝐶𝑜𝑚𝑚𝑒𝑛𝑡𝑠) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' 𝑊𝑜𝑟𝑑𝐿𝑒𝑚𝑎𝑡𝑖𝑎𝑡𝑖𝑜𝑛(𝑈𝑠𝑒𝑟𝐶𝑜𝑚𝑚𝑒𝑛𝑡𝑠) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' 𝐹𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 𝑜𝑓 𝑊𝑜𝑟𝑑𝑠(𝑈𝑠𝑒𝑟𝐶𝑜𝑚𝑚𝑒𝑛𝑡𝑠) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' 𝑇𝑒𝑥𝑡𝐵𝑙𝑜𝑏(𝑈𝑠𝑒𝑟𝐶𝑜𝑚𝑚𝑒𝑛𝑡𝑠) Combined User Rating Acquisition To have a more accurate rating for drugs, we considered the combined user comments and ratings from different sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' This overall rating is called Combined User Rating Acquisition (CUR) parameter and is obtained from analyzing user comments and ratings as follows: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' 𝑂𝑣𝑒𝑟𝑎𝑙𝑙 𝑅𝑎𝑡𝑖𝑛𝑔 ∈ 𝑍, 0 ≤ 𝑂𝑣𝑒𝑟𝑎𝑙𝑙 𝑅𝑎𝑡𝑖𝑛𝑔 ≤ 10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' 𝐸𝑓𝑓𝑒𝑐𝑡𝑖𝑣𝑛𝑒𝑠𝑠 ∈ 𝐸, 𝐸 = {Ineffective, Marginally Effective, Moderately Effective, Considerably Effective, Highly Effective} 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' 𝑆𝑖𝑑𝑒 𝐸𝑓𝑓𝑒𝑐𝑡 ∈ 𝑆, 𝑆 = {𝑁𝑜 𝑆𝑖𝑑𝑒 𝐸𝑓𝑓𝑒𝑐𝑡, 𝑀𝑖𝑙𝑑 𝑆𝑖𝑑𝑒 𝐸𝑓𝑓𝑒𝑐𝑡, 𝑀𝑜𝑑𝑒𝑟𝑎𝑡𝑒 𝑆𝑖𝑑𝑒 𝐸𝑓𝑓𝑒𝑐𝑡, 𝑆𝑒𝑣𝑒𝑟𝑒 𝑆𝑖𝑑𝑒 𝐸𝑓𝑓𝑒𝑐𝑡, 𝐸𝑥𝑡𝑒𝑟𝑒𝑚𝑙 𝑆𝑒𝑣𝑒𝑟𝑒 𝑆𝑖𝑑𝑒 𝐸𝑓𝑓𝑒𝑐𝑡} 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' User Comment CUR parameter is calculated as equation (6), and the above parameters are replaced by CUR in the user feature matrix: CUR = (𝑂𝑣𝑒𝑟𝑎𝑙𝑙𝑅𝑎𝑡𝑖𝑛𝑔 10 +𝐷𝑂𝐸 4 ) 2 − 𝐷𝑂𝑆 4 +𝑃𝑈𝐶 2 (6) In equation (6), DOE (Degree of Effectiveness) represents the degree of drug effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' The user selects the effectiveness of a drug from a list of five different options: Ineffective, Marginally Effective, Moderately Effective, Considerably Effective, and Highly Effective, and it takes a number in the range [0-4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Similarly, DOS (Degree of Side Effects) is the degree that a drug has a side effect (range [0-4]), and the numbers applied in the denominator are for normalization purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' PUC (Polarity of User Comments) is calculated using Natural Language Processing (NLP), and opinion mining techniques and the nltk library in Python are used in this regard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Algorithm 1 shows the steps of the work for calculating PUC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Normalization- After extracting features from drug and user datasets, these features should also be normalized to perform better in training the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Combined User Rating (CUR) Drug Name Other Drug Other Condition Condition Genus Age Level index Comment Side Effects (DOS) Effectiveness (DOE) Overall Rating Drug Name Other Drug Other Condition Condition Genus Age Level Index 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='05 Mirtazapine None Sleeplesness depression male 22 patient 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='8 (Obtained from PUC) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='75 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='1 Mirtazapine None Sleeplesness depression male 22 patient 1 … … … … … … … … … … … … … … … … … … … … … 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='1 Proscar None None Hair loss male 28 patient 3294 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='4 Proscar None None Hair loss male 28 patient 3294 Figure 3- Combination of different user ratings for a given drug Figure 3 is the final user rating dataset after applying the combined user rating acquisition stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' This stage converts the dataset on the left side into the right side dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Each column in both datasets has a given user’s features along with the drug name they rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' In the left side dataset, we can see different ratings of the user, and then in the right side dataset, these ratings are combined into CUR using equation (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='3 Clustering Clustering is considered one of the main steps in a recommender system for improving the diversity, consistency, and reliability [51], which has been considered in many works in recommender systems, particularly for reducing the sparsity of data [52], [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Due to the sparseness of the rating matrix, we consider a clustering-based approach, and patients are clustered before performing the matrix factorization, which is elaborated in the next part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' This clustering is mostly required because users usually review only one drug corresponding to a specific disease, so the rating matrix is highly sparse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Clustering can help group the users and drugs with similar features and significantly resolve the sparsity problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Users are clustered based on their gender, age, comments, and being patient or caregiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' It is clear that after clustering, each class of users reviews several drugs, which can improve the matrix factorization process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' We used a modified K-means algorithm in [54] to perform this clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=" While the original K- means algorithm is unsupervised, which is used for clustering, the number of clusters is pre- determined, and so it couldn't be utilized in the same way in our proposed drug recommendation system." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Therefore, in this paper, we employed the U-Kmeans method [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' This method performs the unsupervised K-means and determines the best cluster numbers that lead to better classification performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' If each row of the dataset and the center of each cluster are represented by F= {𝑓1, … , 𝑓𝑛} and A= {𝑎1, … , 𝑎𝑘} respectively, the K-means objective function is defined as (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' 𝐽(𝑀, 𝐴) = ∑ ∑ 𝑀𝑖𝑗‖𝑓𝑖 − 𝑎𝑗‖ 𝑘 𝑗=1 𝑛 𝑖=1 (7) Where in (7), 𝑘 is the number of clusters, 𝑛 is the number of dataset features, and 𝑀𝑖𝑗 indicates the membership of 𝐹𝑖 to the 𝑗𝑡ℎ cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' In the K-means algorithm, this objective function must be minimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' In [55], an entropy-based method is proposed to improve K-means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' In this method, to determine the centers of the clusters, Equation (8) is added to the objective function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' 𝐵𝑛 ∑ 𝑎𝑗 𝑘 𝑗=1 ln 𝑎𝑗 (8) In (8), the effect of the cluster imbalance is added to the objective function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' As can be seen in (9), when the 𝐵𝑛 coefficient of the improved objective function is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' The following K-means objective function is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' 𝐽(𝑀, 𝐴) = ∑ ∑ 𝑀𝑖𝑗‖𝑥𝑖 − 𝑎𝑗‖ 𝑘 𝑗=1 𝑛 𝑖=1 − 𝐵 ∑ 𝜂𝑗 𝑘 𝑗=1 ln 𝜂𝑗 (9) Where in this equation, 𝜂𝑗 represents the number of members of a cluster, which is determined by (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' 𝜂𝑗 = ∑ 𝑀𝑖𝑗 𝑛 𝑖=1 𝑥𝑖 ∑ 𝑀𝑖𝑗 𝑛 𝑖=1 (10) In [54], equation (11) is considered to determine the optimized number of clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' By adding this term to equation (10), the final objective function is obtained as (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' L ∑ ∑ 𝑀𝑖𝑗 𝑘 𝑗=1 ln 𝑎𝑗 𝑛 𝑖=1 (11) 𝐽(𝑀, 𝐴, 𝑎) = ∑ ∑ 𝑀𝑖𝑗‖𝑥𝑖 − 𝑎𝑗‖ 𝑘 𝑗=1 𝑛 𝑖=1 − 𝐵 ∑ 𝑎𝑗 𝑘 𝑗=1 ln 𝑎𝑗 − L ∑ ∑ 𝑀𝑖𝑗 𝑘 𝑗=1 ln 𝑎𝑗 𝑛 𝑖=1 (12) The pseudocode of the U-K-means classification method based on the approach in [54] is presented in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Algorithm .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' 2 Our modified Pseudo code of U-Kmeans based on [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑐(0) = 𝑛, 𝛼𝑘 (0) = 1 𝑛 , 𝑎𝑘 (0) = 𝑥𝑖 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' 𝐼𝑛𝑖𝑡𝑖𝑎𝑙 𝑙𝑒𝑎𝑟𝑛𝑖𝑛𝑔 𝑟𝑎𝑡𝑒𝑠 𝐿(0) = 𝐵(0) = 1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' 𝑆𝑒𝑡 𝑡 = 0 , 𝜀 > 0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' 𝑤ℎ𝑖𝑙𝑒 𝑚𝑎𝑥‖𝑎𝑘 𝑡+1 − 𝑎𝑘 𝑡 ‖ < 𝜀 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' If ‖𝑥𝑖 − 𝛼𝑘‖2 − 𝐿𝑙𝑛𝛼𝑘 = min 1≤𝑘≤𝑐‖𝑥𝑖 − 𝑎𝑘‖2 − 𝐿𝑙𝑛𝛼𝑘 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' 𝑀𝑖𝑘 (𝑡+1) = 1 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Else 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' 𝑀𝑖𝑘 (𝑡+1) = 0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' 𝐿(𝑡+1) = 𝑒−𝑐(𝑡+1)/250 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' 𝛼𝑘 (𝑡+1) = ∑ 𝑀𝑖𝑘 𝑛 + ( 𝐵 𝐿) 𝛼𝑘 (𝑡) ln 𝑎𝑘 𝑡 − ∑ 𝛼𝑠 𝑡 𝑐 𝑠=1 𝑛 𝑖=1 ln 𝑎𝑠 𝑡 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' 𝐵𝑡+1 = 𝑚𝑖𝑛 ( ∑ exp (−𝜂𝑛|𝑎𝑘 𝑡+1−𝑎𝑘 𝑡 |) 𝑐 𝑘=1 𝑐 , 1− max 1≤𝑘≤𝑐(1 𝑛 ∑ 𝑀𝑖𝑘 𝑛 𝑖=1 ) − max 1≤𝑘≤𝑐 𝑎𝑘 𝑡 ∑ ln 𝑎𝑘 𝑡 𝑐 𝑘=1 ) 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' 𝑢𝑝𝑑𝑎𝑡𝑒 𝐶𝑡 𝑡𝑜 𝐶𝑡+1 𝑏𝑦 𝑑𝑖𝑠𝑐𝑎𝑟𝑑 𝑡ℎ𝑜𝑠𝑒 𝑐𝑙𝑢𝑠𝑡𝑒𝑟 𝑤𝑖𝑡ℎ 𝑎𝑘 𝑡+1 ≤ 1 𝑛 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' 𝑎𝑘 ∗ = 𝑎𝑘 ∗ ∑ 𝑎𝑠∗ 𝑐(𝑡+1) 𝑠=1 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' 𝑀𝑖𝑘 ∗ = 𝑀𝑖𝑘 ∗ ∑ 𝑀𝑖𝑠 ∗ 𝑐(𝑡+1) 𝑠=1 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' 𝑎𝑘 = ∑ 𝑀𝑖𝑘𝑥𝑖𝑗 𝑛 𝑖=1 ∑ 𝑀𝑖𝑘 𝑛 𝑖=1 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' 𝑖𝑓 𝑡 ≥ 60 𝑎𝑛𝑑 𝑐(𝑡−60) − 𝑐𝑡 = 0 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' 𝐵(𝑡+1) = 0 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' t=t+1 TABLE 8: RATE MATRIX WITHOUT CLASSIFICATION 619 618 617 616 615 … 6 5 4 3 2 1 Drugs Users 1 1 3 3 2 5 3 … 1 979 3 980 3 981 TABLE 9: RATE MATRIX AFTER CLASSIFICATION 619 618 617 616 615 … 6 5 4 3 2 1 Drugs Users 1 1 3 3 2 5 3 … 1 38 3 39 3 40 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='4 Modeling In the next step, the clustering outcome is used to build a recommender system model able to recommend the best drugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=" Later, we filter the model's output with a knowledge-based component for safety reasons." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Neural Network-based Matrix Factorization Matrix factorization is a popular method for recommender systems aiming at finding two rectangular matrices called user and item matrices with smaller sizes than the rating matrix [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' The dot product between these two matrices results in the rating matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' To reduce the computational overhead, copeTo reduces the computational overhead, cope with the sparsity of the ratings, and increase accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' We proposed a neural network-based matrix factorization technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' The first two matrices, Rating and Effectiveness, are constructed by extracting information from Druglib.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' In our model, the rating matrix 𝑅𝑎𝑡𝑖𝑛𝑔 ∈ R𝑛∗𝑚 is estimated as the multiplication of two matrices 𝐶𝑙𝑢𝑠𝑡𝑒𝑟𝑠𝑛∗𝑘 and 𝐷𝑟𝑢𝑔𝑠𝑘∗𝑚 as (13): 𝑅𝑎𝑖𝑛𝑔 ≈ Clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' 𝐷𝑟𝑢𝑔𝑠𝑇 (13) This model applies a neural network algorithm to estimate the users’ comments for each medicine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Clustered users and drugs and users’ and drugs’ features are used in building this new model, as illustrated in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Figure 4- Our proposed customized matrix factorization method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' 𝑏𝑚 𝑏2 𝑏1 Drug Cluster k … Drug Cluster 2 Drug Cluster 1 Drug m … Drug 2 Drug 1 𝑐1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='𝑚 𝑐1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='2 𝑐1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='1 Drug Feature 1 𝑑1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='𝑚 𝑑1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='2 𝑑1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='1 Latent Factor 1 𝑐2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='𝑚 𝑐2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='2 𝑐2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='1 Drug Feature 2 𝑑2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='𝑚 𝑑2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='2 𝑑2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='1 Latent Factor 2 … … 𝑐𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='𝑚 𝑐𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='2 𝑐𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='1 Drug Feature k 𝑑𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='𝑚 𝑑𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='2 𝑑𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='1 Latent Factor k User Feature k … User Feature 2 User Feature 1 User Feature k … User Feature 2 User Feature 1 𝑟1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='𝑚 𝑟1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='2 𝑟1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='1 𝑢1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='𝑘 𝑢1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='2 𝑢1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='1 User 1 𝑏1 𝑟1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='𝑚 𝑟1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='2 𝑟1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='1 𝑐1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='𝑘 𝑐1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='2 𝑐1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='1 User Cluster 1 𝑟1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='𝑚 𝑟2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='2 𝑟2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='1 𝑢2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='𝑘 𝑢2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='2 𝑢2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='1 User 2 𝑏1 𝑟1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='𝑚 𝑟2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='2 𝑟2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='1 𝑐2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='𝑘 𝑐2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='2 𝑐2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='1 User Cluster 2 … … 𝑟𝑛,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='𝑚 𝑟𝑛,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='2 𝑟𝑛,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='1 𝑢𝑛,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='𝑘 𝑢𝑛,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='2 𝑢𝑛,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='1 User n 𝑏𝑛 𝑟𝑛,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='𝑚 𝑟𝑛,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='2 𝑟𝑛,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='1 𝑐𝑛,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='𝑘 𝑐𝑛,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='2 𝑐𝑛,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='1 User Cluster n User Ratings Drug Ratings User embedding (User Weights) Drug Cluster Features sEmbedding Drug Embedding (Drug Weights) Drug Bias User Cluster Features User Bias The input to the neural network is user and drug-clustered features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=" Drug Embedding and User Embedding matrices are the input to this network, and drug and user are the network's outputs." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' With sparse rating matrices, the forward and backward pass calculations are accomplished just for non-zero ratings to reduce the computation load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' The neural network layer output is calculated as: 𝑎𝑧+1 = 𝑓𝑧+1(∑ 𝑤𝑖 𝑧+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' 𝜓𝑧+1(𝑛, 𝑚).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' 𝑎𝑖 𝑧 + 𝑏𝑗 𝑧+1 𝐾 𝑖=1 ) 𝑖 ∈ (1, 𝐾), 𝑗 ∈ (1, 𝑀𝑁), 𝑧 ∈ (0, 𝑍 − 1), 𝑛 ∈ (0, 𝑁), 𝑚 ∈ (0, 𝑀) (14) In this equation, 𝑓𝑧+1 is the activation function, 𝑤𝑖 𝑧+1 are the weights, 𝑏𝑗 𝑧+1 are the biases, 𝑍 is the number of layers, 𝑀 is the number of drugs, 𝑁 is the number of users, 𝑎𝑍 is the output of the network, 𝐾 is the number of the features for drugs or users, and 𝑀𝑁 represents the number of drugs in the user network and represents the number of users in the drugs network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' And finally 𝜓𝑧+1 is the rating existence function defined as: {𝜓𝑧+1(𝑛, 𝑚) = 1 𝑖𝑓(𝑅𝑎𝑡𝑖𝑛𝑔 𝑛, 𝑚 𝑒𝑥𝑖𝑡 𝑜𝑟 𝑧 < 𝑍 − 1) 𝜓𝑧+1(𝑛, 𝑚) = 0 𝑖𝑓(𝑅𝑎𝑡𝑖𝑛𝑔 𝑛, 𝑚 𝑛𝑜𝑡 𝑒𝑥𝑖𝑠𝑡) (15) Also, the backward pass calculations are as equations (16) to (19) for the output and hidden layers respectively: For the output: ∆𝑜𝑢𝑡 = (𝑅(𝑛,𝑚) − 𝑎𝑍).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' 𝜓𝑍(𝑛, 𝑚).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' 𝑓𝑧+1′(𝑎𝑍) 𝑛 ∈ (0, 𝑁), 𝑚 ∈ (0, 𝑀) (16) ∆𝑊𝑍 = ∆𝑜𝑢𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' 𝑎𝑍.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' 𝛾𝑍 (17) For the hidden layers: ∆𝐻𝑖𝑑𝑑𝑒𝑛𝑧 = 𝑓′(𝑎𝑧).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' ∑ ∆𝑜𝑢𝑡𝑖 𝑖 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' 𝑤𝑖 𝑧 𝑖 ∈ (1, 𝐾), 𝑧 ∈ (0, 𝑍 − 1) (18) ∆𝑤𝑧 = ∆𝐻𝑖𝑑𝑑𝑒𝑛𝑧.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' 𝑎𝑧.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' 𝛾𝑧 𝑧 ∈ (0, 𝑍 − 1) (19) In these equations, 𝑓𝑧+1′ is the gradient of the activation function, 𝑅(𝑛,𝑚) is the rating corresponding to the users or drugs, ∆𝑊𝑍 is the error correction for the output layer, ∆𝑤𝑧 are the error corrections for the hidden layers, and 𝛾𝑧 is the learning rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' The weight updates are also according to equation (20): 𝑤𝑛𝑒𝑤 𝑧 = 𝑤𝑜𝑙𝑑 𝑧 + ∆𝑤𝑧 𝑧 ∈ (0, 𝑍) (20) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='5 Knowledge-based component After modeling the recommendation system, several constraints on the model output are applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' The final stage in the recommendation process is based on the knowledge-based technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' The knowledge-based recommendation is a specific recommender system that can be used in combination with other algorithms or alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' The aim of using this module is its huge impact in increasing the safety of the recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' We extracted and gathered rules in the drug recommendation domain as queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' These rules are based on Drug Interactions and Adverse Events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Using these rules, we can prevent recommending drugs that lead to events like death, hospitalization, disability, and life-threatening events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' The flowchart of this component has been extracted from Figure 2 and redrawn in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' The set of these rules which our knowledge-based component considers falls into these two categories: - Based on patients’ features: o Gender is allowed to recommend a drug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' o The age is allowed for recommending a drug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' - Based on drug interactions: o The recommended drug has no interaction with other drugs taken by the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Figure 5- Knowledge-based component of our proposed approach based on the bottom left of Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Table 10 presents knowledge-based rules based on patient’s features that have been considered in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=" For example, according to this table, a drug can only be recommended if the patient's age is in the allowed range and the gender is allowed for recommending the drug." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' TABLE 10- USER FEATURE-BASED RULES Zometa Actemra Abilify Drug Name 31 … 29 24 Minimum Not allowed age ranges 67 … 64 45 Maximum 0 … 1 0 None Allowed gender 0 … 0 0 Female 0 … 0 0 Male 1 … 0 1 Both Start New User Registration User Weights & Biases Sets Drug Weights & Biases Sets Compute 10 High Rated Drug Recommendation Drug[i] Interaction?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' I=0 I<10 I=i+1 Drug Interation Set End Recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='append(Drug[i] Knowledge based Recommendations based on the model User Features Filter Set Drug[i] Allowed?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Drug Recommendation For our proposed knowledge-based component, another adverse events dataset is generated from Druglib.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' The structure of this dataset is presented in Table 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Features in this dataset include age, gender, the name of the drug taken by a given patient, its adverse event, reaction, and other drugs used by the patient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' TABLE11-ADVERSE EVENTS DATASET Other Drug Adverse Event Reaction Genus Age Drug Name Index - Death male 63 Ability (Airipiprazole) 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' … … … … .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' … Insulin Death;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Hospitalization female 47 Acterma 12 … … .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' … … … … Fluticasone propiate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Salmeterol;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=" Carbemazepine Hospitalization male 50 Zyprexa 2486 We used Gaussian and Poisson distribution for patients' age and gender from the above dataset for the adverse events of using a specific drug." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' These adverse events can be death, hospitalization, disability, or other life-threatening events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Since, in this case, we require the average and standard deviation, by using Poisson and Gaussian distribution, it is possible to compute the allowed gender for recommending a drug to a patient using much less memory than machine learning for this specific task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Assume that on average,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' by recommending a drug 𝛾 for 𝜂 times to patients with 𝑔𝑒𝑛𝑑𝑒𝑟 = 𝑓𝑒𝑚𝑎𝑙𝑒 they experience one of the adverse events mentioned in Table 11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' then the probability that by recommending drug 𝛾 to a female patient she experiences one of the adverse events is calculated as equation (21): 𝑃𝑛(𝑥) = 𝑒−𝜆𝐹𝑒𝑚𝑎𝑙𝑒 𝛶 𝜆𝐹𝑒𝑚𝑎𝑙𝑒 𝛶 𝜆𝐹𝑒𝑚𝑎𝑙𝑒 𝛶 = 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝐴𝑑𝑣𝑒𝑟𝑠𝑒 𝐸𝑣𝑒𝑛𝑡 𝜂 (21) And similarly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' for a male patient,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' this probability is calculated as equation (22): 𝑃𝑛(𝑥) = 𝑒−𝜆𝑀𝑎𝑙𝑒 𝛶 𝜆𝑀𝑎𝑙𝑒 𝛶 𝜆𝑀𝑎𝑙𝑒 𝛶 = 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝐴𝑑𝑣𝑒𝑟𝑠𝑒 𝐸𝑣𝑒𝑛𝑡 𝜂 (22) Using the above calculations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' if the probability of an adverse event for each gender and each medicine is more than a given threshold value,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' the medicine is removed from the list and is not recommended to the patient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' In this paper, we set the 𝑡ℎ𝑟𝑒𝑠ℎ𝑜𝑙𝑑 = 50%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Also, normal distribution was used for setting the rules related to the patients ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Suppose the average and standard deviation of a patient’s age who have taken medicine 𝛾 and has an adverse event is represented by 𝜇 and 𝜎, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' In that case, the normal distribution function related to age is as equation (23): 𝑓(𝑥) = 1 √2𝜋σ𝛶 𝑒 −1 2(𝑥−μ𝛶 σ𝛶 ) 2 (23) and so for patients who are taking medicine 𝛾, equation (17) for age range has to be met to minimize the adverse event (24): 𝑋𝜖 (𝜇𝛶 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='96 ( 𝜎𝛶 √𝑛) , 𝜇𝛶 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='96 ( 𝜎𝛶 √𝑛)) , 1 − 𝑎 = 95%, 𝑍0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='975 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='96 (24) In this research, we used the rules related to the users’ features and the medicine rules and drug interactions we are also considering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' In this regard, the drug interactions dataset was used to exclude recommendations for drugs having high interactions with other drugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' 6- RESULTS AND DISCUSSION This section discusses our proposed drug recommendation system implementation and the newly generated datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' First, we explain the extracted and newly generated datasets and then we will demonstrate the results of our implemented system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' The dataset As discussed in the proposed method, we used the information from two databases of drugs Druglib.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='com [7] and Drugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='com [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' The first database Druglib.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='com is a comprehensive resource for drug information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' For each drug, a variety of information such as description, side-effects, drug ratings & reviews by patients, and clinical pharmacology has been provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Also, Drugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='com is another database for drug information, and many recommendation systems have been suggested that use this database to build their models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Both the original and the revised version of Drugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='com have been used in RS to evaluate the performance of the approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' We crawled these pharmaceutical websites to construct our intended datasets with the required features in a structured way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' As a result, we gathered much useful information about drugs and patients’ conditions and collected them into three datasets as follows: - The first extracted dataset is the Rating dataset consists of patients’ features and their ratings on drugs consisting of 3294 samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' - The second dataset consists of Drug features containing drug categories, side effects, and benefits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' - The last dataset is the Interaction dataset containing interactions between drugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' To evaluate the performance of our system, we used the most popular existing machine learning evaluation metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Accuracy, sensitivity (recall), specificity, and precision were the basic metrics that we applied to our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' We used 70 percent of the samples (2304 samples) in the dataset for training our model, 20 percent (660 samples) for evaluation, and 10 percent (330 samples) for the test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' After obtaining the values for true positive (TP), false positive (FP), true negative (TN), and false negative (FN), different metrics can be calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' We compared our results with the existing approaches in [27], [48], [57], [58], and [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' We implemented the algorithms in these papers with the datasets they have applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' In [27], SVM and recurrent neural network (RNN) ve been used to recommend a drug to a patient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' In [48] the authors first considered the clustering of drugs according to the drug information, like the algorithm proposed in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Then collaborative filtering is used to recommend a drug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' But unlike our work, they haven’t considered the classification of users and their features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Finally, in [57], an improved matrix factorization has been used, filters the results using NSGA-III to improve the accuracy, diversity, novelty, and recall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Table 12 represents the comparison results between our work and other drug recommendation systems in terms of important machine learning metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' TABLE 12: COMPARISON RESULT OF OUR PROPOSED RECOMMENDATION SYSTEM WITH OTHER STATE-OF-THE-ART APPROACHES F1-Measure Precision Specificity Sensitivity Accuracy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='34 SVM[26] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='86 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='31 Neural Network [26] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='61 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='55 Kmeans User CF [48] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='63 NSGA III [57] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='49 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='48 Conventional MF [58] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='45 MLP [59] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='69 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='65 Proposed Method Comparison results consist of the F2 measure, ROC, and confusion matrix of different approaches depicted in Figures 6 to 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Figure 6- Comparison result of F2 measure metric 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='6 F2 Measure SVM [26] Backpropagation [26] Kmeans CF [48] NSGA II [57] ConvMF [58] MLP [59] Proposed Method Figure 7 comparison result of ROC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Figure 8- comparison result of the confusion matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' ROC 10 True Positive Rate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='4 ProposedMethod Backpropagation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='2 KmeansUserCF Conventional MF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='0 RandomClassifier 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='8 10 FalsePositiveRateProposed Method 110 100 S proper 108 47 ActualLabels 90 80 70 not proper 113 60 50 PredictedLabelsConventional MF 100 I Labels proper 53 90 Actual 80 not proper 105 108 70 60 Predicted LabelsMLP 120 110 Labels proper 73 47 100 06 Actual 80 not_proper 128 82 70 60 F 50 PredictedLabelsSVM 200 175 Labels proper 10 5 150 125 Actual 100 75 not_proper 208 107 50 25 PredictedLabels Figure 9 -Confusion matrix obtained for the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' The construction of a confusion matrix for different ratings is also shown in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' The predictions are compared with actual ratings of users, and drugs for the case when they are considered separately and combined according to our proposed approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' One of the important components of our recommender system is the final knowledge-based approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' This component prevents death, hospitalization, and disability by considering drug interactions and the user’s age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=" The adverse Events Dataset is used in this regard to our system's performance for recognizing such cases and recommending the appropriate drugs." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' This dataset contains 2486 samples, where 80% of them are used for rule extraction, and the remaining 20% are for the test.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='Actual ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='not ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='proper ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='113 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='PredictedLabelsThe ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='following ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='parameters ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='are ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='considered ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='for ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='evaluation: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='𝐷𝑒𝑎𝑡ℎ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='𝑅𝑎𝑡𝑖𝑜 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='𝑁𝑢𝑚𝑏𝑒𝑟 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='𝑜𝑓 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='𝐷𝑒𝑎𝑡ℎ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='𝑇𝑜𝑡𝑎𝑙 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='𝑁𝑢𝑚𝑏𝑒𝑟 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='𝑜𝑓 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='𝑅𝑒𝑐𝑜𝑚𝑚𝑒𝑛𝑑𝑎𝑡𝑖𝑜𝑛 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='𝐷𝑖𝑠𝑎𝑏𝑖𝑙𝑖𝑡𝑦 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='𝑅𝑎𝑡𝑖𝑜 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='𝑁𝑢𝑚𝑏𝑒𝑟 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='𝑜𝑓 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='𝐷𝑖𝑠𝑎𝑏𝑖𝑙𝑖𝑡𝑦 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='𝑇𝑜𝑡𝑎𝑙 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='𝑁𝑢𝑚𝑏𝑒𝑟 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='𝑜𝑓 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='𝑅𝑒𝑐𝑜𝑚𝑚𝑒𝑛𝑑𝑎𝑡𝑖𝑜𝑛 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='𝐻𝑜𝑠𝑝𝑖𝑡𝑎𝑙𝑖𝑧𝑎𝑡𝑖𝑜𝑛 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='𝑅𝑎𝑡𝑖𝑜 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='𝑁𝑢𝑚𝑏𝑒𝑟 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='𝑜𝑓 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='𝐻𝑜𝑠𝑝𝑖𝑡𝑎𝑙𝑖𝑧𝑎𝑡𝑖𝑜𝑛 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='𝑇𝑜𝑡𝑎𝑙 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='𝑁𝑢𝑚𝑏𝑒𝑟 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='𝑜𝑓 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='𝑅𝑒𝑐𝑜𝑚𝑚𝑒𝑛𝑑𝑎𝑡𝑖𝑜𝑛 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='For ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='comparison,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=" the system's performance for different adverse events was calculated one time without a knowledge-based component and the second time using this component." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Table 13 and Figure 10 represent the results of this comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' TABLE 13- COMPARISON RESULTS OF KNOWLEDGE-BASED COMPONENT Adverse event Without knowledge-based component With knowledge-based component Death rate 44% 6% Hospitalization 15% 2% Disability 4% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='7% Knowledge-based component is an essential part of a drug recommendation system in reducing adverse events and improving the quality of recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Figure 10- Comparison result of adding knowledge-based in the recommendation system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' We also considered one more important metric for recommender systems evaluations: hit rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=" 0 10 20 30 40 50 Death Rate Disability Rate Hospitalization Rate With Knowledge Base Without Knowledge base The data set's testing samples (330) are utilized in hit-rate evaluation." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' The hit-rate in evaluation is calculated by the ratio of the total hits in the top 10 recommended drugs returned for all users and the total testing samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' So if 𝜂 is the number of relevant predicted drugs for all users, and 𝑁 is the total number of testing samples, according to [60], the hit-rate is calculated as equation (25): ℎ𝑖𝑡_𝑟𝑎𝑡𝑒 = 𝜂 𝑁 (25) The result of hit rate evaluation is represented in Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' As it can be seen from this figure, our proposed approach has the ℎ𝑖𝑡 𝑟𝑎𝑡𝑒 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='49, which is better than all other approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Figure 11 Top 10 hit rate recommendation systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' The next evaluation metric is cumulative hit-rate, which represents the number of hits with ratings above a given threshold and ignores the predicted ratings lower than the threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' The result of the cumulative hit-rate with the threshold set to 4 is shown in Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' The cumulative hit-rate is calculated as (26): 𝐶𝑢𝑚𝑢𝑙𝑎𝑡𝑖𝑣𝑒 𝐻𝑖𝑡 − 𝑅𝑎𝑡𝑒 = 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 ℎ𝑖𝑡𝑠 𝑤𝑖𝑡ℎ 𝑟𝑎𝑡𝑖𝑛𝑔 𝑎𝑏𝑜𝑣𝑒 𝑡ℎ𝑟𝑒𝑠ℎ𝑜𝑙𝑑 𝑁 (26) The utilization of this threshold makes a better match with the user’s interest in the recommended drug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='6 Top-10 Hit Rate SVM [26] NeuralNetwork [26] Kmeans Used CF [48] NSGA III [57] Conventional MF [58] MLP [59] Proposed Method Figure 12 Top 10 cumulative hit rate of recommendation systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Our results are encouraging in the field of drug recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' It has combined the benefits of basic recommender approaches with less computational overhead through a novel modeling approach and using statistical methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' It also classifies drugs and users in terms of their features, leading to high accuracy compared to state-of-the-art algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' However, better results can be achieved by considering the characteristics of diseases and recommending drugs based on disease features in addition to the features of patients and drugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' 7- CONCLUSION In this paper, we proposed a comprehensive drug recommender system that takes advantage of all basic recommender system techniques and applies natural language processing, neural network- based matrix factorization, and, more importantly, employing knowledge-based recommendations to recommend the most accurate drugs to patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Compared with conventional matrix factorization, our proposed method improves the accuracy, sensitivity, and hit rate by 26%, 34%, and 40%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' In comparison with other machine learning approaches, we obtained an accuracy, sensitivity, and hit rate by an average of 31%, 29%, and 28%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Our approach can be used as an adjunct tool torecommend drugs to patients and improve the quality of prescriptions and reduce the errors caused by medical practitioners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='35 Top-10 Cumulative Hit Rate SVM [26] NeuralNetwork [26] Kmeans Used CF [48] NSGA III [57] Conventional MF [68] MLP [59] Proposed Method In the future, we will extend the knowledge and information extraction from drug databases and include all existing patient features in the user features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Also, we are going to consider the features of the disease in the recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' These features can be captured by general practitioners and help improve the proposed drug recommender system performance and make more accurate recommendations by having more relevant features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' In the final output of the recommendation, we also include the dosage and effectiveness of a drug in addition to the list of drugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Also, in our future work, we will extract the information from other drug resources like the SIDER database for drug side effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' At the end, it should be noted that a physician should approve the recommended medicines for safety reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Data availability The datasets generated during the current study are available in the https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='com/DatasetsLibrary/RECOMMED repositories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' In addition, preprocessed datasets and source code of this study are also available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content='com/DatasetsLibrary/RECOMMEDTool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' REFERENCES 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' McNee, Sean Michael.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} +page_content=' Meeting user information needs in recommender systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfcPfW/content/2301.00280v1.pdf'} 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Engineering, Nagoya University +3 RIKEN Center for Advanced Intelligence Project +∗ E-mail: inatsu.yu@nitech.ac.jp +ABSTRACT +In this study, we address the problem of optimizing multi-output black-box functions under uncertain +environments. +We formulate this problem as the estimation of the uncertain Pareto-frontier (PF) of a +multi-output Bayesian surrogate model with two types of variables: design variables and environmental +variables. +We consider this problem within the context of Bayesian optimization (BO) under uncertain +environments, where the design variables are controllable, whereas the environmental variables are assumed +to be random and not controllable. The challenge of this problem is to robustly estimate the PF when the +distribution of the environmental variables is unknown, that is, to estimate the PF when the environmental +variables are generated from the worst possible distribution. +We propose a method for solving the BO +problem by appropriately incorporating the uncertainties of the environmental variables and their probability +distribution. +We demonstrate that the proposed method can find an arbitrarily accurate PF with high +probability in a finite number of iterations. +We also evaluate the performance of the proposed method +through numerical experiments. +1. Introduction +In many industrial applications, we encounter the problem of optimizing the multi-output black-box function +under uncertain environments. For example, in the problem of optimizing growing conditions for crops, we want +to optimize several conditions such as fertilizer levels to maximize crop quality and yield under an uncertain +environment such as weather conditions. +To formulate this problem, let f (1)(x, w) and f (2)(x, w) be a pair of outputs of a black-box function that +we want to simultaneously maximize, where x ∈ X and w ∈ Ω are the design variables (such as fertilizer levels) +and environmental variables (such as weather conditions) defined in domains X and Ω, respectively, where the +former is controllable and the latter is not. To characterize the uncertainty of the environmental variables w, +we assume that it is sampled from an unknown probability distribution, P †. Because we do not know P †, we +consider the case where we know only A, which is a family of candidate distributions for w. +This study aims to identify a distributionally robust Pareto-frontier (DR-PF) in the above setting, which is +formulated as a PF of the following two functions: +F (1)(x) = +inf +p(w)∈A +� +Ω +f (1)(x, w)p(w)dw, +F (2)(x) = +inf +p(w)∈A +� +Ω +f (2)(x, w)p(w)dw. +Figure 1 shows an example of the problem setup. +To identify a DR-PF, it is necessary to predict it and quantify its uncertainty. In this study, under the +assumption that f (1)(x, w) and f (2)(x, w) follow a Gaussian process (GP), we developed a Bayesian optimization +(BO) method to find a lower bound of the DR-PF by considering the uncertainty of environmental variables w +and the uncertainty of the probability distribution for w. Specifically, we propose an acquisition function (AF) +that enables us to sequentially select the controllable design variable x in a sample-efficient manner to obtain +the DR-PF. +To this end, various technical challenges need to be resolved. One difficulty is that, even when f (1)(x, w) +and f (2)(x, w) are GPs, F (1)(x) and F (2)(x) are not GPs anymore. Therefore, we derive a non-trivial credible +intervals of F (1)(x) and F (2)(x) considering that they are defined as the infima of integrated GPs. Furthermore, +although a naive formulation of multi-objective BOs is computationally expensive, the proposed AF has the +advantage that it can be evaluated efficiently. We also conducted a theoretical analysis of the proposed BO +method to prove that the proposed BO method can find an arbitrarily accurate DR-PF with a high probability +in a finite number of iterations under mild conditions. +1.1. Related Work +For black-box function optimization problems, BOs have been popularly used [Settles, 2009, Shahriari et al., 2016] +in which GP [Williams and Rasmussen, 2006] is often employed as a surrogate model. An optimization problem +1 +arXiv:2301.11588v1 [stat.ML] 27 Jan 2023 + +Multi-objective function +� �, � = � � �, � , � � �, � +(a) +Objective function 1 +x +w +-4 +-2 +0 +2 +4 +-4 +-2 +0 +2 +4 +0 +5 +10 +15 +20 +25 +30 +35 +40 +Objective function 2 +x +w +-4 +-2 +0 +2 +4 +-4 +-2 +0 +2 +4 +0 +10 +20 +30 +40 +50 +60 +70 +80 +(b) +Candidate distributions of +-4 +-2 +0 +2 +4 +0.0 +0.1 +0.2 +0.3 +0.4 +Candidate distribution of w +w +Density +-4 +-2 +0 +2 +4 +0.0 +0.1 +0.2 +0.3 +0.4 +Candidate distribution of w +w +Density +-4 +-2 +0 +2 +4 +0.0 +0.1 +0.2 +0.3 +0.4 +Candidate distribution of w +w +Density +・ +・ +・ +・ +・ +・ +・・・ +・ +・ +・・・ +・ +・ +・・・ +・ +(c) +Expected value of � � (�, �) +Expected value of � � (�, �) +Candidate Pareto-frontiers +Distributionally robust +Pareto-frontier +0 +1 +2 +3 +4 +0 +1 +2 +3 +4 +5 +6 +0 +1 +2 +3 +4 +0 +1 +2 +3 +4 +5 +6 +0 +1 +2 +3 +4 +0 +1 +2 +3 +4 +5 +6 +0 +1 +2 +3 +4 +0 +1 +2 +3 +4 +5 +6 +0 +1 +2 +3 +4 +0 +1 +2 +3 +4 +5 +6 +0 +1 +2 +3 +4 +0 +1 +2 +3 +4 +5 +6 +0 +1 +2 +3 +4 +0 +1 +2 +3 +4 +5 +6 +0 +1 +2 +3 +4 +0 +1 +2 +3 +4 +5 +6 +0 +1 +2 +3 +4 +0 +1 +2 +3 +4 +5 +6 +0 +1 +2 +3 +4 +0 +1 +2 +3 +4 +5 +6 +0 +1 +2 +3 +4 +0 +1 +2 +3 +4 +5 +6 +0 +1 +2 +3 +4 +0 +1 +2 +3 +4 +5 +6 +0 +1 +2 +3 +4 +0 +1 +2 +3 +4 +5 +6 +0 +1 +2 +3 +4 +0 +1 +2 +3 +4 +5 +6 +0 +1 +2 +3 +4 +0 +1 +2 +3 +4 +5 +6 +0 +1 +2 +3 +4 +0 +1 +2 +3 +4 +5 +6 +0 +1 +2 +3 +4 +0 +1 +2 +3 +4 +5 +6 +0 +1 +2 +3 +4 +0 +1 +2 +3 +4 +5 +6 +0 +1 +2 +3 +4 +0 +1 +2 +3 +4 +5 +6 +0 +1 +2 +3 +4 +0 +1 +2 +3 +4 +5 +6 +0 +1 +2 +3 +4 +0 +1 +2 +3 +4 +5 +6 +0 +1 +2 +3 +4 +0 +1 +2 +3 +4 +5 +6 +0 +1 +2 +3 +4 +0 +1 +2 +3 +4 +5 +6 +0 +1 +2 +3 +4 +0 +1 +2 +3 +4 +5 +6 +0 +1 +2 +3 +4 +0 +1 +2 +3 +4 +5 +6 +0 +1 +2 +3 +4 +0 +1 +2 +3 +4 +5 +6 +0 +1 +2 +3 +4 +0 +1 +2 +3 +4 +5 +6 +0 +1 +2 +3 +4 +0 +1 +2 +3 +4 +5 +6 +0 +1 +2 +3 +4 +0 +1 +2 +3 +4 +5 +6 +0 +1 +2 +3 +4 +0 +1 +2 +3 +4 +5 +6 +0 +1 +2 +3 +4 +0 +1 +2 +3 +4 +5 +6 +0 +1 +2 +3 +4 +0 +1 +2 +3 +4 +5 +6 +0 +1 +2 +3 +4 +0 +1 +2 +3 +4 +5 +6 +0 +1 +2 +3 +4 +0 +1 +2 +3 +4 +5 +6 +0 +1 +2 +3 +4 +0 +1 +2 +3 +4 +5 +6 +0 +1 +2 +3 +4 +0 +1 +2 +3 +4 +5 +6 +0 +1 +2 +3 +4 +0 +1 +2 +3 +4 +5 +6 +0 +1 +2 +3 +4 +0 +1 +2 +3 +4 +5 +6 +0 +1 +2 +3 +4 +0 +1 +2 +3 +4 +5 +6 +0 +1 +2 +3 +4 +0 +1 +2 +3 +4 +5 +6 +0 +1 +2 +3 +4 +0 +1 +2 +3 +4 +5 +6 +0 +1 +2 +3 +4 +0 +1 +2 +3 +4 +5 +6 +0 +1 +2 +3 +4 +0 +1 +2 +3 +4 +5 +6 +0 +1 +2 +3 +4 +0 +1 +2 +3 +4 +5 +6 +0 +1 +2 +3 +4 +0 +1 +2 +3 +4 +5 +6 +0 +1 +2 +3 +4 +0 +1 +2 +3 +4 +5 +6 +0 +1 +2 +3 +4 +0 +1 +2 +3 +4 +5 +6 +0 +1 +2 +3 +4 +0 +1 +2 +3 +4 +5 +6 +0 +1 +2 +3 +4 +0 +1 +2 +3 +4 +5 +6 +0 +1 +2 +3 +4 +0 +1 +2 +3 +4 +5 +6 +0 +1 +2 +3 +4 +0 +1 +2 +3 +4 +5 +6 +Pareto-frontier +Figure +1: Conceptional diagram of a DR-PF. The upper and lower color maps in (a) represent objective +functions f (1)(x, w) and f (2)(x, w), respectively, where x and w are the scalar design and environmental variable, +respectively. The plots within the frame in (b) represent multiple candidate distributions for w. A dashed line +in (c) is the expected PF for each candidate distributions of w. A red solid line is the DR-PF, which is defined +by the worst-case expectation of the candidate distributions, provides a lower bound of uncertain PF. The +objective of this study is to efficiently identify the DR-PF. +of multiple black-box functions is typically formulated as a Pareto optimization problem, and BO methods for +such problems have been also studied [Zuluaga et al., 2016, Suzuki et al., 2020]. +Many studies have been conducted on BOs under uncertain environments. For example, [Bogunovic et al., 2018] +proposed a BO method to maximize the worst-case function value with respect to a shift in the input. In several +studies [Beland and Nair, 2017, Toscano-Palmerin and Frazier, 2018, Oliveira et al., 2019, Fr¨ohlich et al., 2020, +Gessner et al., 2020], a BO problem to maximize the expected value of a black-box function with respect to the +input distribution was considered. Furthermore, several studies considered the simultaneous optimization of +multiple black-box functions under the assumption that the distribution of environmental variables is known. +For example, [Iwazaki et al., 2021] dealt with constrained optimization and Pareto optimization problems for the +mean and variance of a black-box function with respect to environmental variables, [Qing et al., 2022] considered +Pareto optimization problems for the expected values of multiple objective functions, and [Amri et al., 2021] +dealt with chance-constrained optimization problems that is an extension of constrained optimization problems +to the input uncertainty setting. +Distributionally robust optimization (DRO) was first introduced by [Scarf, 1958]. Because DRO is an im- +portant topic in the context of robust optimization and has been the subject of numerous studies, we refer an +exhaustive survey of DRO to [Rahimian and Mehrotra, 2019]. In recent years, several studies on DRO in the +context of BOs (DRO-BOs) have been conducted. [Kirschner et al., 2020] and [Nguyen et al., 2020] proposed +BO methods to efficiently find the design variable that maximizes F (1)(x). DRO-BOs have also been stud- +ied under multiple black-box functions. [Inatsu et al., 2022] proposed a BO method for distributionally robust +chance-constrained problems, which is an extension of the chance-constrained problem to DRO. However, to the +best of our knowledge, there are no prior studies on BOs for Pareto optimization under the DRO framework. +1.2. Contributions +The contributions of this study are as follows: +• We develop a BO method for identifying DR-PFs called DR-PF BO method. Specifically, we propose a +novel AF for the DR-PF BO, which is computationally inexpensive and has theoretical guarantees. +• Under mild conditions, we prove that the DR-PF BO method can find an arbitrarily accurate PF with a +high probability in a finite number of iterations. +• Additionally, we prove that with a specification of an appropriate family of candidate distributions, even +if the true distribution is unknown, the DR-PF BO method can find an arbitrarily accurate expected PF +on the true distribution. +2 + +• We confirm the performance of the proposed method through numerical experiments with benchmark +functions and simulator-based functions. +2. Preliminaries +Let f (1) : X × Ω → R and f (2) : X × Ω → R be the expensive-to-evaluate black-box functions1, where X +and Ω are the finite sets. For each input (x, w) ∈ X × Ω, the values of f (1)(x, w) and f (2)(x, w) are observed +with observation noise as y(1) = f (1)(x, w) + ε(1) and y(2) = f (2)(x, w) + ε(2), where ε(1) and ε(2) are random +samples from independent normal distributions with the mean zero and variances σ(1)2 +noise and σ(2)2 +noise, respectively. +In this study, the environmental variable w ∈ Ω was assumed to be a discrete random variable that follows an +unknown probability distribution P †. The two distinct phases called the development phase and use phase exist +in the literature of BOs under uncertainty. In the development phase, environmental variables are completely +controllable as design variables, whereas they are stochastic and uncontrollable in the use phase. In this study, +we consider the development phase, and the use phase is described in Appendix A. Furthermore, we denote the +family of candidate distributions of w as A and consider the following class of distributions: +A = {p(w) | d(p(w), p∗(w)) ≤ ξ}, +where p∗(w) is a user-specified reference distribution, d(·, ·) is a given distance metric function between dis- +tributions, and ξ ≥ 0. This means that we consider a set of candidate distributions whose distance from the +reference distribution is not larger than a certain threshold. Subsequently, the distributionally robust expecta- +tions F (1)(x) and F (2)(x) for each design variable x ∈ X are defined as follows: +F (1)(x) = +inf +p(w)∈A +� +w∈Ω +f (1)(x, w)p(w), +F (2)(x) = +inf +p(w)∈A +� +w∈Ω +f (2)(x, w)p(w). +The objective of this study is to efficiently identify the PF determined by F (1)(x) and F (2)(x). Let F (x) = +(F (1)(x), F (2)(x)) for each x ∈ X, and let F (E) = {F (x) | x ∈ E} for a set E ⊂ X. Furthermore, for a set +B ⊂ R2, we denote the domain dominated by B and the PF of B, respectively, by +Dom(B) = {y ∈ R2 |∃ y′ ∈ B s.t. y ⪯ y′}, +Par(B) = ∂(Dom(B)). +Here, for a point a = (a1, . . . , am) and b = (b1, . . . , bm), a ⪯ b implies ai ≤ bi for any i ∈ {1, . . . , m}. For a set +C, ∂(C) denotes the boundary of C. The PF determined by F (1)(x) and F (2)(x) can then be written as +Z∗ = Par(F (X)). +2.1. Gaussian Process +In this study, we use GP surrogate models for black-box functions f (1) and f (2). First, we assume that +f (1) and f (2) follow GP priors GP(0, k(1)((x, w), (x′, w′))) and GP(0, k(2)((x, w), (x′, w′))), respectively, where +k(1)((x, w), (x′, w′)) and k(2)((x, w), (x′, w′)) are the positive-definite kernels. For l ∈ {1, 2}, given a dataset +{(xi, wi, y(l) +i )}t +i=1, where t is the number of queried instances, the posterior distribution of f (l) is a GP, and its +posterior mean µ(l) +t (x, w) and posterior variance σ(l)2 +t +(x, w) are given by +µ(l) +t (x, w) = k(l) +t (x, w)⊤(K(l) +t ++ σ(l)2 +noiseIt)−1y(l) +t , +σ(l)2 +t +(x, w) = k(l)((x, w), (x, w)) − k(l) +t (x, w)⊤(K(l) +t ++ σ(l)2 +noiseIt)−1k(l) +t (x, w), +where k(l) +t (x, w) is the t-dimensional vector, whose jth element is k(l)((x, w), (xj, wj)), y(l) +t += (y(l) +1 , . . . , y(l) +t )⊤, +It is the t × t identity matrix, K(l) +t +is the t × t matrix whose (j, k) element is k(l)((xj, wj), (xk, wk)), with a +superscript ⊤ indicating the transpose of vectors or matrices. +3. Proposed Method +Here, we propose a BO method to efficiently identify Z∗. Because the functions f (1)(x, w) and f (2)(x, w) +are random variables following GPs, F (1)(x) and F (2)(x) are also random variables. Therefore, a reasonable +approach is to construct credible intervals for F (1)(x) and F (2)(x), and use them to estimate the PF. However, +1Note that the method proposed in this study can be straightforwardly extended to the case where there are more than three +objective functions f(1), f(2), f(3), . . .. +3 + +unlike f (1)(x, w) and f (2)(x, w), F (1)(x) and F (2)(x) do not follow GPs. Therefore, we cannot directly obtain +credible intervals using the properties of GPs. In Section 3.1, we construct credible intervals for F (1)(x) and +F (2)(x) using the method proposed by [Kirschner et al., 2020] and provide a method for estimating the PF +based on the constructed credible intervals. +3.1. Credible Intervals and PF Estimation +For each input (x, w) ∈ X × Ω and time t, the credible interval of f (1)(x, w) is denoted by Q(f (1)) +t +(x, w) = +[l(f (1)) +t +(x, w), u(f (1)) +t +(x, w)]. Here, the lower value l(f (1)) +t +(x, w) and the upper value u(f (1)) +t +(x, w) are given as +l(f (1)) +t +(x, w) = µ(1) +t (x, w) − β1/2 +1,t σ(1) +t +(x, w), +u(f (1)) +t +(x, w) = µ(1) +t (x, w) + β1/2 +1,t σ(1) +t +(x, w), +where β1/2 +1,t +≥ 0 is a user-specified tradeoff parameter. +We then define the credible interval Q(F (1)) +t +(x) ≡ +[l(F (1)) +t +(x), u(F (1)) +t +(x)] of F (1)(x). Here, the lower and upper values are respectively given by +l(F (1)) +t +(x) = +inf +p(w)∈A +� +w∈Ω +l(f (1)) +t +(x, w)p(w), +u(F (1)) +t +(x) = +inf +p(w)∈A +� +w∈Ω +u(f (1)) +t +(x, w)p(w). +(3.1) +Notably, if the L1- (or L2-) norm is used as the distance d(·, ·) between the distributions, the problem of +obtaining the lower and upper values in (3.1) is reduced to a linear programming problem (or a second-order +cone programming problem). In either case, the existence of a fast solver of these problems enabled us to obtain +Q(F (1)) +t +(x). Similarly, we define credible intervals Q(f (2)) +t +(x, w) = [l(f (2)) +t +(x, w), u(f (2)) +t +(x, w)] for f (2)(x, w) and +Q(F (2)) +t +(x) ≡ [l(F (2)) +t +(x), u(F (2)) +t +(x)] for F (2)(x). Next, for any input x ∈ X and any subset E ⊂ X, we define +LCBt(x), UCBt(x) and LCBt(E) as follows: +LCBt(x) = (l(F (1)) +t +(x), l(F (2)) +t +(x)), +UCBt(x) = (u(F (1)) +t +(x), u(F (2)) +t +(x)), +LCBt(E) = {LCBt(x) | x ∈ E}. +The estimated PF solution set ˆΠt ⊂ X for the design variables is then defined as follows: +ˆΠt = {x ∈ X | LCBt(x) ∈ Par(LCBt(X))}. +Figure 2 (a) shows a conceptual diagram of LCBt(x) and UCBt(x), and (b) shows a conceptual diagram of +Par(LCBt(X)) and ˆΠt. +3.2. Acquisition Function +Here, we propose an AF for determining the next evaluation point. First, for each point a ∈ Rm and subset +B ⊂ Rm, we denote the closeness between them as +dist(a, B) = inf +b∈B d∞(a, b), +where d∞(a, b) denotes the metric function given by d∞(a, b) = max{|a1 − b1|, . . . , |am − bm|}. Using this, we +define AF at(x) for x ∈ X as +at(x) = dist(UCBt(x), Dom(LCBt(ˆΠt))). +We then select the following evaluation points, as described in the following definition: +Definition 3.1. The next design variable, xt+1, to be evaluated is selected as follows: +xt+1 = argmax +x∈X +at(x), +and the next environmental variable, wt+1, to be evaluated is selected as +wt+1 = argmax +w∈Ω +{σ(1)2 +t +(xt+1, w) + σ(2)2 +t +(xt+1, w)}. +4 + +𝐹(2)(𝒙) +𝐹(1)(𝒙) +𝐹(2)(𝒙) +𝐹(1)(𝒙) +𝐹(2)(𝒙) +𝐹(1)(𝒙) +(a) +(b) +(c) +𝒙1 +𝒙2 +𝒙3 +𝒙4 +𝒙5 +𝒙6 +𝒙7 +Figure 2: Conceptual diagrams of LCBt(x), UCBt(x), Par(LCBt(X)), ˆΠt and AFs for seven input points +x1, . . . , x7. At each point x in the left figure (a), LCBt(x) and UCBt(x) indicate the lower left point and the +upper right point of the dashed rectangular region, respectively. In (b), the PF (red line) computed using each +LCBt(x) is Par(LCBt(X)), and because it is constructed by LCBt(x1), LCBt(x2), LCBt(x3), LCBt(x7), ˆΠt +is given by ˆΠt = {x1, x2, x3, x7}. In (c), the light red region indicates Dom(LCBt(ˆΠt)), the region dominated +by the red points (LCBt(ˆΠt)), and at(x) is the closeness between the light red region and UCBt(x) (purple +point). The furthest point is represented by the purple triangle, UCBt(x4). Thus, the next design variable to +be evaluated is x4. +Figure 2 (c) shows a conceptual diagram of the AF at(x). Here, AF at(x) can be computed analytically +using the following lemma: +Lemma 3.1. Let UCBt(x) = (u1, u2) and LCBt(ˆΠt) = {(l(i) +1 , l(i) +2 ) | 1 ≤ i ≤ k}. Then, at(x) can be computed +as follows: +˜at(x) = min +1≤i≤k max{u1 − l(i) +1 , u2 − l(i) +2 }, +at(x) = max{˜at(x), 0}. +Notably, when the number of objective functions is m ≥ 3, at(x) is easily extended as follows: +˜at(x) = min +1≤i≤k′ max{u1 − l(i) +1 , . . . , um − l(i) +m }, +at(x) = max{˜at(x), 0}, +(3.2) +where UCBt(x) = (u1, . . . , um) and LCBt(ˆΠt) = {(l(i) +1 , . . . , l(i) +m ) | 1 ≤ i ≤ k′}. The proofs of Lemma 3.1 and +(3.2) are presented in Appendix B. From Lemma 3.1, once LCBt(ˆΠt) is computed, the maximum value of at(x) +can be analytically obtained by performing 2|X| times inf calculations and computing u(F (1)) +t +(x) and u(F (2)) +t +(x) +for all x ∈ X. On the other hand, an AF based on exact posterior distributions of target functions such as the +expected hypervolume improvement [Emmerich, 2005] for ordinary Pareto optimization, requires approximation +by sampling from f (1)(x, w) and f (2)(x, w) under this problem setting. However, in each posterior sample, the +inf calculation must be performed again to calculate F (1)(x) and F (2)(x) for all design variables. Therefore, if +the number of Monte Carlo samples is M, M times more inf calculations are required compared to the proposed +AF. The comparison of computational times is given in Section 5. +3.3. Stopping Condition +Here, we describe the stopping conditions of the proposed algorithm. From Fig. 2 (c), AF at(x) represents +the closeness of the pessimistic PF and the optimistic predictive value of F (x). +That is, if this value is +sufficiently small, there is little room for improvement in the PF; therefore, it is reasonable to use it as the +stopping condition. Let ϵ > 0 be a user-specified parameter. Then the algorithm is terminated if at(x) ≤ ϵ is +satisfied. The pseudocode for the proposed algorithm is given in Algorithm 1. +4. Theoretical Analysis +Here, we provide the theorems for the accuracy and convergence of the proposed algorithm. The details of +the proofs are presented in Appendix B. First, to provide theoretical results, we assume that f (1) and f (2) follow +5 + +Algorithm 1 BO for identifying DR-PF +Input: GP priors GP(0, k(1)), GP(0, k(2)), parameter ξ ≥ 0, tradeoff parameters {β1,t}t≥0, {β2,t}t≥0, stopping +parameter ϵ > 0 +t ← 1 +while at(x) > ϵ do +Compute Q(F (1)) +t +(x) and Q(F (2)) +t +(x) for each x ∈ X +Select the next evaluation point (xt, wt) +Observe y(1) +t += f (1)(xt, wt) + ε(1) +t +and y(2) +t += f (2)(xt, wt) + ε(2) +t +at the point (xt, wt) +Update GPs by adding observed points +t ← t + 1 +end while +Output: Return ˆΠt as the estimated set of design variables comprising the DR-PF +GPs GP(0, k(1)((x, w), (x′, w′))) and GP(0, k(2)((x, w), (x′, w′))), respectively. Moreover, we assume that the +prior variances k(1)((x, w), (x, w)) ≡ σ(1)2 +0 +(x, w) and k(2)((x, w), (x, w)) ≡ σ(2)2 +0 +(x, w) satisfy +max +(x,w)∈X×Ω σ(1)2 +0 +(x, w) ≤ 1, +max +(x,w)∈X×Ω σ(2)2 +0 +(x, w) ≤ 1. +Furthermore, let κ(1) +T +and κ(2) +T +be the maximum information gains of f (1) and f (2) at time T, respectively. +Notably, the maximum information gain is a measure often used in theoretical analyses of GP-based BO (see, +e.g., [Srinivas et al., 2010]). Here, for each j ∈ {1, 2}, using the mutual information I(y(j); f (j)) between y(j) +and f (j), κ(j) +T +can be expressed as +κ(j) +T += +max +A⊂X×Ω I(y(j) +A ; f (j)). +Next, to quantify the goodness of the predicted ˆΠt, we define an ϵ-accurate Pareto region Zϵ. With user-specified +positive numbers ϵ and ϵ = (ϵ, ϵ), we define Zϵ as +Zϵ = {y ∈ R2 |∃ y′ ∈ Z∗ s.t. y ⪯ y′ and ∃y′′ ∈ Z∗ s.t. y′′ ⪯ y + ϵ}. +That is, Zϵ is the set of points that lie inside Z∗ and within ϵ in the sense of d∞(·, ·)-distance. The concept of +Zϵ was also used in [Zuluaga et al., 2016]. Using Zϵ, we define the accuracy of ˆΠt in terms of the following two +aspects: +Definition 4.1 (Accuracy for ˆΠt). Let ϵ be a positive value. We then define ˆΠt as an ϵ-accurate estimated +solution set if the following holds: +F (ˆΠt) ⊂ Zϵ. +(4.1) +Moreover, we define ˆΠt as an ϵ-accurate estimated Pareto solution set if the following holds: +Par(F (ˆΠt)) ⊂ Zϵ. +(4.2) +It is easy to obtain a set that satisfies either (4.1) or (4.2). Generally, by ignoring F (2)(x) and focusing +only on the maximization of F (1)(x), the maximization point x∗ can be estimated using methods such as +[Kirschner et al., 2020]. Subsequently, by letting ˆΠt = {x∗}, (4.1) is satisfied with high probability. Similarly, +if we predict that all points constitute the PF, that is, ˆΠt = X, then (4.2) is satisfied because Par(F (ˆΠt)) = +Par(F (X)) = Z∗. The following theorem guarantees that the proposed algorithm satisfies both (4.1) and (4.2) +with a high probability. +Theorem 4.1. Let t ≥ 1 and δ ∈ (0, 1) and define β1,t = β2,t = 2 log(2|X × Ω|π2t2/(6δ)) ≡ βt. In addition, let +ϵ > 0 be a user-specified stopping parameter. Then, when Algorithm 1 terminates, ˆΠt satisfies (4.1) and (4.2) +with a probability of at least 1 − δ. +Theorem 4.1 does not indicate whether the algorithm terminates or not. The following theorem guarantees +the convergence of Algorithm 1. +Theorem 4.2. Under the same conditions as those in Theorem 4.1, let T be the smallest positive integer that +satisfies the following inequality: +βT (C1κ(1) +T ++ C2κ(2) +T ) +T +≤ ϵ2 +4 , +where C1 = 2/ log(1+σ(1)−2 +noise ) and C2 = 2/ log(1+σ(2)−2 +noise ). Then, Algorithm 1 terminates after at most T trials. +6 + +0 +100 +200 +300 +400 +500 +0 +5 +10 +15 +20 +25 +30 +35 +Simulator setting +iteration +R1 +0 +100 +200 +300 +400 +500 +0 +5 +10 +15 +20 +25 +30 +35 +0 +100 +200 +300 +400 +500 +0 +5 +10 +15 +20 +25 +30 +35 +0 +100 +200 +300 +400 +500 +0 +5 +10 +15 +20 +25 +30 +35 +0 +100 +200 +300 +400 +500 +0 +5 +10 +15 +20 +25 +30 +35 +0 +100 +200 +300 +400 +500 +0 +5 +10 +15 +20 +25 +30 +35 +Random +UCB_F1 +UCB_F2 +MVA +EHI +Proposed +0 +100 +200 +300 +400 +500 +0 +20 +40 +60 +80 +Simulator setting +iteration +R2 +0 +100 +200 +300 +400 +500 +0 +20 +40 +60 +80 +0 +100 +200 +300 +400 +500 +0 +20 +40 +60 +80 +0 +100 +200 +300 +400 +500 +0 +20 +40 +60 +80 +0 +100 +200 +300 +400 +500 +0 +20 +40 +60 +80 +0 +100 +200 +300 +400 +500 +0 +20 +40 +60 +80 +Random +UCB_F1 +UCB_F2 +MVA +EHI +Proposed +0 +100 +200 +300 +400 +500 +0 +5 +10 +15 +20 +25 +30 +35 +Uncontrollable setting +iteration +R1 +0 +100 +200 +300 +400 +500 +0 +5 +10 +15 +20 +25 +30 +35 +0 +100 +200 +300 +400 +500 +0 +5 +10 +15 +20 +25 +30 +35 +0 +100 +200 +300 +400 +500 +0 +5 +10 +15 +20 +25 +30 +35 +0 +100 +200 +300 +400 +500 +0 +5 +10 +15 +20 +25 +30 +35 +0 +100 +200 +300 +400 +500 +0 +5 +10 +15 +20 +25 +30 +35 +Random +UCB_F1 +UCB_F2 +MVA +EHI +Proposed +0 +100 +200 +300 +400 +500 +0 +20 +40 +60 +80 +Uncontrollable setting +iteration +R2 +0 +100 +200 +300 +400 +500 +0 +20 +40 +60 +80 +0 +100 +200 +300 +400 +500 +0 +20 +40 +60 +80 +0 +100 +200 +300 +400 +500 +0 +20 +40 +60 +80 +0 +100 +200 +300 +400 +500 +0 +20 +40 +60 +80 +0 +100 +200 +300 +400 +500 +0 +20 +40 +60 +80 +Random +UCB_F1 +UCB_F2 +MVA +EHI +Proposed +Figure 3: Average values of R1 and R2 for each method in Simulator and Uncontrollable settings. The length +of each error bar represents twice the standard error. +0 +20 +40 +60 +80 +100 +0 +5 +10 +15 +20 +25 +30 +35 +SIR (case1) +iteration +R1 +0 +20 +40 +60 +80 +100 +0 +5 +10 +15 +20 +25 +30 +35 +0 +20 +40 +60 +80 +100 +0 +5 +10 +15 +20 +25 +30 +35 +0 +20 +40 +60 +80 +100 +0 +5 +10 +15 +20 +25 +30 +35 +0 +20 +40 +60 +80 +100 +0 +5 +10 +15 +20 +25 +30 +35 +0 +20 +40 +60 +80 +100 +0 +5 +10 +15 +20 +25 +30 +35 +Random +UCB_F1 +UCB_F2 +MVA +EHI +Proposed +0 +20 +40 +60 +80 +100 +0 +20 +40 +60 +80 +SIR (case1) +iteration +R2 +0 +20 +40 +60 +80 +100 +0 +20 +40 +60 +80 +0 +20 +40 +60 +80 +100 +0 +20 +40 +60 +80 +0 +20 +40 +60 +80 +100 +0 +20 +40 +60 +80 +0 +20 +40 +60 +80 +100 +0 +20 +40 +60 +80 +0 +20 +40 +60 +80 +100 +0 +20 +40 +60 +80 +Random +UCB_F1 +UCB_F2 +MVA +EHI +Proposed +0 +20 +40 +60 +80 +100 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +SIR (case2) +iteration +R1 +0 +20 +40 +60 +80 +100 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0 +20 +40 +60 +80 +100 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0 +20 +40 +60 +80 +100 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0 +20 +40 +60 +80 +100 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0 +20 +40 +60 +80 +100 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Random +UCB_F1 +UCB_F2 +MVA +EHI +Proposed +0 +20 +40 +60 +80 +100 +0 +10 +20 +30 +40 +SIR (case2) +iteration +R2 +0 +20 +40 +60 +80 +100 +0 +10 +20 +30 +40 +0 +20 +40 +60 +80 +100 +0 +10 +20 +30 +40 +0 +20 +40 +60 +80 +100 +0 +10 +20 +30 +40 +0 +20 +40 +60 +80 +100 +0 +10 +20 +30 +40 +0 +20 +40 +60 +80 +100 +0 +10 +20 +30 +40 +Random +UCB_F1 +UCB_F2 +MVA +EHI +Proposed +Figure 4: Average values of R1 and R2 for each method in Case1 and Case2 in SIR model experiments. The +length of each error bar represents twice the standard error. +Table 1: Computational time (second) and the ratios of the computational time to that of the proposed method +Random +UCB F1 +UCB F2 +MVA +EHI +Proposed +Computational time +0.000 +0.068 +0.067 +0.135 +16.21 +0.139 +(Standard error) +(0.000) +(0.000) +(0.000) +(0.001) +(0.011) +(0.001) +Computational time ratio +0.000 +0.496 +0.488 +0.985 +118.68 +1 +(Standard error) +(0.000) +(0.004) +(0.004) +(0.006) +(0.548) +(0) +Here, because the maximum information gains κ(1) +T +and κ(2) +T +are known to be sublinear with respect to T +under mild assumptions [Srinivas et al., 2010], and the order of β1,T = β2,T is O(log T), the positive integer T +satisfying the inequality in Theorem 4.2 exists. +We emphasize that Theorem 4.1 also holds in the use phase setting and a similar theorem holds for Theorem +4.2 under mild additional conditions. Moreover, by appropriately designing the family of candidate distributions +using the empirical distribution as the reference distribution, the proposed method provides an arbitrarily +accurate solution for the expected PF based on the true distribution, even when the true distribution is unknown. +Details are provided in Appendix A. +5. Numerical Experiments +Here, we confirm the performance of the proposed method using synthetic functions and real-world simulation +models. In this experiment, we used a one-dimensional design variable x and environmental variable w and the +following Gaussian kernels: +k(1)((x, w), (x′, w′)) = σ2 +f,1 exp(−∥ν − ν′∥2 +2/L1), +k(2)((x, w), (x′, w′)) = σ2 +f,2 exp(−∥ν − ν′∥2 +2/L2), +where ν = (x, w) and ν′ = (x′, w′). Moreover, we used the L1-norm as the distance between distributions. In +all experiments except computational time experiments, we used the following two indicators R1 and R2 based +on (4.1) and (4.2) to evaluate the goodness of ˆΠt estimated by each method: +R1 = inf{a ∈ R | F (ˆΠt) ⊂ Za}, +R2 = inf{a ∈ R | Par(F (ˆΠt)) ⊂ Za}. +Experimental details and additional experiments, which are not included in the main body, are described in +Appendix C. +7 + +5.1. Synthetic Function +We evaluate the performance in our proposed method through synthetic functions. First, the input space +X × Ω was a set of 50 × 50 grid points equally spaced in [−10, 10] × [−10, 10]. In this experiment, we used the +following (scaled) Himmelblau’s function f (1)(x, w), which is commonly used as a benchmark function in BO +studies [Andrei, 2008], and sinusoidal function f (2)(x, w) as black-box functions: +f (1)(x, w) = (x2 + w − 11)2 +150 ++ (x + w2 − 7)2 +150 +− C, +f (2)(x, w) = (80 sin(1.5x) − 50 cos(2w))/1.5, +where C = 3321.291/150 is a constant to set the mean to zero. +Under this setting, we compared the following six methods: +Random: Determine the next evaluation point xt+1 at random. +UCB F1: Select the next evaluation point by xt+1 = argmaxx∈X u(F (1)) +t +(x). +UCB F2: Select the next evaluation point by xt+1 = argmaxx∈X u(F (2)) +t +(x). +MVA: Select the next evaluation point by xt+1 = argmaxx∈Mt∪ˆΠt λt(x), where Mt and λt(x) are given in +Section 3.2 of [Iwazaki et al., 2021]. +EHI: Select the next evaluation point xt+1 by maximizing an expected hypervolume improvement for the +DR-PF calculated based on posterior means. +Proposed: Select the next evaluation point xt+1 by Definition 3.1. +Here, UCB F1 (or UCB F2) focuses on the maximization of F (1)(x) (or F (2)(x)) and does not consider the +identification of the DR-PF. In contrast, MVA focuses on reducing the uncertainty of a potential optimal set, +which is a set of input points that may constitute the DR-PF. The EHI method is the strategy that extends the +expected hypervolume improvement strategy, which is commonly used in BO for ordinary Pareto optimization +problems, to the DR-PF identification problem. Because the expected hypervolume improvement for the DR-PF +cannot be calculated analytically, we approximated it using Monte Carlo sampling with a sample size of 100. +Experiments were conducted under the following two settings for the observation of w: +Simulator setting: At each time t, arbitrary w can be selected. +Uncontrollable setting: At each time t, w cannot be selected and is observed as a random sample from the +uniform distribution on Ω. +The Simulator setting and Uncontrollable setting correspond to the development phase and use phase, +respectively. In Simulator setting, we used p∗(w) = 1/50, and the next environmental variable to be evaluate +for each method except Random was selected by +wt+1 = argmax +w∈Ω +(σ(1)2 +t +(xt+1, w) + σ(2)2 +t +(xt+1, w)). +In Random, wt+1 was selected as a random sample from the uniform distribution on Ω. In Uncontrollable +setting, we allowed the use of a different reference distribution p∗ +t (w) for each iteration t and used the empirical +distribution of w as p∗ +t (w). +Under this setup, one initial point was taken at random and the algorithm was run until the number of +iterations reached 500. This simulation was repeated 100 times and the average values of R1 and R2 at each +iteration were calculated. From Fig. 3, it can be confirmed that R1 and R2 in Random are not zero even after +500 iterations for both settings. In UCB F1 and UCB F2, the value of R1 is good but the value of R2 is +not good because they focus on one of the black-box functions. For MVA, EHI, and Proposed, which focus +on improving the DR-PF, R1 and R2 tend to be zero in both settings, but Proposed converges to zero more +quickly. +5.2. Computational Time Experiments +We confirmed the computational time required to select xt+1 and wt+1 using each method. We performed the +same experiment as in Simulator setting in the previous section to evaluate the computational time. Under +this setup, one initial point was taken at random and the algorithm was run until the number of iterations +reached 500. We computed the average computational time over 500 iterations. We also computed the ratio +of the computational time of each method to that of the proposed method. From Table 1, it can be confirmed +that Random, which does not require inf calculations, is faster than the proposed method, and UCB F1 (or +8 + +UCB F2), which uses only u(F (1)) +t +(x) (or u(F (2)) +t +(x)) required inf calculations, is about half the computational +time of the proposed method. +The proposed method and MVA using both u(F (1)) +t +(x) and u(F (2)) +t +(x) have +comparable computational speed. On the other hand, EHI, which performs the same number of inf calculations +for each Monte Carlo sample as the proposed method requires, takes about 100 times longer than the proposed +method because the number of Monte Carlo samples is 100. +5.3. Infection Simulation +We applied the proposed method to the Pareto optimization problem using a simulation model of a real- +world infectious disease. We used the SIR model [Kermack and McKendrick, 1927], which is commonly used +as the infection simulation model. In this experiment, we used the SIR model which has the infection rate +β ∈ [0, 1] and the recovery γ ∈ [0, 1]. The input space X × Ω was defined as the set of grid points when the +region [0.01, 0.5] × [0.01, 0.5] was equally divided into 50 × 50 grid points. Using the SIR model, we defined the +following two risk functions which can be interpreted as economic risks: +r1(β, γ) = n(β, γ) − 450β + 800γ − C1, +r2(β, γ) = n(β, γ) − C2, +where n(β, γ) is the maximum number of infected individuals during a given period, calculated using the SIR +model, and C1 and C2 are constants. Notably, r1(β, γ) and r2(β, γ) were also used in [Inatsu et al., 2022]. In +this experiment, we used the same parameter setting as them. To adapt it to our problem setup, we multiplied +them by minus one because risk functions should be minimized. Because β and γ can be interpreted as both +design variables and environmental variables, we considered the following two cases: +Case1: f (1)(x, w) = −r1(x, w) and f (2)(x, w) = −r2(x, w), where x and w are the infection rate and recovery +rate, respectively. +Case2: f (1)(x, w) = −r1(x, w) and f (2)(x, w) = −r2(x, w), where x and w are the recovery rate and infection +rate, respectively. +In this experiment, we considered Simulator setting as in Section 5.1. +Under this setup, one initial point was taken at random and the algorithm was run until the number of +iterations reached 100. This simulation was repeated 100 times and the average values of R1 and R2 at each +iteration were calculated. From Fig. 4, it can be confirmed that the proposed method achieves equal or better +performance in all settings. +6. Conclusion +In this study, we proposed an efficient BO method for identifying the DR-PF. We proved that the proposed +method has theoretical guarantees on accuracy and convergence. Furthermore, through numerical experiments, +we confirmed that the proposed method outperforms other comparative methods. Future work includes extend- +ing the method to the case where w is a continuous random variable. +Acknowledgement +This work was partially supported by MEXT KAKENHI (20H00601), JST CREST (JPMJCR21D3, JP- +MJCR21D3), JST Moonshot R&D (JPMJMS2033-05), JST AIP Acceleration Research (JPMJCR21U2), NEDO +(JPNP18002, JPNP20006) and RIKEN Center for Advanced Intelligence Project. +References +[Amri et al., 2021] Amri, R. E., Riche, R. L., Helbert, C., Blanchet-Scalliet, C., and Da Veiga, S. (2021). A +sampling criterion for constrained bayesian optimization with uncertainties. arXiv preprint arXiv:2103.05706. +[Andrei, 2008] Andrei, N. (2008). An unconstrained optimization test functions collection. Adv. Model. Optim, +10(1):147–161. +[Beland and Nair, 2017] Beland, J. 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Specif- +ically, we consider the following three settings: +• The case with three or more objective functions. +• Environment variables cannot be controlled even during optimization. +• The setting where reference distributions, control parameter ξ, and candidate distribution family A set +differently at each time. +Furthermore, by combining (ii) and (iii), we show that under appropriate assumptions, the solution to the +distributionally robust Pareto optimization problem is also a good solution to the Pareto optimization problem +defined by the true expectation. +A.1. Extended Problem Setting +Let f (1), . . . , f (m) : X × Ω → R be expensive-to-evaluate black-box functions, where 2 ≤ m. Also let X and +Ω be finite sets. Here, for each (x, w) ∈ X ×Ω and j ∈ {1, . . . , m} ≡ [m], the value of f (j)(x, w) is observed with +Gaussian noise ε(j) as y(j) = f (j)(x, w) + ε(j), where ε(1), . . . , ε(m) are mutually independent, and ε(j) follows +Normal distribution with mean zero and variance σ(j)2 +noise, that is, ε(j) ∼ N(0, σ(j)2 +noise). In this section, we assume +that w ∈ Ω is a discrete random variable and follows some unknown distribution P †. Here, for environmental +variables, we consider either settings in the development phase or settings in the use phase. That is, in the +former, environmental variables are completely controllable as design variables; in the latter, environmental +variables are uncontrollable and observed as realizations from the true distribution. Furthermore, let At denote +the candidate distribution family of P † at each time t, and consider the following At: +At = {p(w) | d(p(w), p∗ +t (w)) ≤ ξt}, +where p∗ +t (w) is a user-specified reference distribution, d(·, ·) is a given distance function between distribu- +tions, and ξt ≥ 0. Then, for each design variable x ∈ X and time t, the distributionally robust expectations +F (1) +t +(x), . . . , F (m) +t +(x) are defined as follows: +F (j) +t +(x) = +inf +p(w)∈At +� +w∈Ω +f (j)(x, w)p(w), j ∈ [m]. +Hereafter, we aim to efficiently identify the PF Z∗ +t determined by F (1) +t +(x), . . . , F (m) +t +(x). For each x ∈ X, subset +E ⊂ X and time t, let Ft(x) = (F (1) +t +(x), . . . , F (m) +t +(x)) and Ft(E) = {Ft(x) | x ∈ E}. Moreover, for a set +B ⊂ Rm, the domain Dom(B) dominated by B and the Pareto-frontier Par(B) of B are given by +Dom(B) = {y ∈ Rm |∃ y′ ∈ B s.t. y ⪯ y′}, +Par(B) = ∂(Dom(B)). +Then, the PF Z∗ +t defined by F (1) +t +(x), . . . , F (m) +t +(x) can be expressed as follows: +Z∗ +t = Par(Ft(X)). +A.1.1. Gaussian Process +Next, we construct predictive models for the black-box functions. +As in the main body, GPs are used to +model the black-box functions f (1), . . . , f (m). First, for each j ∈ [m], assume that f (j) follows a GP prior +GP(0, k(j)((x, w), (x′, w′))), where k(j)((x, w), (x′, w′)) is a positive-definite kernel. +Then, under the given +dataset {(xi, wi, y(j) +i )}t +i=1, the posterior distribution of f (j) is again a GP, and its posterior mean µ(j) +t (x, w) +and posterior variance σ(j)2 +t +(x, w) are given by +µ(j) +t (x, w) = k(j) +t (x, w)⊤(K(j) +t ++ σ(j)2 +noiseIt)−1y(j) +t , +σ(j)2 +t +(x, w) = k(j)((x, w), (x, w)) − k(j) +t (x, w)⊤(K(j) +t ++ σ(j)2 +noiseIt)−1k(j) +t (x, w), +where k(j) +t (x, w) is the t-dimensional vector whose kth element is k(j)((x, w), (xk, wk)), y(j) +t += (y(j) +1 , . . . , y(j) +t )⊤, +K(j) +t +is the t × t matrix whose (k, l)th element is k(j)((xk, wk), (xl, wl)) and It is the t × t identity matrix. +11 + +A.2. Proposed Method in the Generalized Setting +Here, we propose a BO method for efficiently identifying Z∗ +t . Using the same argument as the method used in +the main body, we construct credible intervals interval for F (1) +t +(x), . . . , F (m) +t +(x) using the method proposed by +[Kirschner et al., 2020], and give an estimation method for the PF based on the constructed credible intervals. +A.2.1. Composition of Credible Intervals and Pareto-frontier Estimation +For each (x, w) ∈ X ×Ω, j ∈ [m] and time t, let Q(f (j)) +t +(x, w) = [l(f (j)) +t +(x, w), u(f (j)) +t +(x, w)] be a credible interval +of f (j)(x, w). Here, l(f (j)) +t +(x, w) and u(f (j)) +t +(x, w) are given by +l(f (j)) +t +(x, w) = µ(j) +t (x, w) − β1/2 +j,t σ(j) +t (x, w), +u(f (j)) +t +(x, w) = µ(j) +t (x, w) + β1/2 +j,t σ(j) +t (x, w), +where β1/2 +j,t ≥ 0. Then, the credible interval for F (j) +t +(x) is denoted as Q(F (j) +t +) +t +(x) ≡ [l(F (j) +t +) +t +(x), u(F (j) +t +) +t +(x)], where +its lower and upper are given by +l +(F (j) +j +) +t +(x) = +inf +p(w)∈At +� +w∈Ω +l(f (j)) +t +(x, w)p(w), +u +(F (j) +j +) +t +(x) = +inf +p(w)∈At +� +w∈Ω +u(f (j)) +t +(x, w)p(w). +Next, for each x ∈ X, subset E ⊂ X and time t, we define LCB(m) +t +(x), UCB(m) +t +(x) and LCB(m) +t +(E) as +LCB(m) +t +(x) = (l(F (1) +t +) +t +(x), . . . , l(F (m) +t +) +t +(x)), UCB(m) +t +(x) = (u(F (1) +t +) +t +(x), . . . , u(F (m) +t +) +t +(x)), +LCB(m) +t +(E) = {LCB(m) +t +(x) | x ∈ E}. +Using this, we define the estimated Pareto solutions set ˆΠ(m) +t +⊂ X for design variables as +ˆΠ(m) +t += {x ∈ X | LCB(m) +t +(x) ∈ Par(LCB(m) +t +(X))}. +Hereafter, for simplicity, we denote LCB(m) +t +(x), UCB(m) +t +(x), LCB(m) +t +(E) and ˆΠ(m) +t +as LCBt(x), UCBt(x), +LCBt(E) and ˆΠt, respectively. +A.2.2. Acquisition Function +Here, we propose an AF to determine the next evaluation point. Similar to the main body, we define the AF +at(x) for x ∈ X as +at(x) = dist(UCBt(x), Dom(LCBt(ˆΠt))). +Then, the next evaluation point is selected as follows: +Definition A.1 (For the setting in the development phase). The next design variable xt+1 to be evaluated is +selected by +xt+1 = argmax +x∈X +at(x). +Similarly, the next environmental variable wt+1 to be evaluated is selected by +wt+1 = argmax +w∈Ω +{σ(1)2 +t +(xt+1, w) + · · · + σ(m)2 +t +(xt+1, w)}. +Notably, since wt+1 cannot be selected in the setting at the use phase, wt+1 is the realized value from P † +at time t + 1. Here, at(x) can be computed analytically by the following lemma: +Lemma A.1. Let UCBt(x) = (u1, , . . . , um) and LCBt(ˆΠt) = {(l(i) +1 , . . . , l(i) +m ) | 1 ≤ i ≤ k}. Then, at(x) can +be computed as follows: +at(x) = max{˜at(x), 0}, +˜at(x) = min +1≤i≤k max{u1 − l(i) +1 , . . . , um − l(i) +m }. +A.2.3. Stopping Condition +Here, we give a stopping condition of our proposed algorithm. As in the main body, let ϵ > 0 be a user- +specified stopping parameter. Then, algorithms terminate if at(x) ≤ ϵ is satisfied. Finally, the pseudo-codes +of the proposed algorithm in the development phase and use phase settings are given in Algorithm 2 and 3, +respectively. +12 + +Algorithm 2 BO for identifying DR-PF in the development phase setting +Input: GP prior GP(0, k(j)), candidate distribution family At, tradeoff parameter {βj,t}t≥0, stopping param- +eter ϵ > 0, j ∈ [m] +t ← 1 +while at(x) > ϵ do +Compute Q(F (j) +t +) +t +(x) for each x ∈ X and j ∈ [m] +Select the next evaluation point (xt, wt) +Observe y(j) +t += f (j)(xt, wt) + ε(j) +t +for each j ∈ [m] +Update GPs by adding observed points +t ← t + 1 +end while +Output: Return ˆΠt as the estimated set of design variables comprising the DR-PF +Algorithm 3 BO for identifying DR-PF in the use phase setting +Input: GP prior GP(0, k(j)), candidate distribution family At, tradeoff parameter {βj,t}t≥0, stopping param- +eter ϵ > 0, j ∈ [m] +t ← 1 +while at(x) > ϵ do +Compute Q(F (j) +t +) +t +(x) for each x ∈ X and j ∈ [m] +Select the next evaluation point xt +Generate wt from P † +Observe y(j) +t += f (j)(xt, wt) + ε(j) +t +for each j ∈ [m] +Update GPs by adding observed points +t ← t + 1 +end while +Output: Return ˆΠt as the estimated set of design variables comprising the DR-PF +A.3. Theoretical Analysis +Here, we give theorems on the accuracy and convergence of the proposed algorithms. First, to give theoretical +guarantees, we assume that for each j ∈ [m], f (j) follows GP GP(0, k(j)((x, w), (x′, w′))). In addition, we assume +that each prior variance k(j)((x, w), (x, w)) ≡ σ(j)2 +0 +(x, w) satisfies +max +(x,w)∈X×Ω σ(j)2 +0 +(x, w) ≤ 1, +∀j ∈ [m]. +Furthermore, let κ(j) +T +be a maximum information gain for f (j) at time T. Moreover, as in the main body, +we define an ϵ-accurate Pareto region Zϵ,t to quantify the goodness of the predicted ˆΠt as input points that +constitute the PF. For a positive number ϵ and the m-dimensional vector ϵ = (ϵ, . . . , ϵ), we define Zϵ,t as +Zϵ,t = {y ∈ Rm |∃ y′ ∈ Z∗ +t s.t. y ⪯ y′ and ∃y′′ ∈ Z∗ +t s.t. y′′ ⪯ y + ϵ}. +Using Zϵ,t, we define the accuracy of ˆΠt as follows: +Definition A.2 (Accuracy for ˆΠt). Let ϵ be a positive number. +Then, we define ˆΠt to be an ϵ-accurate +estimated solution set if the following holds: +Ft(ˆΠt) ⊂ Zϵ,t. +(A.1) +In addition, we define ˆΠt to be an ϵ-accurate estimated Pareto solution set if the following holds: +Par(Ft(ˆΠt)) ⊂ Zϵ,t. +(A.2) +Then, the following theorem guarantees that the proposed algorithms satisfy both (A.1) and (A.2) with a +high probability. +Theorem A.1. Let t ≥ 1 and δ ∈ (0, 1), and define β1,t = · · · = βm,t = 2 log(m|X × Ω|π2t2/(6δ)) ≡ βt. +Moreover, let ϵ > 0 be a user-specified stopping parameter. +Then, when Algorithm 2 terminates, with a +probability of at least 1 − δ, ˆΠt satisfies both (A.1) and (A.2) for any At. +Next, we give a theorem on convergence in the development phase setting. The following theorem gives +convergence guarantees for Algorithm 2: +13 + +Theorem A.2. Under the same condition as in Theorem A.1, let T be the smallest positive integer satisfying +the following inequality: +βT (C1κ(1) +T ++ · · · + Cmκ(m) +T +) +T +≤ ϵ2 +4 , +where Cj = 2/ log(1 + σ(j)−2 +noise ) and j ∈ [m]. Then, Algorithm 2 terminates after at most T iterations. +Next, we give the theorem on convergence under the setting in the use phase. Unlike the setting in the +development phase, we cannot control w in the use phase. Therefore, to make a reasonable inference, the +uncertainty at all points must be able to be reduced stochastically. However, if the value of the true probability +function p†(w) at some w ∈ Ω is zero, the uncertainty of f (j)(x, w) containing this point cannot be sufficiently +small. To avoid this problem, we make the following assumption on the true probability function: +min +w∈Ω p†(w) ≡ pmin > 0. +Then, the following theorem holds: +Theorem A.3. Under the same condition as in Theorem A.1, assume that pmin > 0. Let T be the smallest +positive integer satisfying the following inequality: +βT ( ˜C1κ(1) +T ++ · · · + ˜Cmκ(m) +T ++ ˜C) +T +≤ ϵ2 +4 , +where ˜Cj = (4p−1 +min)/ log(1 + σ(j)−2 +noise ), ˜C = 8mp−1 +min log(8m/δ) and j ∈ [m]. Then, with a probability of at least +1 − δ, Algorithm 3 terminates after at most T trials. +Finally, we show that under appropriate assumptions, the solution to the distributionally robust Pareto +optimization problem is also a good solution to the Pareto optimization problem defined by the true expectation +in the use phase setting. Here, the expectation function ˜F (j)(x) determined by the true probability function +p†(w) is given by +˜F (j)(x) = +� +w∈Ω +f (j)(x, w)p†(w), j ∈ [m]. +In addition, for each x ∈ X and E ⊂ X, let ˜F (x) = ( ˜F (1)(x), . . . , ˜F (m)(x)) and ˜F (E) = { ˜F (x) | x ∈ E}. +Then, the PF ˜Z∗ defined by ˜F (1)(x), . . . , ˜F (m)(x) can be expressed as follows: +˜Z∗ = Par( ˜F (X)). +Furthermore, as in the case of Z∗ +t , we define an ϵ-accurate Pareto region ˜Zϵ for ˜Z∗. For a positive number ϵ +and an m-dimensional vector ϵ = (ϵ, . . . , ϵ), we define ˜Zϵ as +˜Zϵ = {y ∈ Rm |∃ y′ ∈ ˜Z∗ s.t. y ⪯ y′ and ∃y′′ ∈ ˜Z∗ s.t. y′′ ⪯ y + ϵ}. +Using ˜Zϵ, we define the accuracy of ˆΠt for ˜Z∗. +Definition A.3 (Accuracy of ˆΠt for ˜Z∗). Let ϵ be a positive number. Then, we define ˆΠt to be an ϵ-accurate +estimated solution set for ˜Z∗ if the following holds: +Ft(ˆΠt) ⊂ ˜Zϵ. +Moreover, we define ˆΠt to be an ϵ-accurate estimated Pareto solution set for ˜Z∗ if the following holds: +Par(Ft(ˆΠt)) ⊂ ˜Zϵ. +Then, the following theorem holds: +Theorem A.4. Under the use phase setting, let t ≥ 1 and δ ∈ (0, 1), and define β1,t = · · · = βm,t = +2 log(m|X × Ω|π2t2/(6δ)) ≡ βt. Moreover, let ϵ > 0 be a user-specified stopping parameter. Furthermore, let +the reference distribution p∗ +t (w) be the empirical distribution function for w, and define ξt and the distance +between distributions d(·, ·) as +ξt = |Ω| +� +1 +2t log +�|Ω|π2t2 +3δ +� +, +d(p1(w), p2(w)) = +� +w∈Ω +|p1(w) − p2(w)|. +Then, if Algorithm 3 terminates at t ≥ T after time T, with a probability of at least 1−2δ, ˆΠt is the 2ϵ-accurate +estimated set and estimated Pareto set, that is, the following holds: +Ft(ˆΠt) ⊂ ˜Z2ϵ, +Par(Ft(ˆΠt)) ⊂ ˜Z2ϵ. +Here, T is the smallest positive integer satisfying the following inequality: +∀n ≥ T, 2β1/2 +1 +ξn ≤ ϵ. +(A.3) +14 + +B. Proofs +Here, we give proofs of Lemma A.1 and Theorem A.1–A.4. Notably, Lemma 3.1 and Theorem 4.1–4.2 in the +main body are special cases of Lemma A.1 and Theorem A.1–A.2, respectively. First, we prove Lemma A.1. +Proof. Let UCBt(x) = (u1, . . . , um) ≡ u and LCBt(ˆΠt) = {(l(i) +1 , . . . , l(i) +m ) | 1 ≤ i ≤ k} ≡ L. +Here, if +u ∈ Dom(L), then the following holds from the definition of dist(a, B): +at(x) = dist(u, Dom(L)) = +inf +b∈Dom(L) d∞(u, b) = d∞(u, u) = 0. +In addition, since u ∈ Dom(L), there exists (l(i) +1 , . . . , l(i) +m ) such that uj ≤ l(i) +j +for any j ∈ [m]. Thus, we have +max{u1 − l(i) +1 , . . . , um − l(i) +m } ≤ 0. This implies that +˜at(x) = min +1≤i≤k max{u1 − l(i) +1 , . . . , um − l(i) +m } ≤ 0 +and max{˜at(x), 0} = 0. Therefore, we get at(x) = max{˜at(x), 0}. Next, we consider the case where u /∈ +Dom(L). Let at(x) = η. Then, noting that u /∈ Dom(L), for any i ∈ {1, . . . , k}, there exists j ∈ [m] such that +uj > l(i) +j . This implies that +˜at(x) = min +1≤i≤k max{u1 − l(i) +1 , . . . , um − l(i) +m } ≡ ˜η > 0 +and max{˜at(x), 0} = ˜at(x) = ˜η. For this ˜η, there exists i such that +uj − l(i) +j +≤ ˜η +∀j ∈ [m]. +Hence, we have ˜u ≡ (u1 − ˜η, . . . , um − ˜η) ∈ Dom(L) because uj − ˜η ≤ l(i) +j +for any j ∈ [m]. Thus, from the +definition of at(x), the following holds: +η = at(x) = dist(u, Dom(L)) = +inf +b∈Dom(L) d∞(u, b) ≤ d∞(u, ˜u) = ˜η. +Here, we assume η < ˜η. Then, noting that Dom(L) is the closed set, there exists ˜l = (˜l1, . . . , ˜lm) ∈ Dom(L) +such that d∞(u, ˜l) = η. Therefore, ˜l can be expressed as ˜l = (u1 − s1, . . . , um − sm), where 0 ≤ |sj| ≤ η and at +least one of s1, . . . , sm is η. Thus, since (u1 − η, . . . , um − η) ⪯ ˜l, noting that (u1 − η, . . . , um − η) ∈ Dom(L) +there exists i such that +uj − η ≤ l(i) +j +∀j ∈ [m]. +This implies that max{u1 − l(i) +1 , . . . , um − l(i) +m } ≤ η. Hence, it follows that +˜η = min +1≤i≤k max{u1 − l(i) +1 , . . . , um − l(i) +m } ≤ η. +However, this is a contradiction with η < ˜η. Consequently, we obtain at(x) = max{˜at(x), 0}. +Next, we prove Theorem A.1. +Proof. First, we prove Ft(ˆΠt) ⊂ Zϵ,t. Since ˆΠt ⊂ X, for any y ∈ Ft(ˆΠt), there exists y′ ∈ Z∗ +t such that +y ⪯ y′. Thus, it is sufficient to show that there exists y′′ ∈ Z∗ +t such that y′′ ⪯ y +ϵ. Here, under the theorem’s +assumption, with a probability of at least 1 − δ, the following holds for any (x, w) ∈ X × Ω, j ∈ [m] and time +t ≥ 1: +f (j)(x, w) ∈ Q(f (j)) +t +(x, w), +where this relation can be derived by using Lemma 5.1 of [Srinivas et al., 2010]. Hence, we have F (j) +t +(x) ∈ +Q(F (j) +t +) +t +(x). In addition, since Z∗ +t is the closed set, for any x ∈ ˆΠt, there exist a ≥ 0 and Ft(x) + (a, . . . , a) ≡ +y′′ ∈ Z∗ +t such that +dist(Ft(x), Z∗ +t ) = d∞(Ft(x), y′′) ≤ d∞((l(F (1) +t +) +t +(x), . . . , l(F (m) +t +) +t +(x)), y′′′), +where y′′′ ∈ Z∗ +t can be given by using s ≥ a ≥ 0 as y′′′ = (l(F (1) +t +) +t +(x), . . . , l(F (m) +t +) +t +(x)) + (s, . . . , s). Here, for some +ˆx ∈ X satisfying y′′′ ⪯ Ft(ˆx), the following holds: +d∞((l(F (1) +t +) +t +(x), . . . , l(F (m) +t +) +t +(x)), y′′′) ≤ dist(Ft(ˆx), Dom(LCBt(ˆΠt))). +15 + +Moreover, the right hand side is bounded from above as +dist(Ft(ˆx), Dom(LCBt(ˆΠt))) +≤ dist(UCBt(ˆx), Dom(LCBt(ˆΠt))) +≤ max +x†∈X dist(UCBt(x†), Dom(LCBt(ˆΠt))) = at(xt+1). +Therefore, if at(xt+1) ≤ ϵ, then d∞(Ft(x), y′′) ≤ ϵ. It follows that y′′ ⪯ Ft(x) + ϵ. Since x is an arbitrary +element of ˆΠt, we have Ft(ˆΠt) ⊂ Zϵ,t. Next, we prove Par(Ft(ˆΠt)) ⊂ Zϵ,t. As before, noting that ˆΠt ⊂ X, for +any y ∈ Par(Ft(ˆΠt)), there exists y′ ∈ Z∗ +t such that y ⪯ y′. In addition, since Z∗ +t is the closed set, for any +y ∈ Par(Ft(ˆΠt)), there exist a ≥ 0 and y + (a, . . . , a) ≡ y′′ ∈ Z∗ +t such that +dist(y, Z∗ +t ) = d∞(y, y′′). +Here, for some ˆx ∈ X satisfying y′′ ⪯ Ft(ˆx), the following holds: +d∞(y, y′′) ≤ d∞(y′′′, Ft(ˆx)) ≤ dist(Ft(ˆx), Dom(LCBt(ˆΠt))), +where y′′′ ∈ Par(Ft(ˆΠt)) can be given by using s′ ≥ a ≥ 0 as y′′′ = Ft(ˆx) − (s′, . . . , s′). Then, we obtain +dist(Ft(ˆx), Dom(LCBt(ˆΠt))) +≤ dist(UCBt(ˆx), Dom(LCBt(ˆΠt))) +≤ max +x†∈X dist(UCBt(x†), Dom(LCBt(ˆΠt))) = at(xt+1). +Thus, if at(xt+1) ≤ ϵ, then d∞(y, y′′) ≤ ϵ. This implies that y′′ ⪯ y + ϵ. Consequently, since y is an arbitrary +element of Par(Ft(ˆΠt)), we have Par(Ft(ˆΠt)) ⊂ Zϵ,t. +Next, we prove Theorem A.2. +Proof. Let xt = argmaxx∈X at−1(x). Here, since LCB t−1(xt) ∈ Dom(LCB t−1(ˆΠt−1)), the following in- +equality holds: +at−1(xt) = dist(UCBt−1(xt), Dom(LCB t−1(ˆΠt−1))) +≤ d∞(UCB t−1(xt), LCB t−1(xt)) += max +1≤j≤m{u +(F (j) +t−1) +t−1 +(xt) − l +(F (j) +t−1) +t−1 +(xt)}. +Therefore, from the definition of u +(F (j) +t−1) +t−1 +(xt) and l +(F (j) +t−1) +t−1 +(xt), we get +u +(F (j) +t−1) +t−1 +(xt) − l +(F (j) +t−1) +t−1 +(xt) ≤ 2β1/2 +t +max +w∈Ω σ(j) +t−1(xt, w). +Hence, the following inequality holds: +a2 +t−1(xt) ≤ 4βt max +1≤j≤m max +w∈Ω σ(j)2 +t−1 (xt, w). +Furthermore, since wt is selected by +wt = argmax +w∈Ω +(σ(1)2 +t−1 (xt, w) + · · · + σ(m)2 +t−1 (xt, w)), +the following holds: +a2 +t−1(xt) ≤ 4βt max +1≤j≤m max +w∈Ω σ(j)2 +t−1 (xt, w) +≤ 4βt(σ(1)2 +t−1 (xt, wt) + · · · + σ(m)2 +t−1 (xt, wt)). +In addition, let T be the number given by Theorem A.2. Then, we get +T min +1≤t≤T a2 +t−1(xt) ≤ +T +� +t=1 +a2 +t−1(xt) +≤ 4βT +m +� +j=1 +T +� +t=1 +σ(j)2 +t−1 (xt, wt) +≤ 4βT (C1κ(1) +T ++ · · · + Cmκ(m) +T +), +16 + +where the last inequality can be derived by using Lemma 5.3 and 5.4 of [Srinivas et al., 2010]. +Therefore, +dividing both sides by T, we obtain +min +1≤t≤T a2 +t−1(xt) ≤ 4βT +C1κ(1) +T ++ · · · + Cmκ(m) +T +T +≤ ϵ2. +Hence, we get min1≤t≤T at−1(xt) ≤ ϵ. This implies that there exists t′ ∈ {1, . . . , T} such that at′−1(xt′) ≤ ϵ. +Next, we prove Theorem A.3. +Proof. Using the same argument as in the proof of Theorem A.2, we have +a2 +t−1(xt) ≤ 4βt max +1≤j≤m max +w∈Ω σ(j)2 +t−1 (xt, w). +Furthermore, since pmin > 0, the following inequality holds: +max +w∈Ω σ(j)2 +t−1 (xt, w) ≤ +� +wΩ +σ(j)2 +t−1 (xt, w) += +� +wΩ +(p†(w))−1p†(w)σ(j)2 +t−1 (xt, w) +≤ p−1 +min +� +wΩ +p†(w)σ(j)2 +t−1 (xt, w) +≡ p−1 +minEw[σ(j)2 +t−1 (xt, w)]. +Using this, we get +max +1≤j≤m max +w∈Ω σ(j)2 +t−1 (xt, w) ≤ +m +� +j=1 +max +w∈Ω σ(j)2 +t−1 (xt, w) +≤ p−1 +min +m +� +j=1 +Ew[σ(j)2 +t−1 (xt, w)] += p−1 +minEw +� +� +m +� +j=1 +σ(j)2 +t−1 (xt, w) +� +� . +Here, let T be the number given by Theorem A.3. Then, we obtain +T min +1≤t≤T a2 +t−1(xt) ≤ +T +� +t=1 +a2 +t−1(xt) +≤ 4βT p−1 +min +T +� +t=1 +Ew +� +� +m +� +j=1 +σ(j)2 +t−1 (xt, w) +� +� . +(B.1) +Noting that �m +j=1 σ(j)2 +t−1 (xt, w) is a non-negative random variable and �m +j=1 σ(j)2 +t−1 (xt, w) ≤ m, from Lemma 3 +of [Kirschner and Krause, 2018], the following holds with a probability of at least 1 − δ: +T +� +t=1 +Ew +� +� +m +� +j=1 +σ(j)2 +t−1 (xt, w) +� +� ≤ 2 +m +� +j=1 +T +� +t=1 +σ(j)2 +t−1 (xt, w) + 4m log 1 +δ + 8m log 4m + 1. +Moreover, since 4m log 1 +δ ≤ 8m log 1 +δ and 1 ≤ 8m log 2, we have +4m log 1 +δ + 8m log 4m + 1 ≤ 8m log 1 +δ + 8m log 4m + 8m log 2 += 8m log 8m +δ +and +T +� +t=1 +Ew +� +� +m +� +j=1 +σ(j)2 +t−1 (xt, w) +� +� ≤ 2 +m +� +j=1 +T +� +t=1 +σ(j)2 +t−1 (xt, w) + 8m log 8m +δ . +(B.2) +17 + +Hence, by substituting (B.2) into (B.1), from Lemma 5.3 and 5.4 of [Srinivas et al., 2010], we get +T min +1≤t≤T a2 +t−1(xt) ≤ 4βT ( ˜C1κ(1) +T ++ · · · + ˜Cmκ(m) +T ++ ˜C). +Therefore, from the definition of T, dividing both sides by T, we obtain +min +1≤t≤T a2 +t−1(xt) ≤ 4βT +˜C1κ(1) +T ++ · · · + ˜Cmκ(m) +T ++ ˜C +T +≤ ϵ2. +Thus, we get min1≤t≤T at−1(xt) ≤ ϵ. This implies that there exists t′ ∈ {1, . . . , T} such that at′−1(xt′) ≤ ϵ. +Finally, we prove Theorem A.4. +Proof. Suppose that p∗ +t (w) is the empirical distribution function of w. Then, from Hoeffding’s inequality, the +following inequality holds for any w: +P(|p∗ +t (w) − p†(w)| ≥ λ) ≤ 2 exp(−2tλ2). +Here, let +λ = +� +1 +2t log +�|Ω|π2t2 +3δ +� +. +Then, with a probability of at least 1 − δ, the following holds for any t ≥ 1 and w ∈ Ω: +|p∗ +t (w) − p†(w)| ≤ λ. +In addition, from the definition of d(·, ·), we get +d(p∗ +t (w), p†(w)) = +� +w∈Ω +|p∗ +t (w) − p†(w)| ≤ |Ω|λ = ξt. +This implies that p†(w) ∈ At. Furthermore, from the definition of ˜F (j)(x) and F (j) +t +(x), the following inequality +holds for any t ≥ 1, j ∈ [m] and x ∈ X: +F (j) +t +(x) ≤ ˜F (j)(x). +Therefore, it follows that +Ft(x) ⪯ ˜F (x) +∀x ∈ X,∀ t ≥ 1. +(B.3) +Here, for any x ∈ X, t ≥ 1 and j ∈ [m], let ¯p(j) +t,x(w) ∈ At be the probability function satisfying +F (j) +t +(x) = +� +w∈Ω +f (j)(x, w)¯p(j) +t,x(w). +Then, the following inequality holds: +| ˜F (j)(x) − F (j) +t +(x)| ≤ +� +w∈Ω +|f (j)(x, w)||p†(w) − ¯p(j) +t,x(w)|. +Moreover, from Lemma 5.1 of [Srinivas et al., 2010], with a probability of at least 1 − δ, the following holds for +any x ∈ X, w ∈ Ω and j ∈ [m]: +|f (j)(x, w)| ≤ β1/2 +1 +σ(j) +0 (x, w) ≤ β1/2 +1 +. +Hence, we have +˜F (j)(x) − F (j) +t +(x) ≤ | ˜F (j)(x) − F (j) +t +(x)| +≤ β1/2 +1 +� +w∈Ω +|p†(w) − ¯p(j) +t,x(w)| += β1/2 +1 +d(p†(w), ¯p(j) +t,x(w)) +≤ β1/2 +1 +(d(p†(w), p∗ +t (w)) + d(p∗ +t (w), ¯p(j) +t,x(w))) ≤ 2β1/2 +1 +ξt. +In addition, let T be the smallest positive integer satisfying (A.3). Then, for any t ≥ T, the following inequality +holds: +˜F (j)(x) ≤ F (j) +t +(x) + ϵ. +18 + +Thus, we obtain +˜F (x) ⪯ Ft(x) + ϵ +∀x ∈ X,∀ t ≥ T. +(B.4) +Therefore, by combining (B.3) and (B.4), we have +Par(Ft(X)) ⊂ ˜Zϵ +∀t ≥ T. +(B.5) +Finally, from Theorem A.1, the following holds at t′, the time at which the algorithm terminates: +Ft′(ˆΠt′) ⊂ Zϵ,t′, Par(Ft′(ˆΠt′)) ⊂ Zϵ,t′. +(B.6) +Consequently, if t′ ≥ T, using (B.5) and (B.6) we get +Ft′(ˆΠt′) ⊂ ˜Z2ϵ, Par(Ft′(ˆΠt′)) ⊂ ˜Z2ϵ. +C. Experimental Details and Additional Experiments +Here, we give experimental details and additional experiments in Section 5. +Experimental Setup +The experimental parameters used in each experiment are described in Table 2. +Table 2: Experimental parameters for each setting +Parameters +Simulator setting +σ2 +f,1 = 1000, L1 = 2, σ(1)2 +noise = 10−4, β1/2 +1,t = 3, σ2 +f,2 = 1000, L2 = 2, σ(2)2 +noise = 10−4, β1/2 +2,t = 3, ξ = 0.05 +Uncontrollable setting +σ2 +f,1 = 1000, L1 = 2, σ(1)2 +noise = 10−4, β1/2 +1,t = 3, σ2 +f,2 = 1000, L2 = 2, σ(2)2 +noise = 10−4, β1/2 +2,t = 3, ξ = 0.05 +SIR (Case1) +σ2 +f,1 = 5000, L1 = 0.1, σ(1)2 +noise = 10−8, β1/2 +1,t = 3, σ2 +f,2 = 105, L2 = 0.01, σ(2)2 +noise = 10−4, β1/2 +2,t = 2, ξ = 0.15 +SIR (Case2) +σ2 +f,1 = 104, L1 = 0.1, σ(1)2 +noise = 10−3, β1/2 +1,t = 2, σ2 +f,2 = 105, L2 = 0.1, σ(2)2 +noise = 10−3, β1/2 +2,t = 3, ξ = 0.15 +MVA +The MVA method is based on reducing the uncertainty in the potential optimal set Mt and the +estimated PF solution set ˆΠt. Using ˆΠt, Mt is defined as follows: +Mt = {x ∈ X \ ˆΠt |∀ x′ ∈ ˆΠt, u(F (1)) +t +(x) > l(F (1)) +t +(x′) or u(F (2)) +t +(x) > l(F (2)) +t +(x′)}. +The uncertainty λt(x) is given by +λt(x) = +� +(u(F (1)) +t +(x) − l(F (1)) +t +(x))2 + (u(F (2)) +t +(x) − l(F (2)) +t +(x))2. +EHI +The EHI method is based on the expected hypervolume improvement for a bounded region defined by +PFs and reference points. Let B ⊂ R2 be a set, and let r = (r1, r2) ∈ R2 be a reference point satisfying r ⪯ B. +Then, let us denote the bounded region dominated by B and r, by +Dom(B; r) = Dom(B) ∩ [r1, ∞) × [r2, ∞). +In EHI, for each j ∈ {1, 2} we calculated the estimated value of F (j)(x) by using posterior means as +µ(F (j)) +t +(x) = +inf +p(w)∈A +� +w∈Ω +µ(j) +t (x, w)p(w). +For a point x ∈ X a subset E ⊂ X, we define µ(F ) +t +(x) and µ(F ) +t +(E) as +µ(F ) +t +(x) = (µ(F (1)) +t +(x), µ(F (2)) +t +(x)), +µ(F ) +t +(E) = {µ(F ) +t +(x) | x ∈ E}. +In our experiments, we defined the reference point rt = (rt,1, rt,2) for each iteration t as +rt,1 = min +x∈X µ(F (1)) +t +(x), +rt,2 = min +x∈X µ(F (2)) +t +(x). +Then, the expected hypervolume improvement for x ∈ X is given by +EHIt(x) = EF (1)(x),F (2)(x)[Vol(Dom(µ(F ) +t +(X) ∪ {(F (1)(x), F (2)(x))}; rt) \ Dom(µ(F ) +t +(X); rt))]. +19 + +Table +3: Computational time (second) and computational time ratio for each setting when Nx = 50 and +Nw = 100 +Random +UCB F1 +UCB F2 +MVA +EHI +Proposed +Computational time +0.000 +0.181 +0.185 +0.375 +45.77 +0.364 +(Standard error) +(0.000) +(0.001) +(0.001) +(0.001) +(0.025) +(0.001) +Computational time ratio +0.000 +0.500 +0.510 +1.031 +126.15 +1 +(Standard error) +(0.000) +(0.002) +(0.002) +(0.003) +(0.327) +(0) +Table +4: Computational time (second) and computational time ratio for each setting when Nx = 100 and +Nw = 50 +Random +UCB F1 +UCB F2 +MVA +EHI +Proposed +Computational time +0.000 +0.134 +0.135 +0.272 +31.97 +0.268 +(Standard error) +(0.000) +(0.001) +(0.001) +(0.001) +(0.015) +(0.001) +Computational time ratio +0.000 +0.503 +0.507 +1.019 +119.92 +1 +(Standard error) +(0.000) +(0.002) +(0.002) +(0.004) +(0.346) +(0) +Table +5: Computational time (second) and computational time ratio for each setting when Nx = 100 and +Nw = 100 +Random +UCB F1 +UCB F2 +MVA +EHI +Proposed +Computational time +0.000 +0.360 +0.362 +0.719 +91.82 +0.726 +(Standard error) +(0.000) +(0.001) +(0.001) +(0.002) +(0.099) +(0.002) +Computational time ratio +0.000 +0.496 +0.500 +0.991 +127.02 +1 +(Standard error) +(0.000) +(0.001) +(0.001) +(0.002) +(0.412) +(0) +Here, for a bounded set A, Vol(A) represents the hypervolume of A. +Because F (1)(x) and F (2)(x) do not +follow GPs, we cannot calculate EHIt(x) analytically. Thus, we approximate it by using samples from posterior +distributions. +Let M be a number of Monte Carlo sampling, and let (f (j) +t,(l)(x, w1), . . . , f (j) +t,(l)(x, w|Ω|)) be an +lth sample from the posterior distribution of (f (j)(x, w1), . . . , f (j)(x, w|Ω|)) at iteration t, where 1 ≤ l ≤ M, +j ∈ {1, 2} and x ∈ X. Then, for each t, we calculate F (1) +t,(l)(x) and F (2) +t,(l)(x) as +F (1) +t,(l)(x) = +inf +p(w)∈A +� +w∈Ω +f (1) +t,(l)(x, w)p(w), +F (2) +t,(l)(x) = +inf +p(w)∈A +� +w∈Ω +f (2) +t,(l)(x, w)p(w). +Using this, we approximate EHIt(x) as +1 +M +M +� +l=1 +Vol(Dom(µ(F ) +t +(X) ∪ {(F (1) +t,(l)(x), F (2) +t,(l)(x))}; rt) \ Dom(µ(F ) +t +(X); rt)). +Additional Computational Time Experiments +We also compared the computational time of each method +by changing input space settings conducted in Section 5.2. In this experiment, the input space X × Ω was a set +of grid points divided into [−10, 10] × [−10, 10] equally spaced at Nx × Nw. We compared computational times +using (50, 100), (100, 50) and (100, 100) as (Nx, Nw). From Table 3–5, it can be confirmed that the results are +similar to the experimental results conducted in Section 5.2. +20 + diff --git a/7dFJT4oBgHgl3EQfmywg/content/tmp_files/load_file.txt b/7dFJT4oBgHgl3EQfmywg/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..fd23179ce1050a4b4f57d067dc1bff9aa8cce4b4 --- /dev/null +++ b/7dFJT4oBgHgl3EQfmywg/content/tmp_files/load_file.txt @@ -0,0 +1,2314 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf,len=2313 +page_content='Distributionally Robust Multi-objective Bayesian Optimization under Uncertain Environments Yu Inatsu1,∗ Ichiro Takeuchi2,3 1 Department of Computer Science, Nagoya Institute of Technology 2 Department of Mechanical Systems Engineering, Nagoya University 3 RIKEN Center for Advanced Intelligence Project ∗ E-mail: inatsu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='yu@nitech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='jp ABSTRACT In this study, we address the problem of optimizing multi-output black-box functions under uncertain environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' We formulate this problem as the estimation of the uncertain Pareto-frontier (PF) of a multi-output Bayesian surrogate model with two types of variables: design variables and environmental variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' We consider this problem within the context of Bayesian optimization (BO) under uncertain environments, where the design variables are controllable, whereas the environmental variables are assumed to be random and not controllable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' The challenge of this problem is to robustly estimate the PF when the distribution of the environmental variables is unknown, that is, to estimate the PF when the environmental variables are generated from the worst possible distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' We propose a method for solving the BO problem by appropriately incorporating the uncertainties of the environmental variables and their probability distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' We demonstrate that the proposed method can find an arbitrarily accurate PF with high probability in a finite number of iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' We also evaluate the performance of the proposed method through numerical experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Introduction In many industrial applications, we encounter the problem of optimizing the multi-output black-box function under uncertain environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' For example, in the problem of optimizing growing conditions for crops, we want to optimize several conditions such as fertilizer levels to maximize crop quality and yield under an uncertain environment such as weather conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' To formulate this problem, let f (1)(x, w) and f (2)(x, w) be a pair of outputs of a black-box function that we want to simultaneously maximize, where x ∈ X and w ∈ Ω are the design variables (such as fertilizer levels) and environmental variables (such as weather conditions) defined in domains X and Ω, respectively, where the former is controllable and the latter is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' To characterize the uncertainty of the environmental variables w, we assume that it is sampled from an unknown probability distribution, P †.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Because we do not know P †, we consider the case where we know only A, which is a family of candidate distributions for w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' This study aims to identify a distributionally robust Pareto-frontier (DR-PF) in the above setting, which is formulated as a PF of the following two functions: F (1)(x) = inf p(w)∈A � Ω f (1)(x, w)p(w)dw, F (2)(x) = inf p(w)∈A � Ω f (2)(x, w)p(w)dw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Figure 1 shows an example of the problem setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' To identify a DR-PF, it is necessary to predict it and quantify its uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' In this study, under the assumption that f (1)(x, w) and f (2)(x, w) follow a Gaussian process (GP), we developed a Bayesian optimization (BO) method to find a lower bound of the DR-PF by considering the uncertainty of environmental variables w and the uncertainty of the probability distribution for w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Specifically, we propose an acquisition function (AF) that enables us to sequentially select the controllable design variable x in a sample-efficient manner to obtain the DR-PF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' To this end, various technical challenges need to be resolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' One difficulty is that, even when f (1)(x, w) and f (2)(x, w) are GPs, F (1)(x) and F (2)(x) are not GPs anymore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Therefore, we derive a non-trivial credible intervals of F (1)(x) and F (2)(x) considering that they are defined as the infima of integrated GPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Furthermore, although a naive formulation of multi-objective BOs is computationally expensive, the proposed AF has the advantage that it can be evaluated efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' We also conducted a theoretical analysis of the proposed BO method to prove that the proposed BO method can find an arbitrarily accurate DR-PF with a high probability in a finite number of iterations under mild conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Related Work For black-box function optimization problems, BOs have been popularly used [Settles, 2009, Shahriari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=', 2016] in which GP [Williams and Rasmussen, 2006] is often employed as a surrogate model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' An optimization problem 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='11588v1 [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='ML] 27 Jan 2023 Multi-objective function � �, � = � � �, � , � � �, � (a) Objective function 1 x w 4 2 0 2 4 4 2 0 2 4 0 5 10 15 20 25 30 35 40 Objective function 2 x w 4 2 0 2 4 4 2 0 2 4 0 10 20 30 40 50 60 70 80 (b) Candidate distributions of 4 2 0 2 4 0.' metadata={'source': 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+page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='4 Candidate distribution of w w Density ・ ・ ・ ・ ・ ・ ・・・ ・ ・ ・・・ ・ ・ ・・・ ・ (c) Expected value of � � (�,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' �) Expected value of � � (�,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' �) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='Candidate Pareto-frontiers ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='Distributionally robust ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='Pareto-frontier ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='1 ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='Pareto-frontier ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='Figure ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='1: Conceptional diagram of a DR-PF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' The upper and lower color maps in (a) represent objective functions f (1)(x, w) and f (2)(x, w), respectively, where x and w are the scalar design and environmental variable, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' The plots within the frame in (b) represent multiple candidate distributions for w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' A dashed line in (c) is the expected PF for each candidate distributions of w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' A red solid line is the DR-PF, which is defined by the worst-case expectation of the candidate distributions, provides a lower bound of uncertain PF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' The objective of this study is to efficiently identify the DR-PF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' of multiple black-box functions is typically formulated as a Pareto optimization problem, and BO methods for such problems have been also studied [Zuluaga et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=', 2016, Suzuki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=', 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Many studies have been conducted on BOs under uncertain environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' For example, [Bogunovic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=', 2018] proposed a BO method to maximize the worst-case function value with respect to a shift in the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' In several studies [Beland and Nair, 2017, Toscano-Palmerin and Frazier, 2018, Oliveira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=', 2019, Fr¨ohlich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=', 2020, Gessner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=', 2020], a BO problem to maximize the expected value of a black-box function with respect to the input distribution was considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Furthermore, several studies considered the simultaneous optimization of multiple black-box functions under the assumption that the distribution of environmental variables is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' For example, [Iwazaki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=', 2021] dealt with constrained optimization and Pareto optimization problems for the mean and variance of a black-box function with respect to environmental variables, [Qing et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=', 2022] considered Pareto optimization problems for the expected values of multiple objective functions, and [Amri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=', 2021] dealt with chance-constrained optimization problems that is an extension of constrained optimization problems to the input uncertainty setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Distributionally robust optimization (DRO) was first introduced by [Scarf, 1958].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Because DRO is an im- portant topic in the context of robust optimization and has been the subject of numerous studies, we refer an exhaustive survey of DRO to [Rahimian and Mehrotra, 2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' In recent years, several studies on DRO in the context of BOs (DRO-BOs) have been conducted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' [Kirschner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=', 2020] and [Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=', 2020] proposed BO methods to efficiently find the design variable that maximizes F (1)(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' DRO-BOs have also been stud- ied under multiple black-box functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' [Inatsu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=', 2022] proposed a BO method for distributionally robust chance-constrained problems, which is an extension of the chance-constrained problem to DRO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' However, to the best of our knowledge, there are no prior studies on BOs for Pareto optimization under the DRO framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Contributions The contributions of this study are as follows: We develop a BO method for identifying DR-PFs called DR-PF BO method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Specifically, we propose a novel AF for the DR-PF BO, which is computationally inexpensive and has theoretical guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Under mild conditions, we prove that the DR-PF BO method can find an arbitrarily accurate PF with a high probability in a finite number of iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Additionally, we prove that with a specification of an appropriate family of candidate distributions, even if the true distribution is unknown, the DR-PF BO method can find an arbitrarily accurate expected PF on the true distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' 2 We confirm the performance of the proposed method through numerical experiments with benchmark functions and simulator-based functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Preliminaries Let f (1) : X × Ω → R and f (2) : X × Ω → R be the expensive-to-evaluate black-box functions1, where X and Ω are the finite sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' For each input (x, w) ∈ X × Ω, the values of f (1)(x, w) and f (2)(x, w) are observed with observation noise as y(1) = f (1)(x, w) + ε(1) and y(2) = f (2)(x, w) + ε(2), where ε(1) and ε(2) are random samples from independent normal distributions with the mean zero and variances σ(1)2 noise and σ(2)2 noise, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' In this study, the environmental variable w ∈ Ω was assumed to be a discrete random variable that follows an unknown probability distribution P †.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' The two distinct phases called the development phase and use phase exist in the literature of BOs under uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' In the development phase, environmental variables are completely controllable as design variables, whereas they are stochastic and uncontrollable in the use phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' In this study, we consider the development phase, and the use phase is described in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Furthermore, we denote the family of candidate distributions of w as A and consider the following class of distributions: A = {p(w) | d(p(w), p∗(w)) ≤ ξ}, where p∗(w) is a user-specified reference distribution, d(·, ·) is a given distance metric function between dis- tributions, and ξ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' This means that we consider a set of candidate distributions whose distance from the reference distribution is not larger than a certain threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Subsequently, the distributionally robust expecta- tions F (1)(x) and F (2)(x) for each design variable x ∈ X are defined as follows: F (1)(x) = inf p(w)∈A � w∈Ω f (1)(x, w)p(w), F (2)(x) = inf p(w)∈A � w∈Ω f (2)(x, w)p(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' The objective of this study is to efficiently identify the PF determined by F (1)(x) and F (2)(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Let F (x) = (F (1)(x), F (2)(x)) for each x ∈ X, and let F (E) = {F (x) | x ∈ E} for a set E ⊂ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Furthermore, for a set B ⊂ R2, we denote the domain dominated by B and the PF of B, respectively, by Dom(B) = {y ∈ R2 |∃ y′ ∈ B s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' y ⪯ y′}, Par(B) = ∂(Dom(B)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Here, for a point a = (a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' , am) and b = (b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' , bm), a ⪯ b implies ai ≤ bi for any i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' , m}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' For a set C, ∂(C) denotes the boundary of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' The PF determined by F (1)(x) and F (2)(x) can then be written as Z∗ = Par(F (X)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Gaussian Process In this study, we use GP surrogate models for black-box functions f (1) and f (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' First, we assume that f (1) and f (2) follow GP priors GP(0, k(1)((x, w), (x′, w′))) and GP(0, k(2)((x, w), (x′, w′))), respectively, where k(1)((x, w), (x′, w′)) and k(2)((x, w), (x′, w′)) are the positive-definite kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' For l ∈ {1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' 2},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' given a dataset {(xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' wi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' y(l) i )}t i=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' where t is the number of queried instances,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' the posterior distribution of f (l) is a GP,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' and its posterior mean µ(l) t (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' w) and posterior variance σ(l)2 t (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' w) are given by µ(l) t (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' w) = k(l) t (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' w)⊤(K(l) t + σ(l)2 noiseIt)−1y(l) t ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' σ(l)2 t (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' w) = k(l)((x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' w),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' w)) − k(l) t (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' w)⊤(K(l) t + σ(l)2 noiseIt)−1k(l) t (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' w),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' where k(l) t (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' w) is the t-dimensional vector,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' whose jth element is k(l)((x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' w),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' (xj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' wj)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' y(l) t = (y(l) 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' , y(l) t )⊤, It is the t × t identity matrix, K(l) t is the t × t matrix whose (j, k) element is k(l)((xj, wj), (xk, wk)), with a superscript ⊤ indicating the transpose of vectors or matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Proposed Method Here, we propose a BO method to efficiently identify Z∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Because the functions f (1)(x, w) and f (2)(x, w) are random variables following GPs, F (1)(x) and F (2)(x) are also random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Therefore, a reasonable approach is to construct credible intervals for F (1)(x) and F (2)(x), and use them to estimate the PF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' However, 1Note that the method proposed in this study can be straightforwardly extended to the case where there are more than three objective functions f(1), f(2), f(3), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='. 3 unlike f (1)(x, w) and f (2)(x, w), F (1)(x) and F (2)(x) do not follow GPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Therefore, we cannot directly obtain credible intervals using the properties of GPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' In Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='1, we construct credible intervals for F (1)(x) and F (2)(x) using the method proposed by [Kirschner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=', 2020] and provide a method for estimating the PF based on the constructed credible intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Credible Intervals and PF Estimation For each input (x, w) ∈ X × Ω and time t, the credible interval of f (1)(x, w) is denoted by Q(f (1)) t (x, w) = [l(f (1)) t (x, w), u(f (1)) t (x, w)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Here, the lower value l(f (1)) t (x, w) and the upper value u(f (1)) t (x, w) are given as l(f (1)) t (x, w) = µ(1) t (x, w) − β1/2 1,t σ(1) t (x, w), u(f (1)) t (x, w) = µ(1) t (x, w) + β1/2 1,t σ(1) t (x, w), where β1/2 1,t ≥ 0 is a user-specified tradeoff parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' We then define the credible interval Q(F (1)) t (x) ≡ [l(F (1)) t (x), u(F (1)) t (x)] of F (1)(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Here, the lower and upper values are respectively given by l(F (1)) t (x) = inf p(w)∈A � w∈Ω l(f (1)) t (x, w)p(w), u(F (1)) t (x) = inf p(w)∈A � w∈Ω u(f (1)) t (x, w)p(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='1) Notably, if the L1- (or L2-) norm is used as the distance d(·, ·) between the distributions, the problem of obtaining the lower and upper values in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='1) is reduced to a linear programming problem (or a second-order cone programming problem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' In either case, the existence of a fast solver of these problems enabled us to obtain Q(F (1)) t (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Similarly, we define credible intervals Q(f (2)) t (x, w) = [l(f (2)) t (x, w), u(f (2)) t (x, w)] for f (2)(x, w) and Q(F (2)) t (x) ≡ [l(F (2)) t (x), u(F (2)) t (x)] for F (2)(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Next, for any input x ∈ X and any subset E ⊂ X, we define LCBt(x), UCBt(x) and LCBt(E) as follows: LCBt(x) = (l(F (1)) t (x), l(F (2)) t (x)), UCBt(x) = (u(F (1)) t (x), u(F (2)) t (x)), LCBt(E) = {LCBt(x) | x ∈ E}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' The estimated PF solution set ˆΠt ⊂ X for the design variables is then defined as follows: ˆΠt = {x ∈ X | LCBt(x) ∈ Par(LCBt(X))}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Figure 2 (a) shows a conceptual diagram of LCBt(x) and UCBt(x), and (b) shows a conceptual diagram of Par(LCBt(X)) and ˆΠt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Acquisition Function Here, we propose an AF for determining the next evaluation point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' First, for each point a ∈ Rm and subset B ⊂ Rm, we denote the closeness between them as dist(a, B) = inf b∈B d∞(a, b), where d∞(a, b) denotes the metric function given by d∞(a, b) = max{|a1 − b1|, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' , |am − bm|}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Using this, we define AF at(x) for x ∈ X as at(x) = dist(UCBt(x), Dom(LCBt(ˆΠt))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' We then select the following evaluation points, as described in the following definition: Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' The next design variable, xt+1, to be evaluated is selected as follows: xt+1 = argmax x∈X at(x), and the next environmental variable, wt+1, to be evaluated is selected as wt+1 = argmax w∈Ω {σ(1)2 t (xt+1, w) + σ(2)2 t (xt+1, w)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' 4 𝐹(2)(𝒙) 𝐹(1)(𝒙) 𝐹(2)(𝒙) 𝐹(1)(𝒙) 𝐹(2)(𝒙) 𝐹(1)(𝒙) (a) (b) (c) 𝒙1 𝒙2 𝒙3 𝒙4 𝒙5 𝒙6 𝒙7 Figure 2: Conceptual diagrams of LCBt(x), UCBt(x), Par(LCBt(X)), ˆΠt and AFs for seven input points x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' , x7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' At each point x in the left figure (a), LCBt(x) and UCBt(x) indicate the lower left point and the upper right point of the dashed rectangular region, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' In (b), the PF (red line) computed using each LCBt(x) is Par(LCBt(X)), and because it is constructed by LCBt(x1), LCBt(x2), LCBt(x3), LCBt(x7), ˆΠt is given by ˆΠt = {x1, x2, x3, x7}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' In (c), the light red region indicates Dom(LCBt(ˆΠt)), the region dominated by the red points (LCBt(ˆΠt)), and at(x) is the closeness between the light red region and UCBt(x) (purple point).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' The furthest point is represented by the purple triangle, UCBt(x4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Thus, the next design variable to be evaluated is x4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Figure 2 (c) shows a conceptual diagram of the AF at(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Here, AF at(x) can be computed analytically using the following lemma: Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Let UCBt(x) = (u1, u2) and LCBt(ˆΠt) = {(l(i) 1 , l(i) 2 ) | 1 ≤ i ≤ k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Then, at(x) can be computed as follows: ˜at(x) = min 1≤i≤k max{u1 − l(i) 1 , u2 − l(i) 2 }, at(x) = max{˜at(x), 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Notably, when the number of objective functions is m ≥ 3, at(x) is easily extended as follows: ˜at(x) = min 1≤i≤k′ max{u1 − l(i) 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' , um − l(i) m }, at(x) = max{˜at(x), 0}, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='2) where UCBt(x) = (u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' , um) and LCBt(ˆΠt) = {(l(i) 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' , l(i) m ) | 1 ≤ i ≤ k′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' The proofs of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='1 and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='2) are presented in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' From Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='1, once LCBt(ˆΠt) is computed, the maximum value of at(x) can be analytically obtained by performing 2|X| times inf calculations and computing u(F (1)) t (x) and u(F (2)) t (x) for all x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' On the other hand, an AF based on exact posterior distributions of target functions such as the expected hypervolume improvement [Emmerich, 2005] for ordinary Pareto optimization, requires approximation by sampling from f (1)(x, w) and f (2)(x, w) under this problem setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' However, in each posterior sample, the inf calculation must be performed again to calculate F (1)(x) and F (2)(x) for all design variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Therefore, if the number of Monte Carlo samples is M, M times more inf calculations are required compared to the proposed AF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' The comparison of computational times is given in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Stopping Condition Here, we describe the stopping conditions of the proposed algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' 2 (c), AF at(x) represents the closeness of the pessimistic PF and the optimistic predictive value of F (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' That is, if this value is sufficiently small, there is little room for improvement in the PF;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' therefore, it is reasonable to use it as the stopping condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Let ϵ > 0 be a user-specified parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Then the algorithm is terminated if at(x) ≤ ϵ is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' The pseudocode for the proposed algorithm is given in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Theoretical Analysis Here, we provide the theorems for the accuracy and convergence of the proposed algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' The details of the proofs are presented in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' First,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' to provide theoretical results,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' we assume that f (1) and f (2) follow 5 Algorithm 1 BO for identifying DR-PF Input: GP priors GP(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' k(1)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' GP(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' k(2)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' parameter ξ ≥ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' tradeoff parameters {β1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='t}t≥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' {β2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='t}t≥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' stopping parameter ϵ > 0 t ← 1 while at(x) > ϵ do Compute Q(F (1)) t (x) and Q(F (2)) t (x) for each x ∈ X Select the next evaluation point (xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' wt) Observe y(1) t = f (1)(xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' wt) + ε(1) t and y(2) t = f (2)(xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' wt) + ε(2) t at the point (xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' wt) Update GPs by adding observed points t ← t + 1 end while Output: Return ˆΠt as the estimated set of design variables comprising the DR-PF GPs GP(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' k(1)((x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' w),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' (x′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' w′))) and GP(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' k(2)((x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' w),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' (x′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' w′))),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Moreover, we assume that the prior variances k(1)((x, w), (x, w)) ≡ σ(1)2 0 (x, w) and k(2)((x, w), (x, w)) ≡ σ(2)2 0 (x, w) satisfy max (x,w)∈X×Ω σ(1)2 0 (x, w) ≤ 1, max (x,w)∈X×Ω σ(2)2 0 (x, w) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Furthermore, let κ(1) T and κ(2) T be the maximum information gains of f (1) and f (2) at time T, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Notably, the maximum information gain is a measure often used in theoretical analyses of GP-based BO (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=', [Srinivas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=', 2010]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Here, for each j ∈ {1, 2}, using the mutual information I(y(j);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' f (j)) between y(j) and f (j), κ(j) T can be expressed as κ(j) T = max A⊂X×Ω I(y(j) A ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' f (j)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Next, to quantify the goodness of the predicted ˆΠt, we define an ϵ-accurate Pareto region Zϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' With user-specified positive numbers ϵ and ϵ = (ϵ, ϵ), we define Zϵ as Zϵ = {y ∈ R2 |∃ y′ ∈ Z∗ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' y ⪯ y′ and ∃y′′ ∈ Z∗ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' y′′ ⪯ y + ϵ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' That is, Zϵ is the set of points that lie inside Z∗ and within ϵ in the sense of d∞(·, ·)-distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' The concept of Zϵ was also used in [Zuluaga et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=', 2016].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Using Zϵ, we define the accuracy of ˆΠt in terms of the following two aspects: Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='1 (Accuracy for ˆΠt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Let ϵ be a positive value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' We then define ˆΠt as an ϵ-accurate estimated solution set if the following holds: F (ˆΠt) ⊂ Zϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='1) Moreover, we define ˆΠt as an ϵ-accurate estimated Pareto solution set if the following holds: Par(F (ˆΠt)) ⊂ Zϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='2) It is easy to obtain a set that satisfies either (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='1) or (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Generally, by ignoring F (2)(x) and focusing only on the maximization of F (1)(x), the maximization point x∗ can be estimated using methods such as [Kirschner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=', 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Subsequently, by letting ˆΠt = {x∗}, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='1) is satisfied with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Similarly, if we predict that all points constitute the PF, that is, ˆΠt = X, then (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='2) is satisfied because Par(F (ˆΠt)) = Par(F (X)) = Z∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' The following theorem guarantees that the proposed algorithm satisfies both (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='1) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='2) with a high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Let t ≥ 1 and δ ∈ (0, 1) and define β1,t = β2,t = 2 log(2|X × Ω|π2t2/(6δ)) ≡ βt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' In addition, let ϵ > 0 be a user-specified stopping parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Then, when Algorithm 1 terminates, ˆΠt satisfies (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='1) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='2) with a probability of at least 1 − δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='1 does not indicate whether the algorithm terminates or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' The following theorem guarantees the convergence of Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Under the same conditions as those in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='1, let T be the smallest positive integer that satisfies the following inequality: βT (C1κ(1) T + C2κ(2) T ) T ≤ ϵ2 4 , where C1 = 2/ log(1+σ(1)−2 noise ) and C2 = 2/ log(1+σ(2)−2 noise ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Then, Algorithm 1 terminates after at most T trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='35 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='Simulator setting ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='iteration ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='R1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='0 ' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='0 0 20 40 60 80 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='0 Random UCB_F1 UCB_F2 MVA EHI Proposed 0 20 40 60 80 100 0 10 20 30 40 SIR (case2) iteration R2 0 20 40 60 80 100 0 10 20 30 40 0 20 40 60 80 100 0 10 20 30 40 0 20 40 60 80 100 0 10 20 30 40 0 20 40 60 80 100 0 10 20 30 40 0 20 40 60 80 100 0 10 20 30 40 Random UCB_F1 UCB_F2 MVA EHI Proposed Figure 4: Average values of R1 and R2 for each method in Case1 and Case2 in SIR model experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' The length of each error bar represents twice the standard error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Table 1: Computational time (second) and the ratios of the computational time to that of the proposed method Random UCB F1 UCB F2 MVA EHI Proposed Computational time 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='068 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='067 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='135 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='139 (Standard error) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='000) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='000) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='000) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='001) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='011) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='001) Computational time ratio 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='496 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='488 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='985 118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='68 1 (Standard error) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='000) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='004) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='004) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='006) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='548) (0) Here, because the maximum information gains κ(1) T and κ(2) T are known to be sublinear with respect to T under mild assumptions [Srinivas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=', 2010], and the order of β1,T = β2,T is O(log T), the positive integer T satisfying the inequality in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='2 exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' We emphasize that Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='1 also holds in the use phase setting and a similar theorem holds for Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='2 under mild additional conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Moreover, by appropriately designing the family of candidate distributions using the empirical distribution as the reference distribution, the proposed method provides an arbitrarily accurate solution for the expected PF based on the true distribution, even when the true distribution is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Details are provided in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Numerical Experiments Here, we confirm the performance of the proposed method using synthetic functions and real-world simulation models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' In this experiment, we used a one-dimensional design variable x and environmental variable w and the following Gaussian kernels: k(1)((x, w), (x′, w′)) = σ2 f,1 exp(−∥ν − ν′∥2 2/L1), k(2)((x, w), (x′, w′)) = σ2 f,2 exp(−∥ν − ν′∥2 2/L2), where ν = (x, w) and ν′ = (x′, w′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Moreover, we used the L1-norm as the distance between distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' In all experiments except computational time experiments, we used the following two indicators R1 and R2 based on (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='1) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='2) to evaluate the goodness of ˆΠt estimated by each method: R1 = inf{a ∈ R | F (ˆΠt) ⊂ Za}, R2 = inf{a ∈ R | Par(F (ˆΠt)) ⊂ Za}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Experimental details and additional experiments, which are not included in the main body, are described in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' 7 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Synthetic Function We evaluate the performance in our proposed method through synthetic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' First, the input space X × Ω was a set of 50 × 50 grid points equally spaced in [−10, 10] × [−10, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' In this experiment, we used the following (scaled) Himmelblau’s function f (1)(x, w), which is commonly used as a benchmark function in BO studies [Andrei, 2008], and sinusoidal function f (2)(x, w) as black-box functions: f (1)(x, w) = (x2 + w − 11)2 150 + (x + w2 − 7)2 150 − C, f (2)(x, w) = (80 sin(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='5x) − 50 cos(2w))/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='5, where C = 3321.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='291/150 is a constant to set the mean to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Under this setting, we compared the following six methods: Random: Determine the next evaluation point xt+1 at random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' UCB F1: Select the next evaluation point by xt+1 = argmaxx∈X u(F (1)) t (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' UCB F2: Select the next evaluation point by xt+1 = argmaxx∈X u(F (2)) t (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' MVA: Select the next evaluation point by xt+1 = argmaxx∈Mt∪ˆΠt λt(x), where Mt and λt(x) are given in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='2 of [Iwazaki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=', 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' EHI: Select the next evaluation point xt+1 by maximizing an expected hypervolume improvement for the DR-PF calculated based on posterior means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Proposed: Select the next evaluation point xt+1 by Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Here, UCB F1 (or UCB F2) focuses on the maximization of F (1)(x) (or F (2)(x)) and does not consider the identification of the DR-PF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' In contrast, MVA focuses on reducing the uncertainty of a potential optimal set, which is a set of input points that may constitute the DR-PF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' The EHI method is the strategy that extends the expected hypervolume improvement strategy, which is commonly used in BO for ordinary Pareto optimization problems, to the DR-PF identification problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Because the expected hypervolume improvement for the DR-PF cannot be calculated analytically, we approximated it using Monte Carlo sampling with a sample size of 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Experiments were conducted under the following two settings for the observation of w: Simulator setting: At each time t, arbitrary w can be selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Uncontrollable setting: At each time t, w cannot be selected and is observed as a random sample from the uniform distribution on Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' The Simulator setting and Uncontrollable setting correspond to the development phase and use phase, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' In Simulator setting, we used p∗(w) = 1/50, and the next environmental variable to be evaluate for each method except Random was selected by wt+1 = argmax w∈Ω (σ(1)2 t (xt+1, w) + σ(2)2 t (xt+1, w)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' In Random, wt+1 was selected as a random sample from the uniform distribution on Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' In Uncontrollable setting, we allowed the use of a different reference distribution p∗ t (w) for each iteration t and used the empirical distribution of w as p∗ t (w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Under this setup, one initial point was taken at random and the algorithm was run until the number of iterations reached 500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' This simulation was repeated 100 times and the average values of R1 and R2 at each iteration were calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' 3, it can be confirmed that R1 and R2 in Random are not zero even after 500 iterations for both settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' In UCB F1 and UCB F2, the value of R1 is good but the value of R2 is not good because they focus on one of the black-box functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' For MVA, EHI, and Proposed, which focus on improving the DR-PF, R1 and R2 tend to be zero in both settings, but Proposed converges to zero more quickly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Computational Time Experiments We confirmed the computational time required to select xt+1 and wt+1 using each method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' We performed the same experiment as in Simulator setting in the previous section to evaluate the computational time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Under this setup, one initial point was taken at random and the algorithm was run until the number of iterations reached 500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' We computed the average computational time over 500 iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' We also computed the ratio of the computational time of each method to that of the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' From Table 1, it can be confirmed that Random, which does not require inf calculations, is faster than the proposed method, and UCB F1 (or 8 UCB F2), which uses only u(F (1)) t (x) (or u(F (2)) t (x)) required inf calculations, is about half the computational time of the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' The proposed method and MVA using both u(F (1)) t (x) and u(F (2)) t (x) have comparable computational speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' On the other hand, EHI, which performs the same number of inf calculations for each Monte Carlo sample as the proposed method requires, takes about 100 times longer than the proposed method because the number of Monte Carlo samples is 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Infection Simulation We applied the proposed method to the Pareto optimization problem using a simulation model of a real- world infectious disease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' We used the SIR model [Kermack and McKendrick, 1927], which is commonly used as the infection simulation model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' In this experiment, we used the SIR model which has the infection rate β ∈ [0, 1] and the recovery γ ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' The input space X × Ω was defined as the set of grid points when the region [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='5] × [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='5] was equally divided into 50 × 50 grid points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Using the SIR model, we defined the following two risk functions which can be interpreted as economic risks: r1(β, γ) = n(β, γ) − 450β + 800γ − C1, r2(β, γ) = n(β, γ) − C2, where n(β, γ) is the maximum number of infected individuals during a given period, calculated using the SIR model, and C1 and C2 are constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Notably, r1(β, γ) and r2(β, γ) were also used in [Inatsu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=', 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' In this experiment, we used the same parameter setting as them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' To adapt it to our problem setup, we multiplied them by minus one because risk functions should be minimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Because β and γ can be interpreted as both design variables and environmental variables, we considered the following two cases: Case1: f (1)(x, w) = −r1(x, w) and f (2)(x, w) = −r2(x, w), where x and w are the infection rate and recovery rate, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Case2: f (1)(x, w) = −r1(x, w) and f (2)(x, w) = −r2(x, w), where x and w are the recovery rate and infection rate, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' In this experiment, we considered Simulator setting as in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Under this setup, one initial point was taken at random and the algorithm was run until the number of iterations reached 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' This simulation was repeated 100 times and the average values of R1 and R2 at each iteration were calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' 4, it can be confirmed that the proposed method achieves equal or better performance in all settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Conclusion In this study, we proposed an efficient BO method for identifying the DR-PF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' We proved that the proposed method has theoretical guarantees on accuracy and convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Furthermore, through numerical experiments, we confirmed that the proposed method outperforms other comparative methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Future work includes extend- ing the method to the case where w is a continuous random variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Acknowledgement This work was partially supported by MEXT KAKENHI (20H00601), JST CREST (JPMJCR21D3, JP- MJCR21D3), JST Moonshot R&D 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Multi- objective bayesian optimization using pareto-frontier entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' In International Conference on Machine Learn- ing, pages 9279–9288.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' PMLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' [Toscano-Palmerin and Frazier, 2018] Toscano-Palmerin, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' and Frazier, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Bayesian optimization with expensive integrands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' arXiv preprint arXiv:1803.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='08661.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' [Williams and Rasmussen, 2006] Williams, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' and Rasmussen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Gaussian processes for machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' the MIT Press, 2(3):4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' [Zuluaga et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=', 2016] Zuluaga, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=', Krause, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=', and P¨uschel, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' ε-pal: an active learning approach to the multi-objective optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' The Journal of Machine Learning Research, 17(1):3619–3650.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' 10 Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Generalization of Problem Setup Here, we consider a more general problem setting, including the setting introduced in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Specif- ically, we consider the following three settings: The case with three or more objective functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Environment variables cannot be controlled even during optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' The setting where reference distributions, control parameter ξ, and candidate distribution family A set differently at each time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Furthermore, by combining (ii) and (iii), we show that under appropriate assumptions, the solution to the distributionally robust Pareto optimization problem is also a good solution to the Pareto optimization problem defined by the true expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Extended Problem Setting Let f (1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' , f (m) : X × Ω → R be expensive-to-evaluate black-box functions, where 2 ≤ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Also let X and Ω be finite sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Here, for each (x, w) ∈ X ×Ω and j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' , m} ≡ [m], the value of f (j)(x, w) is observed with Gaussian noise ε(j) as y(j) = f (j)(x, w) + ε(j), where ε(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' , ε(m) are mutually independent, and ε(j) follows Normal distribution with mean zero and variance σ(j)2 noise, that is, ε(j) ∼ N(0, σ(j)2 noise).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' In this section, we assume that w ∈ Ω is a discrete random variable and follows some unknown distribution P †.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Here, for environmental variables, we consider either settings in the development phase or settings in the use phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' That is, in the former, environmental variables are completely controllable as design variables;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' in the latter, environmental variables are uncontrollable and observed as realizations from the true distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Furthermore, let At denote the candidate distribution family of P † at each time t, and consider the following At: At = {p(w) | d(p(w), p∗ t (w)) ≤ ξt}, where p∗ t (w) is a user-specified reference distribution, d(·, ·) is a given distance function between distribu- tions, and ξt ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Then, for each design variable x ∈ X and time t, the distributionally robust expectations F (1) t (x), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' , F (m) t (x) are defined as follows: F (j) t (x) = inf p(w)∈At � w∈Ω f (j)(x, w)p(w), j ∈ [m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Hereafter, we aim to efficiently identify the PF Z∗ t determined by F (1) t (x), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' , F (m) t (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' For each x ∈ X, subset E ⊂ X and time t, let Ft(x) = (F (1) t (x), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' , F (m) t (x)) and Ft(E) = {Ft(x) | x ∈ E}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Moreover, for a set B ⊂ Rm, the domain Dom(B) dominated by B and the Pareto-frontier Par(B) of B are given by Dom(B) = {y ∈ Rm |∃ y′ ∈ B s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' y ⪯ y′}, Par(B) = ∂(Dom(B)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Then, the PF Z∗ t defined by F (1) t (x), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' , F (m) t (x) can be expressed as follows: Z∗ t = Par(Ft(X)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Gaussian Process Next, we construct predictive models for the black-box functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' As in the main body, GPs are used to model the black-box functions f (1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' , f (m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' First, for each j ∈ [m], assume that f (j) follows a GP prior GP(0, k(j)((x, w), (x′, w′))), where k(j)((x, w), (x′, w′)) is a positive-definite kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Then, under the given dataset {(xi, wi, y(j) i )}t i=1, the posterior distribution of f (j) is again a GP, and its posterior mean µ(j) t (x, w) and posterior variance σ(j)2 t (x, w) are given by µ(j) t (x, w) = k(j) t (x, w)⊤(K(j) t + σ(j)2 noiseIt)−1y(j) t , σ(j)2 t (x, w) = k(j)((x, w), (x, w)) − k(j) t (x, w)⊤(K(j) t + σ(j)2 noiseIt)−1k(j) t (x, w), where k(j) t (x, w) is the t-dimensional vector whose kth element is k(j)((x, w), (xk, wk)), y(j) t = (y(j) 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' , y(j) t )⊤, K(j) t is the t × t matrix whose (k, l)th element is k(j)((xk, wk), (xl, wl)) and It is the t × t identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' 11 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Proposed Method in the Generalized Setting Here, we propose a BO method for efficiently identifying Z∗ t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Using the same argument as the method used in the main body, we construct credible intervals interval for F (1) t (x), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' , F (m) t (x) using the method proposed by [Kirschner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=', 2020], and give an estimation method for the PF based on the constructed credible intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Composition of Credible Intervals and Pareto-frontier Estimation For each (x, w) ∈ X ×Ω, j ∈ [m] and time t, let Q(f (j)) t (x, w) = [l(f (j)) t (x, w), u(f (j)) t (x, w)] be a credible interval of f (j)(x, w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Here, l(f (j)) t (x, w) and u(f (j)) t (x, w) are given by l(f (j)) t (x, w) = µ(j) t (x, w) − β1/2 j,t σ(j) t (x, w), u(f (j)) t (x, w) = µ(j) t (x, w) + β1/2 j,t σ(j) t (x, w), where β1/2 j,t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Then, the credible interval for F (j) t (x) is denoted as Q(F (j) t ) t (x) ≡ [l(F (j) t ) t (x), u(F (j) t ) t (x)], where its lower and upper are given by l (F (j) j ) t (x) = inf p(w)∈At � w∈Ω l(f (j)) t (x, w)p(w), u (F (j) j ) t (x) = inf p(w)∈At � w∈Ω u(f (j)) t (x, w)p(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Next, for each x ∈ X, subset E ⊂ X and time t, we define LCB(m) t (x), UCB(m) t (x) and LCB(m) t (E) as LCB(m) t (x) = (l(F (1) t ) t (x), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' , l(F (m) t ) t (x)), UCB(m) t (x) = (u(F (1) t ) t (x), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' , u(F (m) t ) t (x)), LCB(m) t (E) = {LCB(m) t (x) | x ∈ E}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Using this, we define the estimated Pareto solutions set ˆΠ(m) t ⊂ X for design variables as ˆΠ(m) t = {x ∈ X | LCB(m) t (x) ∈ Par(LCB(m) t (X))}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Hereafter, for simplicity, we denote LCB(m) t (x), UCB(m) t (x), LCB(m) t (E) and ˆΠ(m) t as LCBt(x), UCBt(x), LCBt(E) and ˆΠt, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Acquisition Function Here, we propose an AF to determine the next evaluation point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Similar to the main body, we define the AF at(x) for x ∈ X as at(x) = dist(UCBt(x), Dom(LCBt(ˆΠt))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Then, the next evaluation point is selected as follows: Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='1 (For the setting in the development phase).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' The next design variable xt+1 to be evaluated is selected by xt+1 = argmax x∈X at(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Similarly, the next environmental variable wt+1 to be evaluated is selected by wt+1 = argmax w∈Ω {σ(1)2 t (xt+1, w) + · · · + σ(m)2 t (xt+1, w)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Notably, since wt+1 cannot be selected in the setting at the use phase, wt+1 is the realized value from P † at time t + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Here, at(x) can be computed analytically by the following lemma: Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Let UCBt(x) = (u1, , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' , um) and LCBt(ˆΠt) = {(l(i) 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' , l(i) m ) | 1 ≤ i ≤ k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Then, at(x) can be computed as follows: at(x) = max{˜at(x), 0}, ˜at(x) = min 1≤i≤k max{u1 − l(i) 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' , um − l(i) m }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Stopping Condition Here, we give a stopping condition of our proposed algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' As in the main body, let ϵ > 0 be a user- specified stopping parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Then, algorithms terminate if at(x) ≤ ϵ is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Finally, the pseudo-codes of the proposed algorithm in the development phase and use phase settings are given in Algorithm 2 and 3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' 12 Algorithm 2 BO for identifying DR-PF in the development phase setting Input: GP prior GP(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' k(j)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' candidate distribution family At,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' tradeoff parameter {βj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='t}t≥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' stopping param- eter ϵ > 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' j ∈ [m] t ← 1 while at(x) > ϵ do Compute Q(F (j) t ) t (x) for each x ∈ X and j ∈ [m] Select the next evaluation point (xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' wt) Observe y(j) t = f (j)(xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' wt) + ε(j) t for each j ∈ [m] Update GPs by adding observed points t ← t + 1 end while Output: Return ˆΠt as the estimated set of design variables comprising the DR-PF Algorithm 3 BO for identifying DR-PF in the use phase setting Input: GP prior GP(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' k(j)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' candidate distribution family At,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' tradeoff parameter {βj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='t}t≥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' stopping param- eter ϵ > 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' j ∈ [m] t ← 1 while at(x) > ϵ do Compute Q(F (j) t ) t (x) for each x ∈ X and j ∈ [m] Select the next evaluation point xt Generate wt from P † Observe y(j) t = f (j)(xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' wt) + ε(j) t for each j ∈ [m] Update GPs by adding observed points t ← t + 1 end while Output: Return ˆΠt as the estimated set of design variables comprising the DR-PF A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Theoretical Analysis Here, we give theorems on the accuracy and convergence of the proposed algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' First, to give theoretical guarantees, we assume that for each j ∈ [m], f (j) follows GP GP(0, k(j)((x, w), (x′, w′))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' In addition, we assume that each prior variance k(j)((x, w), (x, w)) ≡ σ(j)2 0 (x, w) satisfies max (x,w)∈X×Ω σ(j)2 0 (x, w) ≤ 1, ∀j ∈ [m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Furthermore, let κ(j) T be a maximum information gain for f (j) at time T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Moreover, as in the main body, we define an ϵ-accurate Pareto region Zϵ,t to quantify the goodness of the predicted ˆΠt as input points that constitute the PF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' For a positive number ϵ and the m-dimensional vector ϵ = (ϵ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' , ϵ), we define Zϵ,t as Zϵ,t = {y ∈ Rm |∃ y′ ∈ Z∗ t s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' y ⪯ y′ and ∃y′′ ∈ Z∗ t s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' y′′ ⪯ y + ϵ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Using Zϵ,t, we define the accuracy of ˆΠt as follows: Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='2 (Accuracy for ˆΠt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Let ϵ be a positive number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Then, we define ˆΠt to be an ϵ-accurate estimated solution set if the following holds: Ft(ˆΠt) ⊂ Zϵ,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='1) In addition, we define ˆΠt to be an ϵ-accurate estimated Pareto solution set if the following holds: Par(Ft(ˆΠt)) ⊂ Zϵ,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='2) Then, the following theorem guarantees that the proposed algorithms satisfy both (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='1) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='2) with a high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Let t ≥ 1 and δ ∈ (0, 1), and define β1,t = · · · = βm,t = 2 log(m|X × Ω|π2t2/(6δ)) ≡ βt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Moreover, let ϵ > 0 be a user-specified stopping parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Then, when Algorithm 2 terminates, with a probability of at least 1 − δ, ˆΠt satisfies both (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='1) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='2) for any At.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Next, we give a theorem on convergence in the development phase setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' The following theorem gives convergence guarantees for Algorithm 2: 13 Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Under the same condition as in Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='1, let T be the smallest positive integer satisfying the following inequality: βT (C1κ(1) T + · · · + Cmκ(m) T ) T ≤ ϵ2 4 , where Cj = 2/ log(1 + σ(j)−2 noise ) and j ∈ [m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Then, Algorithm 2 terminates after at most T iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Next, we give the theorem on convergence under the setting in the use phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Unlike the setting in the development phase, we cannot control w in the use phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Therefore, to make a reasonable inference, the uncertainty at all points must be able to be reduced stochastically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' However, if the value of the true probability function p†(w) at some w ∈ Ω is zero, the uncertainty of f (j)(x, w) containing this point cannot be sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' To avoid this problem, we make the following assumption on the true probability function: min w∈Ω p†(w) ≡ pmin > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Then, the following theorem holds: Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Under the same condition as in Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='1, assume that pmin > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Let T be the smallest positive integer satisfying the following inequality: βT ( ˜C1κ(1) T + · · · + ˜Cmκ(m) T + ˜C) T ≤ ϵ2 4 , where ˜Cj = (4p−1 min)/ log(1 + σ(j)−2 noise ), ˜C = 8mp−1 min log(8m/δ) and j ∈ [m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Then, with a probability of at least 1 − δ, Algorithm 3 terminates after at most T trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Finally, we show that under appropriate assumptions, the solution to the distributionally robust Pareto optimization problem is also a good solution to the Pareto optimization problem defined by the true expectation in the use phase setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Here, the expectation function ˜F (j)(x) determined by the true probability function p†(w) is given by ˜F (j)(x) = � w∈Ω f (j)(x, w)p†(w), j ∈ [m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' In addition, for each x ∈ X and E ⊂ X, let ˜F (x) = ( ˜F (1)(x), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' , ˜F (m)(x)) and ˜F (E) = { ˜F (x) | x ∈ E}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Then, the PF ˜Z∗ defined by ˜F (1)(x), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' , ˜F (m)(x) can be expressed as follows: ˜Z∗ = Par( ˜F (X)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Furthermore, as in the case of Z∗ t , we define an ϵ-accurate Pareto region ˜Zϵ for ˜Z∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' For a positive number ϵ and an m-dimensional vector ϵ = (ϵ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' , ϵ), we define ˜Zϵ as ˜Zϵ = {y ∈ Rm |∃ y′ ∈ ˜Z∗ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' y ⪯ y′ and ∃y′′ ∈ ˜Z∗ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' y′′ ⪯ y + ϵ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Using ˜Zϵ, we define the accuracy of ˆΠt for ˜Z∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='3 (Accuracy of ˆΠt for ˜Z∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Let ϵ be a positive number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Then, we define ˆΠt to be an ϵ-accurate estimated solution set for ˜Z∗ if the following holds: Ft(ˆΠt) ⊂ ˜Zϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Moreover, we define ˆΠt to be an ϵ-accurate estimated Pareto solution set for ˜Z∗ if the following holds: Par(Ft(ˆΠt)) ⊂ ˜Zϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Then, the following theorem holds: Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Under the use phase setting, let t ≥ 1 and δ ∈ (0, 1), and define β1,t = · · · = βm,t = 2 log(m|X × Ω|π2t2/(6δ)) ≡ βt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Moreover, let ϵ > 0 be a user-specified stopping parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Furthermore, let the reference distribution p∗ t (w) be the empirical distribution function for w, and define ξt and the distance between distributions d(·, ·) as ξt = |Ω| � 1 2t log �|Ω|π2t2 3δ � , d(p1(w), p2(w)) = � w∈Ω |p1(w) − p2(w)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Then, if Algorithm 3 terminates at t ≥ T after time T, with a probability of at least 1−2δ, ˆΠt is the 2ϵ-accurate estimated set and estimated Pareto set, that is, the following holds: Ft(ˆΠt) ⊂ ˜Z2ϵ, Par(Ft(ˆΠt)) ⊂ ˜Z2ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Here, T is the smallest positive integer satisfying the following inequality: ∀n ≥ T, 2β1/2 1 ξn ≤ ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='3) 14 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Proofs Here, we give proofs of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='1 and Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='1–A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Notably, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='1 and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='1–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='2 in the main body are special cases of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='1 and Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='1–A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' First, we prove Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Let UCBt(x) = (u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' , um) ≡ u and LCBt(ˆΠt) = {(l(i) 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' , l(i) m ) | 1 ≤ i ≤ k} ≡ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Here, if u ∈ Dom(L), then the following holds from the definition of dist(a, B): at(x) = dist(u, Dom(L)) = inf b∈Dom(L) d∞(u, b) = d∞(u, u) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' In addition, since u ∈ Dom(L), there exists (l(i) 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' , l(i) m ) such that uj ≤ l(i) j for any j ∈ [m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Thus, we have max{u1 − l(i) 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' , um − l(i) m } ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' This implies that ˜at(x) = min 1≤i≤k max{u1 − l(i) 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' , um − l(i) m } ≤ 0 and max{˜at(x), 0} = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Therefore, we get at(x) = max{˜at(x), 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Next, we consider the case where u /∈ Dom(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Let at(x) = η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Then, noting that u /∈ Dom(L), for any i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' , k}, there exists j ∈ [m] such that uj > l(i) j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' This implies that ˜at(x) = min 1≤i≤k max{u1 − l(i) 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' , um − l(i) m } ≡ ˜η > 0 and max{˜at(x), 0} = ˜at(x) = ˜η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' For this ˜η, there exists i such that uj − l(i) j ≤ ˜η ∀j ∈ [m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Hence, we have ˜u ≡ (u1 − ˜η, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' , um − ˜η) ∈ Dom(L) because uj − ˜η ≤ l(i) j for any j ∈ [m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Thus, from the definition of at(x), the following holds: η = at(x) = dist(u, Dom(L)) = inf b∈Dom(L) d∞(u, b) ≤ d∞(u, ˜u) = ˜η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Here, we assume η < ˜η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Then, noting that Dom(L) is the closed set, there exists ˜l = (˜l1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' , ˜lm) ∈ Dom(L) such that d∞(u, ˜l) = η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Therefore, ˜l can be expressed as ˜l = (u1 − s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' , um − sm), where 0 ≤ |sj| ≤ η and at least one of s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' , sm is η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Thus, since (u1 − η, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' , um − η) ⪯ ˜l, noting that (u1 − η, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' , um − η) ∈ Dom(L) there exists i such that uj − η ≤ l(i) j ∀j ∈ [m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' This implies that max{u1 − l(i) 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' , um − l(i) m } ≤ η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Hence, it follows that ˜η = min 1≤i≤k max{u1 − l(i) 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' , um − l(i) m } ≤ η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' However, this is a contradiction with η < ˜η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Consequently, we obtain at(x) = max{˜at(x), 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Next, we prove Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' First, we prove Ft(ˆΠt) ⊂ Zϵ,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Since ˆΠt ⊂ X, for any y ∈ Ft(ˆΠt), there exists y′ ∈ Z∗ t such that y ⪯ y′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Thus, it is sufficient to show that there exists y′′ ∈ Z∗ t such that y′′ ⪯ y +ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Here, under the theorem’s assumption, with a probability of at least 1 − δ, the following holds for any (x, w) ∈ X × Ω, j ∈ [m] and time t ≥ 1: f (j)(x, w) ∈ Q(f (j)) t (x, w), where this relation can be derived by using Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='1 of [Srinivas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=', 2010].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Hence, we have F (j) t (x) ∈ Q(F (j) t ) t (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' In addition, since Z∗ t is the closed set, for any x ∈ ˆΠt, there exist a ≥ 0 and Ft(x) + (a, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' , a) ≡ y′′ ∈ Z∗ t such that dist(Ft(x), Z∗ t ) = d∞(Ft(x), y′′) ≤ d∞((l(F (1) t ) t (x), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' , l(F (m) t ) t (x)), y′′′), where y′′′ ∈ Z∗ t can be given by using s ≥ a ≥ 0 as y′′′ = (l(F (1) t ) t (x), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' , l(F (m) t ) t (x)) + (s, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' , s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Here, for some ˆx ∈ X satisfying y′′′ ⪯ Ft(ˆx), the following holds: d∞((l(F (1) t ) t (x), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' , l(F (m) t ) t (x)), y′′′) ≤ dist(Ft(ˆx), Dom(LCBt(ˆΠt))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' 15 Moreover, the right hand side is bounded from above as dist(Ft(ˆx), Dom(LCBt(ˆΠt))) ≤ dist(UCBt(ˆx), Dom(LCBt(ˆΠt))) ≤ max x†∈X dist(UCBt(x†), Dom(LCBt(ˆΠt))) = at(xt+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Therefore, if at(xt+1) ≤ ϵ, then d∞(Ft(x), y′′) ≤ ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' It follows that y′′ ⪯ Ft(x) + ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Since x is an arbitrary element of ˆΠt, we have Ft(ˆΠt) ⊂ Zϵ,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Next, we prove Par(Ft(ˆΠt)) ⊂ Zϵ,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' As before, noting that ˆΠt ⊂ X, for any y ∈ Par(Ft(ˆΠt)), there exists y′ ∈ Z∗ t such that y ⪯ y′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' In addition, since Z∗ t is the closed set, for any y ∈ Par(Ft(ˆΠt)), there exist a ≥ 0 and y + (a, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' , a) ≡ y′′ ∈ Z∗ t such that dist(y, Z∗ t ) = d∞(y, y′′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Here, for some ˆx ∈ X satisfying y′′ ⪯ Ft(ˆx), the following holds: d∞(y, y′′) ≤ d∞(y′′′, Ft(ˆx)) ≤ dist(Ft(ˆx), Dom(LCBt(ˆΠt))), where y′′′ ∈ Par(Ft(ˆΠt)) can be given by using s′ ≥ a ≥ 0 as y′′′ = Ft(ˆx) − (s′, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' , s′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Then, we obtain dist(Ft(ˆx), Dom(LCBt(ˆΠt))) ≤ dist(UCBt(ˆx), Dom(LCBt(ˆΠt))) ≤ max x†∈X dist(UCBt(x†), Dom(LCBt(ˆΠt))) = at(xt+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Thus, if at(xt+1) ≤ ϵ, then d∞(y, y′′) ≤ ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' This implies that y′′ ⪯ y + ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Consequently, since y is an arbitrary element of Par(Ft(ˆΠt)), we have Par(Ft(ˆΠt)) ⊂ Zϵ,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Next, we prove Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Let xt = argmaxx∈X at−1(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Here, since LCB t−1(xt) ∈ Dom(LCB t−1(ˆΠt−1)), the following in- equality holds: at−1(xt) = dist(UCBt−1(xt), Dom(LCB t−1(ˆΠt−1))) ≤ d∞(UCB t−1(xt), LCB t−1(xt)) = max 1≤j≤m{u (F (j) t−1) t−1 (xt) − l (F (j) t−1) t−1 (xt)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Therefore, from the definition of u (F (j) t−1) t−1 (xt) and l (F (j) t−1) t−1 (xt), we get u (F (j) t−1) t−1 (xt) − l (F (j) t−1) t−1 (xt) ≤ 2β1/2 t max w∈Ω σ(j) t−1(xt, w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Hence, the following inequality holds: a2 t−1(xt) ≤ 4βt max 1≤j≤m max w∈Ω σ(j)2 t−1 (xt, w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Furthermore, since wt is selected by wt = argmax w∈Ω (σ(1)2 t−1 (xt, w) + · · · + σ(m)2 t−1 (xt, w)), the following holds: a2 t−1(xt) ≤ 4βt max 1≤j≤m max w∈Ω σ(j)2 t−1 (xt, w) ≤ 4βt(σ(1)2 t−1 (xt, wt) + · · · + σ(m)2 t−1 (xt, wt)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' In addition, let T be the number given by Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Then, we get T min 1≤t≤T a2 t−1(xt) ≤ T � t=1 a2 t−1(xt) ≤ 4βT m � j=1 T � t=1 σ(j)2 t−1 (xt, wt) ≤ 4βT (C1κ(1) T + · · · + Cmκ(m) T ), 16 where the last inequality can be derived by using Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='3 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='4 of [Srinivas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=', 2010].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Therefore, dividing both sides by T, we obtain min 1≤t≤T a2 t−1(xt) ≤ 4βT C1κ(1) T + · · · + Cmκ(m) T T ≤ ϵ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Hence, we get min1≤t≤T at−1(xt) ≤ ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' This implies that there exists t′ ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' , T} such that at′−1(xt′) ≤ ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Next, we prove Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Using the same argument as in the proof of Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='2, we have a2 t−1(xt) ≤ 4βt max 1≤j≤m max w∈Ω σ(j)2 t−1 (xt, w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Furthermore, since pmin > 0, the following inequality holds: max w∈Ω σ(j)2 t−1 (xt, w) ≤ � wΩ σ(j)2 t−1 (xt, w) = � wΩ (p†(w))−1p†(w)σ(j)2 t−1 (xt, w) ≤ p−1 min � wΩ p†(w)σ(j)2 t−1 (xt, w) ≡ p−1 minEw[σ(j)2 t−1 (xt, w)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Using this, we get max 1≤j≤m max w∈Ω σ(j)2 t−1 (xt, w) ≤ m � j=1 max w∈Ω σ(j)2 t−1 (xt, w) ≤ p−1 min m � j=1 Ew[σ(j)2 t−1 (xt, w)] = p−1 minEw � � m � j=1 σ(j)2 t−1 (xt, w) � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Here, let T be the number given by Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Then, we obtain T min 1≤t≤T a2 t−1(xt) ≤ T � t=1 a2 t−1(xt) ≤ 4βT p−1 min T � t=1 Ew � � m � j=1 σ(j)2 t−1 (xt, w) � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='1) Noting that �m j=1 σ(j)2 t−1 (xt, w) is a non-negative random variable and �m j=1 σ(j)2 t−1 (xt, w) ≤ m, from Lemma 3 of [Kirschner and Krause, 2018], the following holds with a probability of at least 1 − δ: T � t=1 Ew � � m � j=1 σ(j)2 t−1 (xt, w) � � ≤ 2 m � j=1 T � t=1 σ(j)2 t−1 (xt, w) + 4m log 1 δ + 8m log 4m + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Moreover, since 4m log 1 δ ≤ 8m log 1 δ and 1 ≤ 8m log 2, we have 4m log 1 δ + 8m log 4m + 1 ≤ 8m log 1 δ + 8m log 4m + 8m log 2 = 8m log 8m δ and T � t=1 Ew � � m � j=1 σ(j)2 t−1 (xt, w) � � ≤ 2 m � j=1 T � t=1 σ(j)2 t−1 (xt, w) + 8m log 8m δ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='2) 17 Hence, by substituting (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='2) into (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='1), from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='3 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='4 of [Srinivas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=', 2010], we get T min 1≤t≤T a2 t−1(xt) ≤ 4βT ( ˜C1κ(1) T + · · · + ˜Cmκ(m) T + ˜C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Therefore, from the definition of T, dividing both sides by T, we obtain min 1≤t≤T a2 t−1(xt) ≤ 4βT ˜C1κ(1) T + · · · + ˜Cmκ(m) T + ˜C T ≤ ϵ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Thus, we get min1≤t≤T at−1(xt) ≤ ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' This implies that there exists t′ ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' , T} such that at′−1(xt′) ≤ ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Finally, we prove Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Suppose that p∗ t (w) is the empirical distribution function of w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Then, from Hoeffding’s inequality, the following inequality holds for any w: P(|p∗ t (w) − p†(w)| ≥ λ) ≤ 2 exp(−2tλ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Here, let λ = � 1 2t log �|Ω|π2t2 3δ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Then, with a probability of at least 1 − δ, the following holds for any t ≥ 1 and w ∈ Ω: |p∗ t (w) − p†(w)| ≤ λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' In addition, from the definition of d(·, ·), we get d(p∗ t (w), p†(w)) = � w∈Ω |p∗ t (w) − p†(w)| ≤ |Ω|λ = ξt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' This implies that p†(w) ∈ At.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Furthermore, from the definition of ˜F (j)(x) and F (j) t (x), the following inequality holds for any t ≥ 1, j ∈ [m] and x ∈ X: F (j) t (x) ≤ ˜F (j)(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Therefore, it follows that Ft(x) ⪯ ˜F (x) ∀x ∈ X,∀ t ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='3) Here, for any x ∈ X, t ≥ 1 and j ∈ [m], let ¯p(j) t,x(w) ∈ At be the probability function satisfying F (j) t (x) = � w∈Ω f (j)(x, w)¯p(j) t,x(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Then, the following inequality holds: | ˜F (j)(x) − F (j) t (x)| ≤ � w∈Ω |f (j)(x, w)||p†(w) − ¯p(j) t,x(w)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Moreover, from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='1 of [Srinivas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=', 2010], with a probability of at least 1 − δ, the following holds for any x ∈ X, w ∈ Ω and j ∈ [m]: |f (j)(x, w)| ≤ β1/2 1 σ(j) 0 (x, w) ≤ β1/2 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Hence, we have ˜F (j)(x) − F (j) t (x) ≤ | ˜F (j)(x) − F (j) t (x)| ≤ β1/2 1 � w∈Ω |p†(w) − ¯p(j) t,x(w)| = β1/2 1 d(p†(w), ¯p(j) t,x(w)) ≤ β1/2 1 (d(p†(w), p∗ t (w)) + d(p∗ t (w), ¯p(j) t,x(w))) ≤ 2β1/2 1 ξt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' In addition, let T be the smallest positive integer satisfying (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Then, for any t ≥ T, the following inequality holds: ˜F (j)(x) ≤ F (j) t (x) + ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' 18 Thus, we obtain ˜F (x) ⪯ Ft(x) + ϵ ∀x ∈ X,∀ t ≥ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='4) Therefore, by combining (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='3) and (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='4), we have Par(Ft(X)) ⊂ ˜Zϵ ∀t ≥ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='5) Finally, from Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='1, the following holds at t′, the time at which the algorithm terminates: Ft′(ˆΠt′) ⊂ Zϵ,t′, Par(Ft′(ˆΠt′)) ⊂ Zϵ,t′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='6) Consequently, if t′ ≥ T, using (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='5) and (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='6) we get Ft′(ˆΠt′) ⊂ ˜Z2ϵ, Par(Ft′(ˆΠt′)) ⊂ ˜Z2ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Experimental Details and Additional Experiments Here, we give experimental details and additional experiments in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Experimental Setup The experimental parameters used in each experiment are described in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Table 2: Experimental parameters for each setting Parameters Simulator setting σ2 f,1 = 1000, L1 = 2, σ(1)2 noise = 10−4, β1/2 1,t = 3, σ2 f,2 = 1000, L2 = 2, σ(2)2 noise = 10−4, β1/2 2,t = 3, ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='05 Uncontrollable setting σ2 f,1 = 1000, L1 = 2, σ(1)2 noise = 10−4, β1/2 1,t = 3, σ2 f,2 = 1000, L2 = 2, σ(2)2 noise = 10−4, β1/2 2,t = 3, ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='05 SIR (Case1) σ2 f,1 = 5000, L1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='1, σ(1)2 noise = 10−8, β1/2 1,t = 3, σ2 f,2 = 105, L2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='01, σ(2)2 noise = 10−4, β1/2 2,t = 2, ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='15 SIR (Case2) σ2 f,1 = 104, L1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='1, σ(1)2 noise = 10−3, β1/2 1,t = 2, σ2 f,2 = 105, L2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='1, σ(2)2 noise = 10−3, β1/2 2,t = 3, ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='15 MVA The MVA method is based on reducing the uncertainty in the potential optimal set Mt and the estimated PF solution set ˆΠt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Using ˆΠt, Mt is defined as follows: Mt = {x ∈ X \\ ˆΠt |∀ x′ ∈ ˆΠt, u(F (1)) t (x) > l(F (1)) t (x′) or u(F (2)) t (x) > l(F (2)) t (x′)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' The uncertainty λt(x) is given by λt(x) = � (u(F (1)) t (x) − l(F (1)) t (x))2 + (u(F (2)) t (x) − l(F (2)) t (x))2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' EHI The EHI method is based on the expected hypervolume improvement for a bounded region defined by PFs and reference points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Let B ⊂ R2 be a set, and let r = (r1, r2) ∈ R2 be a reference point satisfying r ⪯ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Then, let us denote the bounded region dominated by B and r, by Dom(B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' r) = Dom(B) ∩ [r1, ∞) × [r2, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' In EHI, for each j ∈ {1, 2} we calculated the estimated value of F (j)(x) by using posterior means as µ(F (j)) t (x) = inf p(w)∈A � w∈Ω µ(j) t (x, w)p(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' For a point x ∈ X a subset E ⊂ X, we define µ(F ) t (x) and µ(F ) t (E) as µ(F ) t (x) = (µ(F (1)) t (x), µ(F (2)) t (x)), µ(F ) t (E) = {µ(F ) t (x) | x ∈ E}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' In our experiments, we defined the reference point rt = (rt,1, rt,2) for each iteration t as rt,1 = min x∈X µ(F (1)) t (x), rt,2 = min x∈X µ(F (2)) t (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Then, the expected hypervolume improvement for x ∈ X is given by EHIt(x) = EF (1)(x),F (2)(x)[Vol(Dom(µ(F ) t (X) ∪ {(F (1)(x), F (2)(x))};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' rt) \\ Dom(µ(F ) t (X);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' rt))].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' 19 Table 3: Computational time (second) and computational time ratio for each setting when Nx = 50 and Nw = 100 Random UCB F1 UCB F2 MVA EHI Proposed Computational time 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='181 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='185 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='375 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='77 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='364 (Standard error) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='000) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='001) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='001) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='001) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='025) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='001) Computational time ratio 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='510 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='031 126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='15 1 (Standard error) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='000) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='002) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='002) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='003) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='327) (0) Table 4: Computational time (second) and computational time ratio for each setting when Nx = 100 and Nw = 50 Random UCB F1 UCB F2 MVA EHI Proposed Computational time 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='134 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='135 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='272 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='268 (Standard error) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='000) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='001) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='001) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='001) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='015) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='001) Computational time ratio 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='503 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='507 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='019 119.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='92 1 (Standard error) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='000) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='002) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='002) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='004) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='346) (0) Table 5: Computational time (second) and computational time ratio for each setting when Nx = 100 and Nw = 100 Random UCB F1 UCB F2 MVA EHI Proposed Computational time 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='360 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='362 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='719 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='82 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='726 (Standard error) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='000) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='001) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='001) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='002) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='099) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='002) Computational time ratio 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='496 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='991 127.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='02 1 (Standard error) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='000) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='001) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='001) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='002) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='412) (0) Here, for a bounded set A, Vol(A) represents the hypervolume of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Because F (1)(x) and F (2)(x) do not follow GPs, we cannot calculate EHIt(x) analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Thus, we approximate it by using samples from posterior distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Let M be a number of Monte Carlo sampling, and let (f (j) t,(l)(x, w1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' , f (j) t,(l)(x, w|Ω|)) be an lth sample from the posterior distribution of (f (j)(x, w1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' , f (j)(x, w|Ω|)) at iteration t, where 1 ≤ l ≤ M, j ∈ {1, 2} and x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Then, for each t, we calculate F (1) t,(l)(x) and F (2) t,(l)(x) as F (1) t,(l)(x) = inf p(w)∈A � w∈Ω f (1) t,(l)(x, w)p(w), F (2) t,(l)(x) = inf p(w)∈A � w∈Ω f (2) t,(l)(x, w)p(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Using this, we approximate EHIt(x) as 1 M M � l=1 Vol(Dom(µ(F ) t (X) ∪ {(F (1) t,(l)(x), F (2) t,(l)(x))};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' rt) \\ Dom(µ(F ) t (X);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' rt)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' Additional Computational Time Experiments We also compared the computational time of each method by changing input space settings conducted in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' In this experiment, the input space X × Ω was a set of grid points divided into [−10, 10] × [−10, 10] equally spaced at Nx × Nw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' We compared computational times using (50, 100), (100, 50) and (100, 100) as (Nx, Nw).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' From Table 3–5, it can be confirmed that the results are similar to the experimental results conducted in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} +page_content=' 20' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFJT4oBgHgl3EQfmywg/content/2301.11588v1.pdf'} diff --git a/8tE1T4oBgHgl3EQfUAOX/content/tmp_files/2301.03085v1.pdf.txt b/8tE1T4oBgHgl3EQfUAOX/content/tmp_files/2301.03085v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..6c17b390880f647a525339075221209867315281 --- /dev/null +++ b/8tE1T4oBgHgl3EQfUAOX/content/tmp_files/2301.03085v1.pdf.txt @@ -0,0 +1,726 @@ +Granger causality test for heteroskedastic and +structural-break time series using generalized least squares +Hugo J. Bello ∗ +January 10, 2023 +Abstract +This paper proposes a novel method (GLS Granger test) to determine causal relation- +ships between time series based on the estimation of the autocovariance matrix and gener- +alized least squares. We show the effectiveness of proposed autocovariance matrix estimator +(the sliding autocovariance matrix) and we compare the proposed method with the classical +Granger F-test with via a synthetic dataset and a real dataset composed by cryptocurren- +cies. The simulations show that the proposed GLS Granger test captures causality more +accurately than Granger F-tests in the cases of heteroskedastic or structural-break residu- +als. Finally, we use the proposed method to unravel unknown causal relationships between +cryptocurrencies. +Keywords: +Granger Test, Causality, Generalized Least Squares, Wald Test. +1 +Introduction +Granger causality is a statistical concept that helps us determine whether the information in +one time series is useful in predicting another. It is widely used in numerous fields, including +economics, finance, geology, genetics and neuroscience, to understand the relationships between +variables and to identify possible causal connections. +The main tool for studying time series causality is the Granger causality F-test ([Gra69], +[Gra80]) which is based on ordinary least squares (OLS) estimation and therefore is tied to the +assumptions of OLS (among them homocedasticity of the residuals) +In many real-world applications, we are confronted with time series suffering from a variety +of problems such as heteroscedasticity (non-equal variance amongst time observations) or struc- +tural breaks (the appearance of two or more stationary periods with different means). In these +situations OLS is not recommended and Granger F-test can be inadequate. +The aim of this paper is to present a method for determining Granger causality based on +generalized least squares based on the estimation of the autocovariance matrix of the residuals. +1.1 +Preliminaries +In this section we will present introducing notation and definitions. We also review the notion +of Granger causality. +∗hugojose.bello@uva.es Department of Applied Mathematics, University of Valladolid (Campus Soria) +1 +arXiv:2301.03085v1 [stat.ME] 8 Jan 2023 + +1.1.1 +Granger causality F-test +The Granger causality ([Gra69], [Gra80]) test is a statistical hypothesis test for determining +whether one time series x = (xt)N +t=0 is useful in forecasting another y = (yt)N +t=0. In particular +Granger causality focuses on the possibility of x and y to predict future values of y significatively +better than those of y alone. In this case it is said that x Granger causes y +The idea behind this is that a cause should be helpful in predicting the future effects, beyond +what can be predicted solely based on their own past values. +The test null hypothesis states that the linear regression model +yt = β0 + +p +� +k=1 +βkyt−k + εt +(1) +approximates y significatively better than the model +yt = β0 + +p +� +k=1 +βkyt−k + +p +� +k=1 +β′ +kxt−k + εt +(2) +If the null hypothesis is correct, this will imply that the lagged values of x add explanatory +power to the prediction of y and therefore the process behind x causes y. +To test if model (2) is significatively more accurate than (1) the following statistic is used +(SSRRM − SSRUM)/N +SSEUM/(N − 2)(p − 1) +(3) +where SSRRM and SSRUM are the sum of residuals for the restricted and unrestricted models +respectively +SSRRM = Σt(yt − β0 − +p +� +k=1 +βkyt−k)2 +SSRUM = Σt(yt − β0 − +p +� +k=1 +βkyt−k − +p +� +k=1 +β′ +kxt−k)2 +If the necessary assumptions for ordinary least of squares are satisfied, (3) follows a F(p, N − +2p − 1) distribution under the null hypothesis. +Observation 1. Notice that 1 can be understood as writing y as a linear combination of the +lagged time series Bky (using the backward operatior notation), that is: +y = β0 + +L +� +k=1 +βkBky + ε +Similarly 1 +y = β0 + +L +� +k=1 +βkBky + +L +� +k=1 +βkBkx + ε +Therefore Granger causality can be understood as the use of a F-test to compare two multi- +linear regression models. +2 + +1.1.2 +Generalized Least Squares +For a linear regression model of the form +yi = β1 xi1 + β2 xi2 + · · · + βp xip + εi, +(4) +The following assumptions must be satisfied for ordinary least squares (OLS) to have the +desired asymptotic properties: +(P1) Correct specification. +Te underlying process generating the data must be in esence +linear. +(P2) Strict exogeneity. The errors in the regression must have conditional mean zero: E[ ε | X ] = 0.. +(P3) No linear dependence. The regressors must all be linearly independent. +(P4) Homoscedasticity E[ε2 +i |X] = σ2 ∀i. The error term has the same variance σ2 in each +observation. +(P5) No autocorrelation E[εiεj|X] = 0 ∀i ̸= j. The errors are uncorrelated between observa- +tions. +(P6) Normality. It is sometimes additionally assumed that the errors have normal distribution +conditional on the regressors +If homocedasticity (P4) or non-autocorrelation (P5) assumptions are not satisfied we can use +Generalized Least Squares (GLS) to better approximate the parameters of (4). +Using the standard matricial notation (4) can be written as +y = Xβ + ε +where y = (yi), X = (xT +1 . . . xT +p ) is the design matrix and ε is the error term. if the error +term satisfies that E[ε|X] = 0 and denoting cov[ε|X] = Ω the non-singular covariance matrix of +the residuals. the GLS estimate for β is +�βGLS = argminβ(y − Xβ)T Ω(y − Xβ) = +� +XTΩ−1X +�−1 XTΩ−1y +(5) +(see [Gre03, §9.3]) Notice that for Ω = σ2I (where I is the identity matrix) we are under the +assumptions of OLS and the resulting estimator is +�βOLS = +� +XTX +�−1 XTy +(6) +It is known that +E[�βGLS] = βGLS +(7) +cov[�βGLS] = V = (XT Ω−1X)−1 +(8) +In fact +√ +N(�βGLS − βGLS) −→D N(0, V ) +1.1.3 +Wald Test +The Wald test is a statistical hypothesis test that assesses constrains on statistical parameters +for regression models based on the weighted distance between an unrestricted estimate and its +hypothesized value under the null hypothesis (see [Gre03, §5.3]). +Let �β the sample estimate for the regression model (GLS or OLS) model with covariance V +as described before. If Q hypothesis on the p parameters are expressed in the form of a Q × p +matrix R: +3 + +H0 : Rβ = r +H1 : Rβ ̸= r +The wald test statistic is +(R�β − r)T · +� +R�V RT · 1 +n +�−1 +· (R�β − r) +(9) +Under the null hypothesis, the Wald statistic (9) follows a F(Q, N − p) distribution. +Granger F-test as a Wald Test +The Granger F-Test described in (3) is in fact a particular case of the Wald test. If we consider +�βOLS, then we can stablish the unrestricted regression model +y = β0 + β1By + . . . + βpBpy + β′ +1Bx + . . . + β′ +pBpx +(10) +Where B is the backshift operator Bkx = (xt−k)t. So for y to be caused by x the coefficients +β′ +1, . . . β′ +p must be zero, therefore we need to test the hypothesis +H0 :β′ +k = 0 for all k ≤ p +(11) +H1 :β′ +k ̸= 0 for some k ≤ p +Defining the matrices +R = +� +� +� +� +� +� +� +� +� +� +0 +... +0 +1 +... +1 +� +� +� +� +� +� +� +� +� +� +; β = +� +� +� +� +� +� +� +� +� +� +β1 +... +βp +β′ +1 +... +β′ +p +� +� +� +� +� +� +� +� +� +� +; r = +� +� +� +0 +... +0 +� +� +� +We can codify the test 11 as +H0 :Rβ = 0 +(12) +H1 :Rβ ̸= 0 +It can be shown [Gre03, §5.4], that with this notation the corresponding wald statistic (9) in +fact coincides with the F-test statistic (3) +2 +Methodology +The Granger F-test (3) assumes that conditions (4) are satisfied. We aim to present a version +of the Granger test based on generalized least squares, for that we need �Ω, an estimate for the +covariance matrix cov[ε|X] = Ω. +4 + +In most cases Ω is not known and a reasonable approach is to use the βOLS to obtain the +residuals +rt(β) = yt − β0 − +p +� +k=1 +yt−kβk − +p +� +k=1 +xt−kβ′ +k +And attempt to estimate Ω using the covariance matrix of rt. This procedure is often called +feasible least squares. +The difficulty lies in the fact that the covariance matrix of a time series (which is often called +autocovariance matrix) is not known in general. To overcome this problem in many cases rt is +assumed to follow a known model such as AR(1), whose theoretical autocovariance matrix is +known and can be obtained from the model parameters. This approach is very restrictive since +in general rt can take many forms, for this reason we will first tackle the following problem: +Problem 2. Given a time series x = (xt) how can we estimate the covariance matrix Ω = +(cov(xt, xt′))t,t′? +Since in general the previous problem can be really difficult to tackle we will impose certain +assumptions. +We will focus on the following realm of very general time series: the locally +jointly-stationary which as we see admit a convenient estimation of their covariance (which +we will call the the sliding autocovariance matrix) +2.1 +Locally jointly-stationarity and the sliding autocovariance matrix +Definition 3. A time series x = (xt) is locally jointly-stationary if there exists an increas- +ing sequence of time instances 0 < t1 < t2 < . . . < tn (called time breaks) such that each +subsequence +x(k) = xtk:tk+1 = {xt : t ∈ [tk, tk+1]} +is stationary and jointly-stationary with respect to the rest of the subsequences. +Recall that two time series x, y are jointly-stationary if they satisfy cov(xt, yt) = cov(xt+h, yt+h) +Example 4. Stationary time series are locally jointly-stationary. This is very easy to verify +since every subseries of a stationary time series will be cross stationary with any other subseries. +One can consider any instance T and the initial subseries x0:T is trivially cross stationary with +the rest of the series by definition. +Example 5. A time series (xt) is called stationary with structural breaks if it satisfies +xt = α + δDt + εt +where +Dt = +� +1 +if t ≥ TB + 1 +0 +otherwise +for α, δ ∈ R and εt stationary. These time series were introduced by Perron (see [Per89] and +[LS01]). +Property 6. Stationary time series with structural breaks are locally jointly-stationary. +5 + +Proof. Consider the subsampled time series at = α + εt defined for values of t between 0 and Tb, +and bt = α + δ + εt. Since εt is stationary, this two time series are jointly-stationary: +cov(at1, bt2) = cov(α + εt1, α + δ + εt1) += cov(εt1, εt2) = cov(εt1+h, εt2+h) += cov(α + εt1+h, α + δ + εt1+h) += cov(at1+h, bt2+h) +Therefore x satisfies definition 3, taking the partition 0 ≤ t ≤ Tb and Tb < t. +Definition 7. Given a time series x = (xt), we define the window of length τ at t0 as +wτ(x, t0) = xt0:t0−τ = {xt : t ∈ [t0 − τ, t0]} +Definition 8. Let x = (xt) be a time series, and two time instants t1, t2. Consider the windows +of length τ +w1 = wτ(x, t1) +w2 = wτ(x, t2) +We will call the windowed sample autocovariance of length τ at t1, t2 to +�γτ(t, t′) = �γw1w2(t, t′) +(13) += 1 +τ +τ +� +k=0 +(w1 +k − w1)(w2 +k − w2) +(14) += 1 +τ +τ +� +k=0 +(xt−k − w1)(xt′−k − w2) +(15) +which coincides with the sample cross-covariance 1 between the time series wτ(x, t) and +wτ(x, t′) +Property 9. Let x = (xt) be a locally jointly-stationary time series with time breaks 0 < t1 < +t2 < . . . < tn. Given t, t′, taking +τ = argmin +1≤m≤n +T =t,t′ +|T − tm| +the windowed sample autocovariance of length τ is an estimator for cov(xt, xt′) +Proof. Suppose that the time break immediately lower that t is tm and that the one immediately +lower than t′ is tm′. We will consider first the case that tm and t′ +m are different. +Notice that (following the notation in 3), by the choice of τ +wτ(x, t, τ) = wτ(x(tm), t, τ) +wτ(x, t′, τ) = wτ(x(tm′), t′, τ) +1The sample cross-covariance of two (jointly-stationary) time series x and y is defined as �γxy(h) = �(xt+h − +x)(yt+h − y). See example 1.23 [SSS00] +6 + +Therefore, �γτ(t, t′), the windowed sample autocovariance of length τ coincides with the sample +cross covariance of the previous two subseries windows wτ(x(tm), t, τ), wτ(x(tm′), t′, τ). +Since the subsequences x(tm) and x(tm′) are jointly-stationary, �γτ(t, t′) estimates the covari- +ance +cov(wτ(x(tm), t, τ), wτ(x(tm′), t′, τ)) +which must coincide with cov(x(tm) +t +, x(tm′) +t′ +) = cov(xt, xt′). +If tm = t′ +m then in the previous argument x(tm) = x(tm′) and the same consequence follows +using the stationarity of x(tm). +Observation 10. Notice that since the sample cross-covariance for jointly-stationary time se- +ries is a biased estimator, the windowed sample autocovariance �γτ(t, t′) is a biased estimator. +Nevertheless, in the case that the expected value E[x] is known, replacing the average by the +expected value in the formula the cross-covariance becomes unbiased and therefore the same +holds for �γτ(t, t′) in view of the previous proof. +Definition 11. The sliding autocovariance matrix of length τ is the N × N matrix Ωτ +defined thewindowed sample autocovariance �γτ(t, t′) for every pair of time instances, that is +Ωτ = (�γτ(t, t′))t,t′ +(16) +By prop. 9 Ωτ is an estimator for the covariance matrix Ω in the case that the series x is +locally jointly-stationary. By prop. 6 if x is stationary with structural breaks, Ωτ estimates Ω. +Observation 12. Notice that in (16) for low values of t, t′ the estimation �γτ(t, t′))t,t′ becomes +imprecise due to the small number of values before. One way to fix this problem in certain +situations is to complete the time series with values before 0 using x−t = xt. +Example 13. Consider time series xt = φ1xt−1 + εt with φ1 = 0.9. Since the time series follows +an AR(1) model, we know that the autocovariance +cov(xt, xt+h) = φh +1 · var(xt) +(17) +Figure 1 shows a simulation of this time seres at the top, the theoretical covariance matrix +using the previous formula 17 is shown on the left side and the sliding covariance matrix estimated +using (16) is shown on the left. For the imprecision around low values we used the procedure +described in obs. 12. To calculate the autocovariance matrix we used the value τ = N/3 where +N is the sample size for the time series, lower values of τ produce similar estimations. +7 + +Figure 1: Autocovariance matrix and estimated sliding autocovariance matrix. +2.2 +Generalized least squares Granger causality test +Going back to Granger causality test, in this section we present a novel Granger causality test +based on wald tests and the estimation of the covariance matrix of the residuals via the sliding +autocovariance matrix. +Method 14 (GLS Granger Causality test). Given two time series x, y, in order to assess +whether x causes y with lag L, we follow the following procedure, which is a variation of the +classical Granger causality test: +1. Use OLS to obtain an estimate (βOLS, β′ +OLS) for the model +yt = β0 + +p +� +k=1 +βkyt−k + +p +� +k=1 +β′ +kxt−k + εt +(18) +2. Using the residuals of the previous model +rt = yt − β0 − +p +� +k=1 +βkyt−k − +p +� +k=1 +β′ +kxt−k +estimate their covariance matrix using the sliding autocovariance matrix Ωτ as in (16) +8 + +at=0.9at-1+Et +0.06 +0.04 +0.02 +0.00 +0.02 +0.04 +0.06 +0 +200 +400 +600 +800 +1000 +0 +0 +0.0004 +200 +200 +0.0002 +400 +400 +0.0000 +600 +600 +-0.0002 +800 +800 +-0.0004 +0 +200 +400 +600 +800 +0 +200 +400 +600 +800 +Theorethical autocovariancematrix +Estimatedautocovariancematrix3. Use GLS and the previous covariance matrix to estimate again the parameters (βGLS, β′ +GLS) +in the model (18) as in (5). +4. Use a Wald test with null hypothesis +H0 : β′ +GLS k = 0 for all k ≤ p +If the null hypothesis is rejected conclude that x causes y otherwise conclude the opposite. +3 +Results +We proceed now to assess the efficacy of the proposed GLS Granger causality test (14) in com- +parison with the classical Granger F-Test (sec. 1.1.1). +3.1 +Simulated dataset +Given a time series x we applied the following procedure to consistently generate a caused series +y. The procedure consists of defining y in the following way +yt = +L +� +k=1 +xt−k · βk + εt +(19) +where βk are generated randomly and ε is a time series that can be constructed in several +ways depending in the type of causality that we want to simulate. We can consider: +(M1) ε stationary time series, for instance a white noise εt ∼ N(0, σ2). +(M2) ε is stationary with structural breaks, for instance considering εt =∼ N(µt, σ2) with µt = 0 +for t ≤ tb and µt = µ for t > tb. +(M3) ε non-stationary time series with changing variance for instance εt ∼ N(0, (t · σ)2). +On the other hand, to simulate non causality, we will simply simulate two time series x +and y by using auto-regressive processes with different parameters +xt = xt−1φ + εt +(AR1) +yt = yt−1φ′ + εt +Observation 15. Notice that a regression model applied to predict y from the lagged time +series Bky and Bky (using the backward operator as in obs. 1). The regression parameters will +approximate β1, . . . βL residuals of the regression will approximate ε. +With this in mind, (M1) will produce time series in which classical Granger F-test will be +very effective. In contrast (M2) will give us stationary with structural breaks residuals and (M3) +will produce heteroskedastic residuals, therefore the classical Granger F-test will be less effective +with these time series. +For this reason the introduced dataset (simulated using M1, M2 and M3) will be suitable for +comparing the proposed GLS Granger causality test with the classical Granger F-Test. +9 + +Example 16. In the following figure 2 we show three examples of the generated dataset using +(M1), (M2) and (M3). +Notice that in the graphs of figure 2 we can observe the causality, in the sense that changes +x (shown in blue) cause changes in y after a number of lags. +In the second graph the figure we see the structural break introduced in the residual of y +(M2). +Finally, in the last graph of the figure we appreciate the residual with growing variance +introduced by (M3) in y. +Figure 2: Three pairs of time series generated using the previous method (for L = 15 and random +β1, . . . , βL). In blue, the original time series x generated using an AR(1) process. In red the +caused time series y generated using the three methods (M1), (M2) and (M3) respectively. +Experiment results +We perform four experiments, each with 150 pairs of time series each with 600 points. The lag +used to simulate the causality is L = 15. +The first three experiments consist on testing the performance of Classical Granger against +the proposed GLS Granger by simulating causal relationships using methods (M1), (M2) and +(M3). For the sake of this comparison, we record the percentage of correct predictions by each +method. +The last experiment attempts to search for false positives. In this experiment we generate +non-caused time series using the method (AR1) described before. +Simulated causal +relationships +simulation +procedure +% of correct classical +Granger F-test +% of correct +GLS-Granger +y caused by x +(M1) stationary residual +75.0% +96.6% +y caused by x +(M2) structural breaks residual +57.3% +85.5% +y caused by x +(M3) heteroskedastic residual +32.6% +42.6% +y not caused by x +(AR1) +94.0% +94.7% +Table 1: Experiment results table +The window length τ used for the sliding autocovariance matrix estimation was τ = N/5 +where N is the number of observations. This value was obtained using cross-validation, but +greater values of τ produced very similar results. +In view of table 1, the proposed method gets more accurate results than the Granger F-test +in every one of the datasets simulated. +10 + +4 - +10 +3 - +2 - +-5 +-10 +-1 +-15 +-20 +0 +25 +50 +75 +100 +125 +150 +175 +200 +0 +25 +50 +100 +125 +150 +175 +200 +25 +50 +75 +100 +125 +150 +175 +2003.1.1 +Real dataset +Cryptocoins are known for their volatility and interdependence. Granger causality is a known tool +to study the interdependence of cryptocoins, for instance [KCP21] found a strong relationship +between Bitcoin (BTN) and Ethereum (ETH) and [Yav22] points out complex interdependence +between the main cryptocoins. +We will use a dataset composed by the values of the main 10 cryptocurrencies (Bitcoin,Ethereum, +Aave, BinanceCoin, Cardano, ChainLink, Cosmos, CryptocomCoin, Dogecoin, EOS, Iota, Lite- +coin, Monero) from July 2020 to July 2021. +The trend component of these time series was removed applying first order differentiation. +Even after differentiation a progressive change in variance is observed in the time series (see +figure 3). This suggests that even though in some cases the series pass a Augmented Dickey- +Fuller stationarity test, the OLS estimation performed in every Granger F-test will be imprecise +or problematic. This situation has many similarities to the simulation preformed in (M3), for +this reason our proposed method is more suitable to deal with the heteroskedastic behavior of +the residuals. (see sec. 3.1 and fig. 2). +Figure 3: Differentated data of the cryptocoin Ethereum from July 2020 to July 2021 +We applied a Granger F-test and our proposed GLS Granger test on each pair of cryptocoins +considered composing causals graphs, i.e. a graph that has as nodes all the time series and as +edges the causal relationships (if x causes y we draw x → y) +We used the lag L = 1, the optimal lag was obtained using Akaike Information Criterion +(AIC). +The result is shown in Figure 4. The left graph of 4 shows the Granger F-test causal graph, +whereas the right graph show the resulting GLS Granger Graph. We obtained a very connected +causal network as it is to expect from the behavior of cryptocurrencies. Interestingly, the pro- +posed method was able to capture more causal relationships, showing an even more connected +network. It also noteworthy that the GLS Granger graph shows many causal relationship that +connect the two leading cryptocoins Bitcoin and Ethereum with the rest of them. For instance +the proposed method finds the relations Ethereum → cardano, Bitcoin → EOS, Bitcoin → +ChainLink, Iota → Bitcoin. +11 + +600 +400 +200 +-200 +-400 +-600 +-800Figure 4: Causal graphs +4 +Conclusions and Future work +In this paper, we propose a generalization of the Granger F-test to uncover the temporal causal +structures from heteroskedastic and structural-breaks time series trough the estimation of the +residual autocovariance matrix and GLS. +We demonstrate its effectiveness on four simulation datasets and one real application dataset. +For future work, we are interested in researching other uses of the sliding covariance matrix +in the field of time series classification and machine learning. +Code availability +Datasets and scripts for this article are available at github: https://github.com/Granger-Causality- +GLS +References +[Gra69] +Clive WJ Granger, Investigating causal relations by econometric models and cross- +spectral methods, Econometrica: journal of the Econometric Society (1969), 424–438. +[Gra80] +, Testing for causality: a personal viewpoint, Journal of Economic Dynamics +and control 2 (1980), 329–352. +[Gre03] +William H Greene, Econometric analysis, Pearson Education India, 2003. +[KCP21] Myeong Jun Kim, Nguyen Phuc Canh, and Sung Y Park, Causal relationship among +cryptocurrencies: A conditional quantile approach, Finance Research Letters 42 (2021), +101879. +[LS01] +Junsoo Lee and Mark Strazicich, Testing the null of stationarity in the presence of a +structural break, Applied Economics Letters 8 (2001), no. 6, 377–382. +[Per89] +Pierre Perron, The great crash, the oil price shock, and the unit root hypothesis, Econo- +metrica: journal of the Econometric Society (1989), 1361–1401. +[SSS00] +Robert H Shumway, David S Stoffer, and David S Stoffer, Time series analysis and its +applications, vol. 3, Springer, 2000. +12 + +cosms +binance +cosmes +binance +opero +oherc +dogec +dogec新 +chainL +chainLink +ave +cryptocomcpin +cryptocomcin +etherum +etherum +cardao +ec +oin +cal +iota +ota +Titecotn +Classical Granger F-test causal graph +GLS Granger test causal graph[Yav22] +G¨UL Yavuz, Causality and cointegration in cryptocurrency markets, Uluslararası +˙Iktisadi ve ˙Idari ˙Incelemeler Dergisi (2022), no. 34, 129–142. +13 + diff --git a/8tE1T4oBgHgl3EQfUAOX/content/tmp_files/load_file.txt b/8tE1T4oBgHgl3EQfUAOX/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5aa630c56532ee9f1f120b4fcece5cdb17bb7e31 --- /dev/null +++ b/8tE1T4oBgHgl3EQfUAOX/content/tmp_files/load_file.txt @@ -0,0 +1,252 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf,len=251 +page_content='Granger causality test for heteroskedastic and structural-break time series using generalized least squares Hugo J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' Bello ∗ January 10, 2023 Abstract This paper proposes a novel method (GLS Granger test) to determine causal relation- ships between time series based on the estimation of the autocovariance matrix and gener- alized least squares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' We show the effectiveness of proposed autocovariance matrix estimator (the sliding autocovariance matrix) and we compare the proposed method with the classical Granger F-test with via a synthetic dataset and a real dataset composed by cryptocurren- cies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' The simulations show that the proposed GLS Granger test captures causality more accurately than Granger F-tests in the cases of heteroskedastic or structural-break residu- als.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' Finally, we use the proposed method to unravel unknown causal relationships between cryptocurrencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' Keywords: Granger Test, Causality, Generalized Least Squares, Wald Test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' 1 Introduction Granger causality is a statistical concept that helps us determine whether the information in one time series is useful in predicting another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' It is widely used in numerous fields, including economics, finance, geology, genetics and neuroscience, to understand the relationships between variables and to identify possible causal connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' The main tool for studying time series causality is the Granger causality F-test ([Gra69],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' [Gra80]) which is based on ordinary least squares (OLS) estimation and therefore is tied to the assumptions of OLS (among them homocedasticity of the residuals) In many real-world applications,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' we are confronted with time series suffering from a variety of problems such as heteroscedasticity (non-equal variance amongst time observations) or struc- tural breaks (the appearance of two or more stationary periods with different means).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' In these situations OLS is not recommended and Granger F-test can be inadequate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' The aim of this paper is to present a method for determining Granger causality based on generalized least squares based on the estimation of the autocovariance matrix of the residuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content='1 Preliminaries In this section we will present introducing notation and definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' We also review the notion of Granger causality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' ∗hugojose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content='bello@uva.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content='es Department of Applied Mathematics, University of Valladolid (Campus Soria) 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content='03085v1 [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content='ME] 8 Jan 2023 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content='1 Granger causality F-test The Granger causality ([Gra69], [Gra80]) test is a statistical hypothesis test for determining whether one time series x = (xt)N t=0 is useful in forecasting another y = (yt)N t=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' In particular Granger causality focuses on the possibility of x and y to predict future values of y significatively better than those of y alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' In this case it is said that x Granger causes y The idea behind this is that a cause should be helpful in predicting the future effects, beyond what can be predicted solely based on their own past values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' The test null hypothesis states that the linear regression model yt = β0 + p � k=1 βkyt−k + εt (1) approximates y significatively better than the model yt = β0 + p � k=1 βkyt−k + p � k=1 β′ kxt−k + εt (2) If the null hypothesis is correct, this will imply that the lagged values of x add explanatory power to the prediction of y and therefore the process behind x causes y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' To test if model (2) is significatively more accurate than (1) the following statistic is used (SSRRM − SSRUM)/N SSEUM/(N − 2)(p − 1) (3) where SSRRM and SSRUM are the sum of residuals for the restricted and unrestricted models respectively SSRRM = Σt(yt − β0 − p � k=1 βkyt−k)2 SSRUM = Σt(yt − β0 − p � k=1 βkyt−k − p � k=1 β′ kxt−k)2 If the necessary assumptions for ordinary least of squares are satisfied,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' (3) follows a F(p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' N − 2p − 1) distribution under the null hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' Observation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' Notice that 1 can be understood as writing y as a linear combination of the lagged time series Bky (using the backward operatior notation), that is: y = β0 + L � k=1 βkBky + ε Similarly 1 y = β0 + L � k=1 βkBky + L � k=1 βkBkx + ε Therefore Granger causality can be understood as the use of a F-test to compare two multi- linear regression models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content='2 Generalized Least Squares For a linear regression model of the form yi = β1 xi1 + β2 xi2 + · · · + βp xip + εi, (4) The following assumptions must be satisfied for ordinary least squares (OLS) to have the desired asymptotic properties: (P1) Correct specification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' Te underlying process generating the data must be in esence linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' (P2) Strict exogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' The errors in the regression must have conditional mean zero: E[ ε | X ] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content='. (P3) No linear dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' The regressors must all be linearly independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' (P4) Homoscedasticity E[ε2 i |X] = σ2 ∀i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' The error term has the same variance σ2 in each observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' (P5) No autocorrelation E[εiεj|X] = 0 ∀i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' The errors are uncorrelated between observa- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' (P6) Normality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' It is sometimes additionally assumed that the errors have normal distribution conditional on the regressors If homocedasticity (P4) or non-autocorrelation (P5) assumptions are not satisfied we can use Generalized Least Squares (GLS) to better approximate the parameters of (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' Using the standard matricial notation (4) can be written as y = Xβ + ε where y = (yi), X = (xT 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' xT p ) is the design matrix and ε is the error term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' if the error term satisfies that E[ε|X] = 0 and denoting cov[ε|X] = Ω the non-singular covariance matrix of the residuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' the GLS estimate for β is �βGLS = argminβ(y − Xβ)T Ω(y − Xβ) = � XTΩ−1X �−1 XTΩ−1y (5) (see [Gre03, §9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content='3]) Notice that for Ω = σ2I (where I is the identity matrix) we are under the assumptions of OLS and the resulting estimator is �βOLS = � XTX �−1 XTy (6) It is known that E[�βGLS] = βGLS (7) cov[�βGLS] = V = (XT Ω−1X)−1 (8) In fact √ N(�βGLS − βGLS) −→D N(0, V ) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content='3 Wald Test The Wald test is a statistical hypothesis test that assesses constrains on statistical parameters for regression models based on the weighted distance between an unrestricted estimate and its hypothesized value under the null hypothesis (see [Gre03, §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' Let �β the sample estimate for the regression model (GLS or OLS) model with covariance V as described before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' If Q hypothesis on the p parameters are expressed in the form of a Q × p matrix R: 3 H0 : Rβ = r H1 : Rβ ̸= r The wald test statistic is (R�β − r)T · � R�V RT · 1 n �−1 (R�β − r) (9) Under the null hypothesis, the Wald statistic (9) follows a F(Q, N − p) distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' Granger F-test as a Wald Test The Granger F-Test described in (3) is in fact a particular case of the Wald test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' If we consider �βOLS, then we can stablish the unrestricted regression model y = β0 + β1By + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' + βpBpy + β′ 1Bx + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' + β′ pBpx (10) Where B is the backshift operator Bkx = (xt−k)t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' So for y to be caused by x the coefficients β′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' β′ p must be zero, therefore we need to test the hypothesis H0 :β′ k = 0 for all k ≤ p (11) H1 :β′ k ̸= 0 for some k ≤ p Defining the matrices R = � � � � � � � � � � 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' 0 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' 1 � � � � � � � � � � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' β = � � � � � � � � � � β1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' βp β′ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' β′ p � � � � � � � � � � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' r = � � � 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' 0 � � � We can codify the test 11 as H0 :Rβ = 0 (12) H1 :Rβ ̸= 0 It can be shown [Gre03, §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content='4], that with this notation the corresponding wald statistic (9) in fact coincides with the F-test statistic (3) 2 Methodology The Granger F-test (3) assumes that conditions (4) are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' We aim to present a version of the Granger test based on generalized least squares, for that we need �Ω, an estimate for the covariance matrix cov[ε|X] = Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' 4 In most cases Ω is not known and a reasonable approach is to use the βOLS to obtain the residuals rt(β) = yt − β0 − p � k=1 yt−kβk − p � k=1 xt−kβ′ k And attempt to estimate Ω using the covariance matrix of rt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' This procedure is often called feasible least squares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' The difficulty lies in the fact that the covariance matrix of a time series (which is often called autocovariance matrix) is not known in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' To overcome this problem in many cases rt is assumed to follow a known model such as AR(1), whose theoretical autocovariance matrix is known and can be obtained from the model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' This approach is very restrictive since in general rt can take many forms, for this reason we will first tackle the following problem: Problem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' Given a time series x = (xt) how can we estimate the covariance matrix Ω = (cov(xt, xt′))t,t′?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' Since in general the previous problem can be really difficult to tackle we will impose certain assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' We will focus on the following realm of very general time series: the locally jointly-stationary which as we see admit a convenient estimation of their covariance (which we will call the the sliding autocovariance matrix) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content='1 Locally jointly-stationarity and the sliding autocovariance matrix Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' A time series x = (xt) is locally jointly-stationary if there exists an increas- ing sequence of time instances 0 < t1 < t2 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' < tn (called time breaks) such that each subsequence x(k) = xtk:tk+1 = {xt : t ∈ [tk, tk+1]} is stationary and jointly-stationary with respect to the rest of the subsequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' Recall that two time series x, y are jointly-stationary if they satisfy cov(xt, yt) = cov(xt+h, yt+h) Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' Stationary time series are locally jointly-stationary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' This is very easy to verify since every subseries of a stationary time series will be cross stationary with any other subseries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' One can consider any instance T and the initial subseries x0:T is trivially cross stationary with the rest of the series by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' A time series (xt) is called stationary with structural breaks if it satisfies xt = α + δDt + εt where Dt = � 1 if t ≥ TB + 1 0 otherwise for α, δ ∈ R and εt stationary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' These time series were introduced by Perron (see [Per89] and [LS01]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' Property 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' Stationary time series with structural breaks are locally jointly-stationary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' 5 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' Consider the subsampled time series at = α + εt defined for values of t between 0 and Tb, and bt = α + δ + εt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' Since εt is stationary, this two time series are jointly-stationary: cov(at1, bt2) = cov(α + εt1, α + δ + εt1) = cov(εt1, εt2) = cov(εt1+h, εt2+h) = cov(α + εt1+h, α + δ + εt1+h) = cov(at1+h, bt2+h) Therefore x satisfies definition 3, taking the partition 0 ≤ t ≤ Tb and Tb < t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' Given a time series x = (xt), we define the window of length τ at t0 as wτ(x, t0) = xt0:t0−τ = {xt : t ∈ [t0 − τ, t0]} Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' Let x = (xt) be a time series, and two time instants t1, t2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' Consider the windows of length τ w1 = wτ(x, t1) w2 = wτ(x, t2) We will call the windowed sample autocovariance of length τ at t1, t2 to �γτ(t, t′) = �γw1w2(t, t′) (13) = 1 τ τ � k=0 (w1 k − w1)(w2 k − w2) (14) = 1 τ τ � k=0 (xt−k − w1)(xt′−k − w2) (15) which coincides with the sample cross-covariance 1 between the time series wτ(x, t) and wτ(x, t′) Property 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' Let x = (xt) be a locally jointly-stationary time series with time breaks 0 < t1 < t2 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' < tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' Given t, t′, taking τ = argmin 1≤m≤n T =t,t′ |T − tm| the windowed sample autocovariance of length τ is an estimator for cov(xt, xt′) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' Suppose that the time break immediately lower that t is tm and that the one immediately lower than t′ is tm′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' We will consider first the case that tm and t′ m are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' Notice that (following the notation in 3), by the choice of τ wτ(x, t, τ) = wτ(x(tm), t, τ) wτ(x, t′, τ) = wτ(x(tm′), t′, τ) 1The sample cross-covariance of two (jointly-stationary) time series x and y is defined as �γxy(h) = �(xt+h − x)(yt+h − y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' See example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content='23 [SSS00] 6 Therefore, �γτ(t, t′), the windowed sample autocovariance of length τ coincides with the sample cross covariance of the previous two subseries windows wτ(x(tm), t, τ), wτ(x(tm′), t′, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' Since the subsequences x(tm) and x(tm′) are jointly-stationary, �γτ(t, t′) estimates the covari- ance cov(wτ(x(tm), t, τ), wτ(x(tm′), t′, τ)) which must coincide with cov(x(tm) t , x(tm′) t′ ) = cov(xt, xt′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' If tm = t′ m then in the previous argument x(tm) = x(tm′) and the same consequence follows using the stationarity of x(tm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' Observation 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' Notice that since the sample cross-covariance for jointly-stationary time se- ries is a biased estimator, the windowed sample autocovariance �γτ(t, t′) is a biased estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' Nevertheless, in the case that the expected value E[x] is known, replacing the average by the expected value in the formula the cross-covariance becomes unbiased and therefore the same holds for �γτ(t, t′) in view of the previous proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' Definition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' The sliding autocovariance matrix of length τ is the N × N matrix Ωτ defined thewindowed sample autocovariance �γτ(t, t′) for every pair of time instances, that is Ωτ = (�γτ(t, t′))t,t′ (16) By prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' 9 Ωτ is an estimator for the covariance matrix Ω in the case that the series x is locally jointly-stationary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' By prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' 6 if x is stationary with structural breaks, Ωτ estimates Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' Observation 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' Notice that in (16) for low values of t, t′ the estimation �γτ(t, t′))t,t′ becomes imprecise due to the small number of values before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' One way to fix this problem in certain situations is to complete the time series with values before 0 using x−t = xt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' Example 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' Consider time series xt = φ1xt−1 + εt with φ1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' Since the time series follows an AR(1) model, we know that the autocovariance cov(xt, xt+h) = φh 1 · var(xt) (17) Figure 1 shows a simulation of this time seres at the top, the theoretical covariance matrix using the previous formula 17 is shown on the left side and the sliding covariance matrix estimated using (16) is shown on the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' For the imprecision around low values we used the procedure described in obs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' To calculate the autocovariance matrix we used the value τ = N/3 where N is the sample size for the time series, lower values of τ produce similar estimations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' 7 Figure 1: Autocovariance matrix and estimated sliding autocovariance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content='2 Generalized least squares Granger causality test Going back to Granger causality test, in this section we present a novel Granger causality test based on wald tests and the estimation of the covariance matrix of the residuals via the sliding autocovariance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' Method 14 (GLS Granger Causality test).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' Given two time series x, y, in order to assess whether x causes y with lag L, we follow the following procedure, which is a variation of the classical Granger causality test: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' Use OLS to obtain an estimate (βOLS, β′ OLS) for the model yt = β0 + p � k=1 βkyt−k + p � k=1 β′ kxt−k + εt (18) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' Using the residuals of the previous model rt = yt − β0 − p � k=1 βkyt−k − p � k=1 β′ kxt−k estimate their covariance matrix using the sliding autocovariance matrix Ωτ as in (16) 8 at=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content='9at-1+Et 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content='06 0 200 400 600 800 1000 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content='0004 200 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content='0002 400 400 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content='0000 600 600 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content='0002 800 800 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content='0004 0 200 400 600 800 0 200 400 600 800 Theorethical autocovariancematrix Estimatedautocovariancematrix3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' Use GLS and the previous covariance matrix to estimate again the parameters (βGLS, β′ GLS) in the model (18) as in (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' Use a Wald test with null hypothesis H0 : β′ GLS k = 0 for all k ≤ p If the null hypothesis is rejected conclude that x causes y otherwise conclude the opposite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' 3 Results We proceed now to assess the efficacy of the proposed GLS Granger causality test (14) in com- parison with the classical Granger F-Test (sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content='1 Simulated dataset Given a time series x we applied the following procedure to consistently generate a caused series y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' The procedure consists of defining y in the following way yt = L � k=1 xt−k · βk + εt (19) where βk are generated randomly and ε is a time series that can be constructed in several ways depending in the type of causality that we want to simulate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' We can consider: (M1) ε stationary time series, for instance a white noise εt ∼ N(0, σ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' (M2) ε is stationary with structural breaks, for instance considering εt =∼ N(µt, σ2) with µt = 0 for t ≤ tb and µt = µ for t > tb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' (M3) ε non-stationary time series with changing variance for instance εt ∼ N(0, (t · σ)2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' On the other hand, to simulate non causality, we will simply simulate two time series x and y by using auto-regressive processes with different parameters xt = xt−1φ + εt (AR1) yt = yt−1φ′ + εt Observation 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' Notice that a regression model applied to predict y from the lagged time series Bky and Bky (using the backward operator as in obs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' The regression parameters will approximate β1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' βL residuals of the regression will approximate ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' With this in mind, (M1) will produce time series in which classical Granger F-test will be very effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' In contrast (M2) will give us stationary with structural breaks residuals and (M3) will produce heteroskedastic residuals, therefore the classical Granger F-test will be less effective with these time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' For this reason the introduced dataset (simulated using M1, M2 and M3) will be suitable for comparing the proposed GLS Granger causality test with the classical Granger F-Test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' 9 Example 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' In the following figure 2 we show three examples of the generated dataset using (M1), (M2) and (M3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' Notice that in the graphs of figure 2 we can observe the causality, in the sense that changes x (shown in blue) cause changes in y after a number of lags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' In the second graph the figure we see the structural break introduced in the residual of y (M2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' Finally, in the last graph of the figure we appreciate the residual with growing variance introduced by (M3) in y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' Figure 2: Three pairs of time series generated using the previous method (for L = 15 and random β1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' , βL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' In blue, the original time series x generated using an AR(1) process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' In red the caused time series y generated using the three methods (M1), (M2) and (M3) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' Experiment results We perform four experiments, each with 150 pairs of time series each with 600 points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' The lag used to simulate the causality is L = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' The first three experiments consist on testing the performance of Classical Granger against the proposed GLS Granger by simulating causal relationships using methods (M1), (M2) and (M3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' For the sake of this comparison, we record the percentage of correct predictions by each method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' The last experiment attempts to search for false positives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' In this experiment we generate non-caused time series using the method (AR1) described before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' Simulated causal relationships simulation procedure % of correct classical Granger F-test % of correct GLS-Granger y caused by x (M1) stationary residual 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content='0% 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content='6% y caused by x (M2) structural breaks residual 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content='3% 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content='5% y caused by x (M3) heteroskedastic residual 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content='6% 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content='6% y not caused by x (AR1) 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content='0% 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content='7% Table 1: Experiment results table The window length τ used for the sliding autocovariance matrix estimation was τ = N/5 where N is the number of observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' This value was obtained using cross-validation, but greater values of τ produced very similar results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' In view of table 1, the proposed method gets more accurate results than the Granger F-test in every one of the datasets simulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' 10 4 - 10 3 - 2 - 5 10 1 15 20 0 25 50 75 100 125 150 175 200 0 25 50 100 125 150 175 200 25 50 75 100 125 150 175 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content='1 Real dataset Cryptocoins are known for their volatility and interdependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' Granger causality is a known tool to study the interdependence of cryptocoins, for instance [KCP21] found a strong relationship between Bitcoin (BTN) and Ethereum (ETH) and [Yav22] points out complex interdependence between the main cryptocoins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' We will use a dataset composed by the values of the main 10 cryptocurrencies (Bitcoin,Ethereum, Aave, BinanceCoin, Cardano, ChainLink, Cosmos, CryptocomCoin, Dogecoin, EOS, Iota, Lite- coin, Monero) from July 2020 to July 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' The trend component of these time series was removed applying first order differentiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' Even after differentiation a progressive change in variance is observed in the time series (see figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' This suggests that even though in some cases the series pass a Augmented Dickey- Fuller stationarity test, the OLS estimation performed in every Granger F-test will be imprecise or problematic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' This situation has many similarities to the simulation preformed in (M3), for this reason our proposed method is more suitable to deal with the heteroskedastic behavior of the residuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' (see sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content='1 and fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' Figure 3: Differentated data of the cryptocoin Ethereum from July 2020 to July 2021 We applied a Granger F-test and our proposed GLS Granger test on each pair of cryptocoins considered composing causals graphs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' a graph that has as nodes all the time series and as edges the causal relationships (if x causes y we draw x → y) We used the lag L = 1, the optimal lag was obtained using Akaike Information Criterion (AIC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' The result is shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' The left graph of 4 shows the Granger F-test causal graph, whereas the right graph show the resulting GLS Granger Graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' We obtained a very connected causal network as it is to expect from the behavior of cryptocurrencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' Interestingly, the pro- posed method was able to capture more causal relationships, showing an even more connected network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' It also noteworthy that the GLS Granger graph shows many causal relationship that connect the two leading cryptocoins Bitcoin and Ethereum with the rest of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' For instance the proposed method finds the relations Ethereum → cardano, Bitcoin → EOS, Bitcoin → ChainLink, Iota → Bitcoin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' 11 600 400 200 200 400 600 800Figure 4: Causal graphs 4 Conclusions and Future work In this paper, we propose a generalization of the Granger F-test to uncover the temporal causal structures from heteroskedastic and structural-breaks time series trough the estimation of the residual autocovariance matrix and GLS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' We demonstrate its effectiveness on four simulation datasets and one real application dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' For future work, we are interested in researching other uses of the sliding covariance matrix in the field of time series classification and machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' Code availability Datasets and scripts for this article are available at github: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content='com/Granger-Causality- GLS References [Gra69] Clive WJ Granger, Investigating causal relations by econometric models and cross- spectral methods, Econometrica: journal of the Econometric Society (1969), 424–438.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' [Gra80] , Testing for causality: a personal viewpoint, Journal of Economic Dynamics and control 2 (1980), 329–352.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' [Gre03] William H Greene, Econometric analysis, Pearson Education India, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' [KCP21] Myeong Jun Kim, Nguyen Phuc Canh, and Sung Y Park, Causal relationship among cryptocurrencies: A conditional quantile approach, Finance Research Letters 42 (2021), 101879.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' [LS01] Junsoo Lee and Mark Strazicich, Testing the null of stationarity in the presence of a structural break, Applied Economics Letters 8 (2001), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' 6, 377–382.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' [Per89] Pierre Perron, The great crash, the oil price shock, and the unit root hypothesis, Econo- metrica: journal of the Econometric Society (1989), 1361–1401.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' [SSS00] Robert H Shumway, David S Stoffer, and David S Stoffer, Time series analysis and its applications, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' 3, Springer, 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' 12 cosms binance cosmes binance opero oherc dogec dogec新 chainL chainLink ave cryptocomcpin cryptocomcin etherum etherum cardao ec oin cal iota ota Titecotn Classical Granger F-test causal graph GLS Granger test causal graph[Yav22] G¨UL Yavuz, Causality and cointegration in cryptocurrency markets, Uluslararası ˙Iktisadi ve ˙Idari ˙Incelemeler Dergisi (2022), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' 34, 129–142.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} +page_content=' 13' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'} diff --git a/9dE0T4oBgHgl3EQfwwE1/content/tmp_files/2301.02636v1.pdf.txt b/9dE0T4oBgHgl3EQfwwE1/content/tmp_files/2301.02636v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..dd5196cd8ce7fe5d771373ec1e7519ef6e09a3a0 --- /dev/null +++ b/9dE0T4oBgHgl3EQfwwE1/content/tmp_files/2301.02636v1.pdf.txt @@ -0,0 +1,1293 @@ +CENTRAL H-SPACES AND BANDED TYPES +ULRIK BUCHHOLTZ, J. DANIEL CHRISTENSEN, JARL G. TAXER˚AS FLATEN, AND EGBERT RIJKE +Abstract. We introduce and study central types, which are generalizations of Eilenberg–Mac Lane +spaces. A type is central when it is equivalent to the component of the identity among its own self- +equivalences. From centrality alone we construct an infinite delooping in terms of a tensor product +of banded types, which are the appropriate notion of torsor for a central type. Our constructions are +carried out in homotopy type theory, and therefore hold in any ∞-topos. +Even when interpreted into the ∞-topos of spaces, our main results and constructions are new. +In particular, we give a description of the moduli space of H-space structures on an H-space which +generalizes a formula of Arkowitz–Curjel and Copeland which counts the number of path components +of this moduli space. +Contents +1. +Introduction +1 +2. +H-spaces and evaluation fibrations +3 +2.1. +H-space structures +3 +2.2. +Evaluation fibrations +7 +2.3. +Unique H-space structures +9 +3. +Central types +10 +4. +Bands and torsors +13 +4.1. +Types banded by a central type +13 +4.2. +Tensoring bands +15 +4.3. +Bands and torsors +17 +5. +Examples and non-examples +18 +5.1. +The H-space of G-torsors +19 +5.2. +Eilenberg–Mac Lane spaces +20 +5.3. +Products of Eilenberg–Mac Lane spaces +21 +References +21 +1. Introduction +In this paper we study H-spaces and their deloopings. We work in homotopy type theory, so our +results apply to any ∞-topos. Many of our results are new, even for the ∞-topos of spaces. +A key concept is that of a central type. A pointed type A is central if the map (A → A)(id) → A +sending a function f to f(pt) is an equivalence. Here (A → A)(id) denotes the identity component of +the type of all self-maps of A, and pt denotes the base point of A. Every central type is a connected +H-space, and a connected H-space is central precisely when the type A →∗ A of pointed self-maps is +a set. We prove this and other characterizations of central types in Proposition 3.6. It follows, for +example, that every Eilenberg–Mac Lane space K(G, n), with G abelian and n ≥ 1, is central. We +show in Section 5.3 that some, but not all, products of Eilenberg–Mac Lane spaces are central. We +don’t know whether every central type is a product of Eilenberg–Mac Lane spaces. +Our first result is: +Date: January 6, 2023. +1 +arXiv:2301.02636v1 [math.AT] 6 Jan 2023 + +2 +BUCHHOLTZ, CHRISTENSEN, FLATEN, AND RIJKE +Theorem 4.6. Let A be a central type. Then A has a unique delooping. +The key ingredient of this result and much of the paper is that we have a concrete description of +the delooping of A. It is given by the type BAut1(A) :≡ ΣX:U∥A = X∥0 of types banded by A, which +is the 1-connected cover of BAut(A). As an example, since K(G, n) is central for G abelian and n ≥ 1, +this gives an alternative way to define K(G, n + 1) in terms of K(G, n), as previously indicated by the +first author [Buc19]. +We also show: +Theorem 4.10. Let A be a central type. Then every pointed map f : A →∗ A is uniquely deloopable +to a map Bf : BAut1(A) →∗ BAut1(A). +It follows that the type of pointed self-maps of BAut1(A) is a set, since it is equivalent to A →∗ A. +One of the motivations for studying BAut1(A) is that one can define a tensoring operation. Given +two banded types X and Y in BAut1(A), the type X∗ = Y has a natural banding, where X∗ is a certain +dual of X. We write X ⊗ Y for this banded type, and show in Theorem 4.19 that it makes BAut1(A) +into an abelian H-space. Combined with Theorem 4.6, Theorem 4.10, and the characterization of +central types mentioned earlier, we therefore deduce: +Corollary 4.20. For a central type A, the type BAut1(A) is again central. Therefore, A is an infinite +loop space, in a unique way. Moreover, every pointed map A →∗ A is infinitely deloopable, in a unique +way. +Our tensoring operation gives a new description of the H-space structure on K(G, n), which will +be helpful for calculations of Euler classes in work in progress and is what originally motivated this +work. +We also give an alternate description of the delooping of a central type A as a certain type of +A-torsors, and give an analogous description of K(G, 1) for any group G. +To prove the above results, we first need to further develop the theory of H-spaces. One difference +between our work and classical work in topology is that we emphasize the moduli space HSpace(A) +of H-space structures on a pointed type A, rather than just the set of components. For example, we +prove: +Theorem 2.27. Let A be an H-space such that for all a : A, the map a · − is an equivalence. Then +the type HSpace(A) of H-space structures on A is equivalent to the type A ∧ A →∗ A of pointed maps. +This generalizes a classical formula of Arkowitz–Curjel and Copeland, which plays a key role in +classical results on the number of H-space structures on various spaces. The classical formula only de- +termines the path components of the type of H-space structures, while our formula gives an equivalence +of types. From our formula it immediately follows, for example, that the type of H-space structures on +the 3-sphere is Ω6S3. The proof of Theorem 2.27 uses evaluation fibrations, which generalize the map +appearing in the definition of “central.” In fact, these evaluation fibrations play an important role in +much of the paper. For example, we include results relating the existence of sections of an evaluation +fibration to the vanishing of Whitehead products, and use this to show that no even spheres besides +S0 admit H-space structures. +In Proposition 3.3 we show that every central type has a unique H-space structure, in the strong +sense that the type HSpace(A) is contractible. We prove several results about types with unique +H-space structures. For example, we show that such H-space structures are associative and coherently +abelian, and that every pointed self-map is an H-space map, a weaker version of the delooping above. +We also give an example showing that not every type with a unique H-space structure is central. +We note that these results rely on us defining “H-space” to include a coherence between the two +unit laws (see Definition 2.1). +Outline. In Section 2, we give results about H-spaces which do not depend on centrality, including a +description of the moduli space of H-space structures, results about Whitehead products and H-space + +CENTRAL H-SPACES AND BANDED TYPES +3 +structures on spheres, and results about unique H-space structures. In Section 3, we define central +types, show that central types have a unique H-space structure, give a characterization of which H- +spaces are central, and prove other results needed in the next section. Section 4 is the heart of the +paper. It defines the type BAut1(A) of bands for a central type A, shows that it is a unique delooping +of A, proves that it is an H-space under a tensoring operation, and shows that central types and their +self-maps are uniquely infinitely deloopable. We also give the alternate description of the delooping +in terms of A-torsors. Finally, Section 5 gives examples and non-examples of central types, mostly +related to Eilenberg–Mac Lane spaces and their products. +Notation and conventions. In general, we follow the notation used in [Uni13]. For example, we +write path composition in diagrammatic order: given paths p : x = y and q : y = z, their composite +is p � q. The reflexivity path is written refl. +Given a type A and an element a : A, we write (A, a) for the type A pointed at a. If A is already a +pointed type with unspecified base point, then we write pt for the base point. If A and B are pointed +types, and f, g : A →∗ B are pointed maps, then f =∗ g is the type of pointed homotopies between f +and g. If A is an H-space, then we write x · y for the product of two elements x, y : A (unless another +notation for the multiplication is given). For a pointed type A, we write HSpace(A) for the type of +H-space structures on A with the basepoint as the identity element (Definition 2.1). We write Sn for +the n-sphere, and U for a fixed universe of types. +Acknowledgements. We would like to thank David Jaz Myers for many lively discussions on the +content of this paper, especially related to bands and torsors. We also thank David W¨arn for fruitful +discussions and for sharing drafts of his forthcoming work. +Egbert Rijke gratefully acknowledges the support by the Air Force Office of Scientific Research +through grant FA9550-21-1-0024, and support by the Slovenian Research Agency research programme +P1-0294. Dan Christensen and Jarl Flaten both acknowledge the support of the Natural Sciences and +Engineering Research Council of Canada (NSERC), RGPIN-2022-04739. +2. H-spaces and evaluation fibrations +In Section 2.1, we begin by recalling the notion of a (coherent) H-space structure on a pointed type +A. We discuss the type of pointed extensions of a map B ∨ C →∗ A to B × C, and show that the type +of H-space structures on A is equivalent to the type of pointed extensions of the fold map. We relate +the existence of extensions to the vanishing of Whitehead products, and use this to show that there +are no H-space structures on even spheres except S0. In addition, we show that for any n-connected +H-space A, the Freudenthal map π2n+1(A) → π2n+2(ΣA) is an isomorphism, not just a surjection. +In Section 2.2, we study evaluation fibrations. We show that the type of H-space structures is +equivalent to a type of sections of an evaluation fibration, and use this to show that the type of +H-space structures on a left-invertible H-space A is equivalent to A ∧ A →∗ A, generalizing a classical +formula of Arkowitz–Curjel and Copeland. It immediately follows, for example, that the type of H- +space structures on the 3-sphere is Ω6S3. We end with a result relating the existence of sections of an +evaluation fibration to the vanishing of Whitehead products. +Section 2.3 is a short section which studies the case when the type of H-space structures is con- +tractible. We stress that this is not the same as HSpace(A) having a single component, which is what +is classically meant by “A has a unique H-space structure.” This situation is interesting in its own +right. We show that such H-space structures are associative and coherently abelian, and we prove +that all pointed self-maps of A are automatically H-space maps. +2.1. H-space structures. We begin by giving the notion of H-space structure that we will consider +in this paper. +Definition 2.1. Let A be a pointed type. +(1) A non-coherent H-space structure on A consists of a binary operation µ : A → A → A, +along with two homotopies µl : µ(pt, −) = idA and µr : µ(−, pt) = idA; + +4 +BUCHHOLTZ, CHRISTENSEN, FLATEN, AND RIJKE +(2) A (coherent) H-space structure on A consists of a non-coherent H-space structure µ on +A along with a coherence µlr : µl(pt) =µ(pt,pt)=pt µr(pt). +(3) We write HSpace(A) for the type of (coherent) H-space structures on A. +When the H-space structure is clear from the context we may write x · y :≡ µ(x, y). Any H-space +structure yields a non-coherent H-space structure by forgetting the coherence. +Suppose A has a +(non)coherent H-space structure µ. +(4) If µ(a, −) : A → A is an equivalence for all a : A, then µ is left-invertible, and we write +x\y :≡ µ(x, −)−1(y). Right-invertible is defined dually, and we write x/y :≡ µ(−, y)−1(x). +(5) The twist µT of µ is the natural (non)coherent H-space structure with operation +µT (a0, a1) :≡ µ(a1, a0). +When we say “H-space” we mean the coherent notion—we will only say “coherent” for emphasis. +The notion of H-space structure considered in [Uni13, Def. 8.5.4] corresponds to our non-coherent H- +space structures. While many constructions can be carried out for non-coherent H-spaces (such as the +Hopf construction), the coherent case is more natural for our purposes. Moreover, any non-coherent +H-space can be made coherent by simply changing one of the unit laws: +Proposition 2.2. Any non-coherent H-space structure on a pointed type A gives rise to a coherent +H-space structure with the same underlying binary operation. +Proof. Let (A, µ, µl, µr) be a non-coherent H-space. We define a new homotopy µ′ +r : µ(−, pt) = id as +the concatenation of paths +µ(x, pt) +µ(x, µ(pt, pt)) +µ(x, pt) +x. +apµ(x)(µr(pt))−1 +apµ(x)(µl(pt)) +µr(x) +We claim that µl(pt) = µ′ +r(pt). To see this, it suffices to show that the square +µ(pt, µ(pt, pt)) +µ(pt, pt) +µ(pt, pt) +pt +apµ(pt)(µl(pt)) +apµ(pt)(µr(pt)) +µr(pt) +µl(pt) +commutes. We will show that the top path is equal to µl(µ(pt, pt)), and this turns the square into a +naturality square for the homotopy µl, which always commutes. To see that +apµ(pt)(µl(pt)) = µl(µ(pt, pt)), +observe that µl is a homotopy µ(pt) = id, and for any homotopy H : f = id we have apf Hx = Hf(x) +for all x. +□ +The proposition implies that the types of non-coherent and coherent H-space structures on a pointed +type are logically equivalent. However, they are not generally equivalent as types (see Remark 3.4). +We’ll be interested in abelian and associative H-spaces later on. +Definition 2.3. Let A be an H-space with multiplication µ. +(1) If there is a homotopy h : Πa,bµ(a, b) = µ(b, a) then µ is abelian. +(2) If µ = µT in HSpace(A) then µ is coherently abelian. +(3) If there is a homotopy α : Πa,b,c:Aµ(µ(a, b), c) = µ(a, µ(b, c)) then µ is associative. +The following lemma gives a convenient way of constructing abelian H-space structures, and will +be used in Theorem 4.19. +Lemma 2.4. Let A be a pointed type with a binary operation µ, a symmetry σa,b : µ(a, b) = µ(b, a) +for every a, b : A such that σpt,pt = refl, and a left unit law µl : µ(pt, −) = idA. Then A becomes an +abelian H-space with the right unit law induced by symmetry. + +CENTRAL H-SPACES AND BANDED TYPES +5 +Proof. For b : A, the right unit law is given by the path σb,pt � µl(b) of type µ(b, pt) = b. For coherence +we need to show that the following triangle commutes: +µ(pt, pt) +µ(pt, pt) +pt . +µl +σpt,pt +µl +By our assumption that σpt,pt = refl, the triangle is filled reflµl. +□ +For any right-invertible H-space A, for b : A one can define the two operations (−)/b and (−)·(pt/b) +of type A → A. If A is associative, then these coincide: +Lemma 2.5. Let A be an associative H-space. For any a, b : A, we have that a/b = a · (pt/b). +Proof. For all a, b : A we have (a · (pt/b)) · b = a · ((pt/b) · b) = a · pt = a. Thus by dividing by b on +the right, we deduce a · (pt/b) = a/b. +□ +We collect a few basic facts about H-spaces. The following lemma generalizes a result of Evan +Cavallo, who formalized the fact that unpointed homotopies between pointed maps into a homogeneous +type A can be upgraded to pointed homotopies. Being a homogeneous type is logically equivalent to +being a left-invertible H-space [Cav21]. Here we do not need to assume left-invertibility, and we factor +this observation through a further generalization. +Lemma 2.6. Let A be a pointed type, and consider the following three conditions: +(1) A is an H-space. +(2) The evaluation map (idA = idA) → (pt = pt) has a section. +(3) For every pointed type B and pointed maps f, g : B →∗ A, there is a map (f = g) → (f =∗ g) +which upgrades unpointed homotopies to pointed homotopies. +Then (1) implies (2) and (2) implies (3). +Proof. To show that (1) implies (2), suppose that A is an H-space, and let p : pt = pt. For any x : A +we define the path px : x = x to be the concatenation +x +x · pt +x · pt +x . +µ−1 +r +apµ(x)(p) +µr +This defines a map s : (pt = pt) → (idA = idA). To see that this map is a section of the evaluation +map, it suffices to show that the square +pt · pt +pt · pt +pt +pt +apµ(pt)(p) +µr +µr +p +commutes. To see this, note that µr = µl. If we replace µr by µl in the above square, we obtain a +naturality square of homotopies, which always commutes. +We next show that (2) implies (3). Let f, g : B →∗ A be pointed maps and let H : f = g be an +unpointed homotopy. By path induction on H, we can assume we have a single function f : B → A +with two pointings, fpt and f ′ +pt : f(pt) = pt. Our goal is to define a homotopy K : f = f such that +Kpt = r, where r :≡ fpt · f ′pt : f(pt) = f(pt). By path induction on fpt, we can assume that the +basepoint of A is f(pt). By (2), we have s : (f(pt) = f(pt)) → (idA = idA) such that s(p, f(pt)) = p +for all p : f(pt) = f(pt). For b : B, define Kb to be s(r, f(b)). Then Kpt = r, as required. +□ +The following result is straightforward and has been formalized, so we do not include a proof. + +6 +BUCHHOLTZ, CHRISTENSEN, FLATEN, AND RIJKE +Proposition 2.7. Suppose A is a (left-invertible) H-space. For any pointed type B, the mapping +type B →∗ A based at the constant map is naturally a (left-invertible) H-space under pointwise mul- +tiplication. +Similarly, for any type B, the mapping type B → A based at the constant map is a +(left-invertible) H-space under pointwise multiplication. +□ +In particular, if A is left-invertible then for any f : B →∗ A there is a self-equivalence of B →∗ A +which sends the constant map to f—namely, the pointwise multiplication by f on the left. +Our next goal is to rule out H-space structures on even spheres using Brunerie’s computation of +Whitehead products. (See [Bru16, Section 3.3] for their definition.) To do so, we prove some results +about Whitehead products from [Whi46] which relate to H-spaces. +Definition 2.8. Let α : B →∗ A and β : C →∗ A be pointed maps. An (α, β)-extension is a +pointed map f : B × C →∗ A equipped with a pointed homotopy filling the following diagram: +B ∨ C +A +B × C . +α∨β +f +Remark 2.9. It is equivalent to consider the type of unpointed (α, β)-extensions consisting of unpointed +maps f : B × C → A and unpointed fillers. The additional data in a pointed extension is a path +fpt : f(pt, pt) = pt and a 2-path that determines fpt in terms of the other data. +These form a +contractible pair. +When α and β are maps between spheres, Whitehead instead says that f is “of type (α, β)” but we +prefer to stress that we work with a structure and not a property, as the following lemma illustrates: +Lemma 2.10. H-space structures on a pointed type A correspond to (idA, idA)-extensions. +□ +The proof consists of straightforward reshuffling of data. +Lemma 2.11. If A is an H-space, then there is an (α, β)-extension for every pair α : B →∗ A and +β : C →∗ A of pointed maps. +Proof. Using naturality of the left and right unit laws and coherence, one can show that the map +(b, c) �→ α(b)·β(c) : B ×C → A is an (α, β)-extension. Alternatively, observe that the (α, β)-extension +problem factors through the (idA, idA)-extension problem via the map α × β : B × C → A × A. +□ +The lemmas explain the relation between H-space structures and (α, β)-extensions, which are in +turn related to Whitehead products via the next two results. +Proposition 2.12 ([Whi46, Corollary 3.5]). Let m, n > 0 be natural numbers and consider two +pointed maps α : Sm →∗ A and β : Sn →∗ A. The type of (α, β)-extensions is equivalent to the type +of witnesses that the map [α, β] : Sm+n−1 →∗ A is constant (as a pointed map). +Proof. Consider the diagram of pointed maps below, where the composite of the top two maps is [α, β] +and the left diamond is a pushout of pointed types: +Sm ∨ Sn +Sm+n−1 +Sm × Sn +A . +1 +α∨β +f +An (α, β)-extension is the same as a pointed map f along with a pointed homotopy filling the top-right +triangle. Since the bottom-right triangle is filled by a unique pointed homotopy, an (α, β)-extension +thus corresponds exactly to the data of a filler in the outer diagram, i.e., a homotopy witnessing that +[α, β] is constant as a pointed map. +□ + +CENTRAL H-SPACES AND BANDED TYPES +7 +With the notation of the previous proposition, we have the following: +Corollary 2.13 ([Whi46, Corollary 3.6]). Suppose A is an H-space. Then [α, β] is constant. +Proof. The follows from Lemma 2.11 and Proposition 2.12. +□ +Using the above results, we can rule out H-space structures on even spheres in positive dimensions. +Proposition 2.14. The n-sphere merely admits an H-space structure if and only if [ιn, ιn] = 0. In +particular, there are no H-space structures on the n-sphere when n > 0 is even. +Proof. The implication (→) is immediate by Corollary 2.13. Conversely, Proposition 2.12 implies that +[ιn, ιn] = 0 if and only if an (idSn, idSn)-extension merely exists, which by Lemma 2.10 happens if and +only if Sn merely admits an H-space structure. +Finally, Brunerie showed that [ιn, ιn] = 2 in π2n−1(Sn) for even n > 0 [Bru16, Proposition 5.4.4], +which by the above implies that Sn cannot admit an H-space structure. +□ +We also record the following result and a corollary. +Proposition 2.15. Let A be a left-invertible H-space. The unit η : A →∗ ΩΣA has a pointed retrac- +tion, given by the connecting map δ : ΩΣA →∗ A associated to the Hopf fibration of A. +Proof. Let δ : ΩΣA →∗ A be the connecting map associated to the Hopf fibration of A. Recall that +for a loop p : N = N, we have δ(p) :≡ p∗(pt) where p∗ : A → A denotes transport and A is the fibre +above N. By definition of the Hopf fibration, a path merid(a) : N =ΣA S sends an element x of the +fibre A to a · x. Now define a homotopy δ ◦ η = id by +δ(η(a)) ≡ δ(merid(a) � merid(pt)−1) = merid(pt)−1 +∗ (merid(a)∗(pt)) ≡ pt\(a · pt) = a. +Finally, we promote this to a pointed homotopy using Lemma 2.6. +□ +It follows that for any n-connected H-space A, the Freudenthal map π2n+1(A) → π2n+2(ΣA) is an +isomorphism, not just a surjection. In particular, we have: +Corollary 2.16. The natural map π5(S3) → π6(S4) is an isomorphism. +□ +The fact that the unit η : A →∗ ΩΣA has a retraction when A is a left-invertible H-space also follows +from James’ reduced product construction, as shown in [Jam55]. Using [Bru16], one can see that this +goes through in homotopy type theory. However, the above argument is much more elementary. We +don’t know if this argument had been observed before. +2.2. Evaluation fibrations. We now begin our study of evaluation fibrations and their relation to +H-space structures and (α, β)-extensions from the previous section. Given a pointed map f : B →∗ A, +we will simply write ev : (B → A, f) →∗ A for the map which evaluates at pt : B. This map is +pointed since f is. If no map f is specified, then we mean that f ≡ id. +In a moment we will define evaluation fibrations to be the restriction of ev to a component, but +first we make a useful observation. +Definition 2.17. Let e : X →∗ A and g : B →∗ A be pointed maps. A pointed lift of g through +e consists of a pointed map s : B →∗ X along with a pointed homotopy e ◦ s =∗ g. If g ≡ id, then s +is more specifically a pointed section of e. +Proposition 2.18. Let f : B →∗ A and g : C →∗ A be pointed maps. The type of (f, g)-extensions +is equivalent to the type of pointed lifts of g through ev : (B → A, f) →∗ A. +□ +We stress that the domain of ev is the type of unpointed maps B → A, pointed by (the underlying +map of) f. The proof of the statement is a straightforward reshuffling of data. Diagrammatically, it +gives a correspondence between the dashed arrows below, with pointed homotopies filling the triangles: +B ∨ C +A +(B → A, f) +B × C +C +A +f∨g +ev +g + +8 +BUCHHOLTZ, CHRISTENSEN, FLATEN, AND RIJKE +Combining Lemma 2.10 with the previous proposition, we deduce: +Corollary 2.19. Let A be a pointed type. The type of H-space structures on A is equivalent to the +type of pointed sections of ev : (A → A, id) →∗ A. +□ +Remark 2.20. Phrased another way, an H-space structure on a pointed type A is equivalent to a family +µ : Π(a:A)(A, pt) →∗ (A, a). +If A is a higher inductive type with a point pt, one can define µ(pt) :≡ id to simplify the task. +Definition 2.21. Let A be a type and a : ∥A∥0. The path component of a in A is +A(a) :≡ Σa′:A(|a′|0 = a). +If a : A then we abuse notation and write A(a) for A(|a|0), and in this case A(a) is pointed at (a, refl). +Definition 2.22. For any pointed map α : B →∗ A, the evaluation fibration (at α) is the pointed +map evα : (B → A)(α) →∗ A induced by evaluating at the base point of B. +Observe that the component (A → A)(id) is equivalent to (A ≃ A)(id), since being an equivalence is +a property of a map. We permit ourselves to pass freely between the two. +Since pointed maps out of connected types land in the component of the base point of the codomain, +we have the following consequence of Corollary 2.19. +Corollary 2.23. Let A be a pointed, connected type. The type of H-space structures on A is equivalent +to the type of pointed sections of evid : (A ≃ A)(id) →∗ A. +□ +For certain H-spaces, various evaluation fibrations become trivial: +Proposition 2.24. Suppose A is a left-invertible H-space. We have a pointed equivalence over A +(A → A) +(A →∗ A) × A +A , +ev +∼ +pr2 +where the mapping spaces are both pointed at their identity maps. This pointed equivalence restricts +to pointed equivalences (A ≃ A) ≃∗ (A ≃∗ A) × A over A, and (A → A)(id) ≃∗ (A →∗ A)(id) × A(pt) +over A(pt). +Proof. Define e : (A → A) → (A →∗ A) × A by e(f) :≡ (a �→ f(pt)\f(a), f(pt)) where the first +component is a pointed map in the obvious way. Clearly e is a map over A, and moreover e is pointed. +It is straightforward to check that the triangle above is filled by a pointed homotopy. (One could also +apply Lemma 2.6, but a direct inspection suffices in this case.) +Finally, it’s straightforward to check that e has an inverse given by +(g, a) �→ (x �→ a · g(x)). +Hence e is an equivalence, as desired. The restrictions to equivalences and path components follow by +functoriality. +□ +The hypotheses of the proposition are satisfied, for example, by connected H-spaces. +Example 2.25. We obtain three pointed equivalences for any abelian group A and n ≥ 1: +� +K(A, n) → K(A, n) +� +≃∗ Ab(A, A) × K(A, n), +� +K(A, n) ≃ K(A, n) +� +≃∗ AutAb(A) × K(A, n), and +� +K(A, n) → K(A, n) +� +(id) ≃∗ K(A, n). + +CENTRAL H-SPACES AND BANDED TYPES +9 +Example 2.26. Taking A :≡ S3 in the previous proposition, by virtue of the H-space structure on +the 3-sphere constructed in [BR18], we get three pointed equivalences: +(S3 → S3) ≃∗ Ω3S3 × S3, +(S3 ≃ S3) ≃∗ Ω3 +±1S3 × S3, +and +(S3 ≃ S3)(id) ≃∗ (S3 ≃∗ S3)(id) × S3, +where Ω3 +±1S3 :≡ (Ω3S3)(1)⊔(Ω3S3)(−1) and 1 and −1 refer to the corresponding elements of π3(S3) = Z. +By combining our results thus far, we obtain the following equivalence which generalizes a classical +formula of [Cop59, Theorem 5.5A], independently shown by [AC63], for counting homotopy classes of +H-space structures on certain spaces. +Theorem 2.27. Let A be a left-invertible H-space. The type HSpace(A) of H-space structures on A +is equivalent to A ∧ A →∗ A. +Proof. By Corollary 2.19, the type of H-space structures on A is equivalent to the type of pointed +sections of ev : (A → A) → A. By Proposition 2.24, this type is equivalent to the type of pointed +sections of pr2 : (A →∗ A) × A → A, which are simply pointed maps A →∗ (A →∗ A, id), where +the codomain is pointed at the identity. The latter type is equivalent to A →∗ (A →∗ A), where +the codomain is pointed at the constant map, by Proposition 2.7. Finally, this type is equivalent to +A ∧ A →∗ A by the smash–hom adjunction for pointed types [vDoo18, Theorem 4.3.28]. +□ +Example 2.28. It follows from the proposition that HSpace(S1) ≃ 1 and HSpace(S3) ≃ Ω6S3. +We record the following result which relates Whitehead products and evaluation fibrations. +Proposition 2.29 ([Han74, Lemma 2.2]). Let n, m ≥ 2 and let α : πm(Sn). +The evaluation +fibration evα : (Sm → Sn)(α) → Sn merely has a section if and only if the Whitehead product +[α, ιn] : πn+m−1(Sn) vanishes. +Proof. As we are proving a proposition, we may pick a representative α : Sm →∗ Sn throughout. +Using Proposition 2.18 and that Sn is connected, we see that [α, ιn] vanishes if and only if there +merely exists a pointed section of evα. The fibre of the forgetful map from pointed sections of evα +to unpointed sections of evα over some section (s, h) is equivalent to +� +k:s(pt,−)=α +h(pt) =s(pt,pt)=pt k(pt) � αpt, +where αpt : α(pt) = pt is the pointing of α. This fibre is (−1)-connected since s lands in the component +of α and the inner part of the Σ-type is a double path space of Sn with n ≥ 2. In other words, this +forgetful map is an epimorphism. A pointed section of evα therefore merely exists if and only if an +unpointed section merely exists, completing the proof. +□ +2.3. Unique H-space structures. We collect results about H-space structures which are unique, in +the sense that the type of H-space structures is contractible. In particular, we give elementary proofs +that such H-space structures are automatically coherently abelian and associative. Moreover, pointed +self-maps of such are automatically H-space self-maps. +Lemma 2.30. Let A be a pointed type and suppose HSpace(A) is contractible. +Then the unique +H-space structure µ on A is coherently abelian. +Proof. Since HSpace(A) is contractible, there is an identification µ = µT of H-space structures. (Here, +µT is the twist, defined in Definition 2.1.) +□ +For the next result, we use the definition of the smash product from [vDoo18, Definition 4.3.6] +(see also [CS20, Definition 2.29]) which avoids higher paths. For pointed types (X, x0) and (Y, y0), +the smash product X ∧ Y is the higher inductive type with point constructors sm : X × Y → +X ∧ Y and auxl, auxr : X ∧ Y , and path constructors gluel : � +y:Y sm(x0, y) = auxl and gluer : +� +x:X sm(x, y0) = auxr. +It is pointed by auxl. +The smash product was shown to be associative +in [vDoo18, Definition 4.3.33]. + +10 +BUCHHOLTZ, CHRISTENSEN, FLATEN, AND RIJKE +Proposition 2.31. Suppose A is a pointed type with a unique H-space structure, and suppose moreover +that this H-space structure is left-invertible. Then any pointed map f : A →∗ A is an H-space map, +i.e., we have f(a · b) = f(a) · f(b) for all a, b : A. +Proof. Let f : A →∗ A be a pointed map. We will define an associated map ν : A ∧ A →∗ A, which +records how f deviates from being an H-space map. We define ν(sm(a, b)) :≡ +� +f(a · b)/f(b) +� +/f(a), +ν(auxl) :≡ pt, and ν(auxr) :≡ pt. For b : A, we have a path ν(sm(pt, b)) ≡ +� +f(pt · b)/f(b) +� +/f(pt) = +� +f(b)/f(b) +� +/pt = pt/pt = pt, and similarly for the other path constructor. Since A admits a unique +H-space structure, the type A∧A →∗ A is contractible by Theorem 2.27. Consequently, ν is constant, +whence for all a, b : A we have +� +f(a · b)/f(b) +� +/f(a) = pt, and therefore +f(a · b) = f(a) · f(b). +□ +Remark 2.32. Note that when A and B are two pointed types, each with unique H-space structures, +it is not necessarily the case that every pointed map f : A →∗ B is an H-space map. For example, the +squaring operation gives a natural transformation H2(X; Z) → H4(X; Z) which is represented by a +map K(Z, 2) →∗ K(Z, 4). But since squaring isn’t a homomorphism, this map isn’t an H-space map. +Proposition 2.33. Suppose A is a pointed type with a unique H-space structure which is left-invertible. +Then the H-space structure is necessarily associative. +Proof. Let a : A. Define a map ν : A ∧ A →∗ A as follows. We let ν(sm(b, c)) :≡ ((a · b) · c)/(a · (b · c)), +ν(auxl) :≡ pt, and ν(auxr) :≡ pt. For c : A, we have a path ν(sm(pt, c)) ≡ ((a · pt) · c)/(a · (pt · c)) = +(a · c)/(a · c) = pt, and similarly for the other path constructor. Since A admits a unique H-space +structure, the type A ∧ A →∗ A is contractible by Theorem 2.27. Consequently, for each a, ν is +constant. It follows that for all a, b, c : A we have ((a · b) · c)/(a · (b · c)) = pt, and therefore +(a · b) · c = a · (b · c). +□ +Note that if A ∧ A →∗ A is contractible, then it follows from the smash-hom adjunction that +A∧n →∗ A is contractible for each n ≥ 2, where A∧n denotes the smash power. +3. Central types +In this and the next section we focus on pointed types which we call central. Centrality is an +elementary property with remarkable consequences. For example, in the next section we will see that +every central type is an infinite loop space (Corollary 4.20). To show this, we require a certain amount +of theory about central types. We first show that every central type has a unique H-space structure. +When A is already known to be an H-space, we give several conditions which are equivalent to A +being central. From this, it follows that every Eilenberg–Mac Lane space K(G, n), with G abelian and +n ≥ 1, is central. We also prove several other results which we will need in the next section. +Definition 3.1. Let A be a pointed type. The center of A is the type ZA :≡ (A → A)(id), which +comes with a natural map evid : ZA →∗ A (see Definition 2.22). If the map evid is an equivalence, +then A is central. +Remark 3.2. The terminology “central” comes from higher group theory. Suppose A :≡ BG is the +delooping of an ∞-group G. The center of G is the ∞-group ZG :≡ Πx:G(x = x) with delooping +BZG :≡ (BG ≃ BG)(id), which is our ZA. +Central types and H-spaces are connected through evaluation fibrations: +Proposition 3.3. Suppose that A is central. Then A admits a unique H-space structure. In addition, +A is connected, so this H-space structure is both left- and right-invertible. +Proof. Since evid is an equivalence, it has a unique section. By Corollary 2.23, we deduce that A has +a unique H-space structure µ. It follows from Lemma 2.30 that it is coherently abelian. Finally, the +equivalence evid : (A → A)(id) ≃ A implies that A is connected. Then, since µ(pt, −) and µ(−, pt) are +both equal to the identity, it follows that µ is left- and right-invertible. +□ + +CENTRAL H-SPACES AND BANDED TYPES +11 +It follows from Proposition 2.33 and Lemma 2.30 that the unique H-space structure on a central +type is associative and coherently abelian. +Remark 3.4. In contrast, the type of non-coherent H-space structures on a central type A is rarely +contractible. We’ll show here that it is equivalent to the loop space ΩA. First consider the type of +binary operations µ : A → (A → A) which merely satisfy the left unit law. This is equivalent to the +type of maps A → (A → A)(id), since A is connected. Such a map µ satisfies the right unit law if and +only if the composite evid ◦µ : A → A is the identity map. In other words, µ must be a section of the +equivalence evid, so there is a contractible type of such µ. +The left unit law says that µ sends pt to id. After post-composing with evid, it therefore says that +it sends pt to id(pt), which equals pt. So the type of left unit laws is pt = pt, i.e., the loop space ΩA. +Note that we imposed the left unit law both merely and purely, but that doesn’t change the type. So +it follows that the type of all non-coherent H-space structures on a central type A is ΩA. +We give conditions for an H-space to be central, in which case the H-space structure is the unique +one coming from centrality. For the next two results, write +F :≡ Σf:A→∗A∥f = id∥ +for the fibre of evid : (A → A)(id) →∗ A over pt : A. Note that the equality f = id is in the type of +unpointed maps A → A. +Lemma 3.5. Suppose that A is a connected H-space. Then F ≃ (A →∗ A)(id). +Proof. By our assumptions, Proposition 2.24 gives a trivialization of evid over A: +t : (A → A)(id) ≃∗ (A →∗ A)(id) × A. +Passing to the fibres of evid and pr2 over pt : A gives the desired equivalence. +□ +The lemma can also be shown using Lemma 2.6. +Proposition 3.6. Let A be a pointed type. Then the following are logically equivalent: +(1) A is central; +(2) A is a connected H-space and A →∗ A is a set; +(3) A is a connected H-space and A ≃∗ A is a set; +(4) A is a connected H-space and A →∗ ΩA is contractible; +(5) A is a connected H-space and ΣA →∗ A is contractible. +Proof. (1) =⇒ (2): Assume that A is central. Then Proposition 3.3 implies that A is a connected H- +space. Since A is a left-invertible H-space, so is A →∗ A, by Proposition 2.7. Therefore all components +of A →∗ A are equivalent to (A →∗ A)(id), and thus to F by Lemma 3.5. Now, F is contractible since +evid is an equivalence, and consequently A →∗ A is a set since all of its components are contractible. +(2) =⇒ (3): This follows from the fact that A ≃∗ A embeds into A →∗ A. +(3) =⇒ (1): If (A ≃∗ A) is a set, then its component (A →∗ A)(id) is contractible. Therefore F is +contractible, by Lemma 3.5. It follows that evid is an equivalence, since A is connected. Hence A is +central. +(3) ⇐⇒ (4): Since A is a left-invertible H-space, so is A →∗ A. The latter is therefore a set if and +only if the component of the constant map is contractible, which is true if and only if the loop space +Ω(A →∗ A) is contractible. Finally, the equivalence Ω(A →∗ A) ≃ (A →∗ ΩA) shows that this is true +if and only if A →∗ ΩA is contractible. +(4) ⇐⇒ (5): This follows from the equivalence (A →∗ ΩA) ≃ (ΣA →∗ A). +□ +Example 3.7. Consider the Eilenberg–Mac Lane space K(G, n) for n ≥ 1 and G an abelian group. +It is a pointed, connected type. +Since K(G, n) ≃ Ω K(G, n + 1), it is an H-space. +By [BvDR18, +Theorem 5.1], K(G, n) ≃∗ K(G, n) is equivalent to the set of automorphisms of G. It therefore follows +from Proposition 3.6 that K(G, n) is central. We will see in Proposition 5.9 a more self-contained +proof of this result. + +12 +BUCHHOLTZ, CHRISTENSEN, FLATEN, AND RIJKE +Example 3.8. Brunerie showed that π4(S3) ≃ Z/2 [Bru16]. Therefore, S4 →∗ S3 is not contractible, +and so S3 is not central, by Proposition 3.6(5). Since this is in the stable range, it follows that Sn is +not central for n ≥ 3. +Remark 3.9. For a pointed type A, we have seen that A being central is logically equivalent to A being +a connected H-space such that A ≃∗ A is a set. It is natural to ask whether the reverse implication +holds without the assumption that A is an H-space. However, this is not the case. Consider, for +example, the pointed, connected type K(G, 1) for a non-abelian group G. Then K(G, 1) ≃∗ K(G, 1) +is equivalent to the set of group automorphisms of G. If K(G, 1) were central, then G would be twice +deloopable, which would contradict G being non-abelian. +By the previous proposition, the type A →∗ A is a set whenever A is central. Presently we observe +that it is in fact a ring. +Corollary 3.10. For any central type A, the set A →∗ A is a ring under pointwise multiplication +and function composition. +Proof. It follows from A being a commutative and associative H-space that the set A →∗ A is an +abelian group. The only nontrivial thing we need to show is that function composition is linear. Let +f, g, φ : A →∗ A, and consider a : A. By Proposition 2.31, φ is an H-space map. Consequently, +� +φ ◦ (f · g) +� +(a) ≡ φ(f(a) · g(a)) = φ(f(a)) · φ(g(a)) ≡ +� +(φ ◦ f) · (φ ◦ g) +� +(a). +□ +The following remark gives some insight into the nature of the ring A →∗ A. +Remark 3.11. If BG is an ∞-group and X is a pointed type, recall that a bundle over X is G-principal +if it is classified by a map X →∗ BG (see e.g. [Sco20, Def. 2.23] for a formal definition which easily +generalizes to arbitrary ∞-groups). In particular, it is not hard to see that the Hopf fibration of G +(as the loop space of BG) is a G-principal bundle, i.e., classified by a map ΣG →∗ BG. +In Proposition 4.4 we will see that any central type A has a delooping BAut1(A). This means we +have equivalences +(A →∗ A) ≃ +� +A →∗ (A ≃ A)(id) +� +≃ (ΣA →∗ BAut1(A)). +Thus we see that A →∗ A is the ring of principal A-bundles over ΣA. The equivalence above maps the +identity id : A →∗ A to the Hopf fibration of A (as a principal A-bundle), meaning the Hopf fibration +is the multiplicative unit from this perspective. +In the remainder of this section we collect various results which are needed later on. The first result +is that “all” of the evaluation fibrations of a central type A are equivalences: +Proposition 3.12. Let A be a central type and let f : A →∗ A be a pointed map. The evaluation +fibration evf : (A → A)(f) →∗ A is an equivalence, with inverse given by a �→ a · f(−). +Proof. The type A → A is a left-invertible H-space via pointwise multiplication, by Proposition 2.7. +So there is an equivalence (A → A)(id) → (A → A)(f) sending g to f · g. Since f is pointed, we have +evf(f · g) ≡ (f · g)(pt) ≡ f(pt) · g(pt) = pt · g(pt) = g(pt) = evid(g). +In other words, evf ◦(f · −) = evid, which shows that evf is an equivalence. Since f is pointed, the +stated map is a section of evf, hence is an inverse. +□ +Corollary 3.13. Let A be a central type, let f : A →∗ A, and let g : (A → A)(f). Then for all a : A, +we have g(a) = g(pt) · f(a). +□ +Any central type has an inversion map, which plays a key role in the next section. +Definition 3.14. Suppose that A is central. The inversion map id∗ : A → A sends a to a∗ :≡ pt/a. + +CENTRAL H-SPACES AND BANDED TYPES +13 +The defining property of a∗ is that a∗·a = pt. Since A is abelian, we also have a·a∗ = pt, so it would +have been equivalent to define the inversion to be a �→ a\pt. From associativity of a central H-space +it follows that pt∗ = pt and a∗∗ = a for all a, so the inversion map id∗ is a pointed self-equivalence of +A and an involution. +A curious property is that on the component of id∗, inversion of equivalences is homotopic to the +identity. This comes up in the next section. +Proposition 3.15. The map φ �→ φ−1 : (A ≃ A)(id∗) → (A ≃ A)(id∗) is homotopic to the identity. +Proof. Let φ : (A ≃ A)(id∗). We need to show that φ = φ−1, or equivalently that φ(φ(pt)) = pt, since +evid is an equivalence. (Note that φ ◦ φ : (A ≃ A)(id).) Using Corollary 3.13, we have that +φ(φ(pt)) = φ(pt) · φ(pt)∗ = pt. +□ +4. Bands and torsors +We begin in Section 4.1 by defining and studying types banded by a central type A, also called +A-bands. We show that the type BAut1(A) of banded types is a delooping of A, that A has a unique +delooping, and that every pointed self-map A →∗ A has a unique delooping. +In Section 4.2, we show that BAut1(A) is itself an H-space under a tensoring operation, from which +it follows that it is again a central type. Thus we may iteratively consider banded types to obtain +an infinite loop space structure on A, which is unique. As a special case, taking A to be K(G, n) for +some abelian group G produces a novel description of the infinite loop space structure on K(G, n), as +described in Section 5.2. +In Section 4.3, we define the type of A-torsors, which we show is equivalent to the type of A-bands +when A is central, thus providing an alternate description of the delooping of A. The type of A-torsors +has been independently studied by David W¨arn, who has shown that it is a delooping of A under the +weaker assumption that A has a unique H-space structure. +4.1. Types banded by a central type. We now study types banded by a central type A. On this +type we will construct an H-space structure, which will be seen to be central. +Definition 4.1. For a type A, let BAut1(A) :≡ ΣX:U∥A = X∥0. The elements of BAut1(A) are types +which are banded by A or A-bands, for short. We denote A-bands by Xp, where p : ∥A = X∥0 is +the band. The type BAut1(A) is pointed by A|refl|0. +Given a band p : ∥A = X∥0, we will write ˜p : ∥X ≃ A∥0 for the associated equivalence. +Remark 4.2. It’s not hard to see that BAut1(A) is a connected, locally small type—hence essentially +small, by the join construction [Rij17]. +The characterization of paths in Σ-types tells us what paths between banded types are. +Lemma 4.3. Consider two A-bands Xp and Yq. A path Xp = Yq of A-bands corresponds to a path +e : X = Y between the underlying types making the following triangle of truncated paths commute: +A +X +Y . +p +q +|e|0 +In other words, there is an equivalence (Xp = Yq) ≃ (X = Y )(¯p � q). +□ +For the remainder of this section, let A be a central type. We begin by showing that the type of +A-bands is a delooping of A. +Proposition 4.4. We have that Ω BAut1(A) ≃ A. +Proof. We have Ω BAut1(A) ≃ (A = A)(refl) ≃ (A ≃ A)(id) ≃ A, where the first equivalence uses +Lemma 4.3 and the last equivalence is by centrality. +□ + +14 +BUCHHOLTZ, CHRISTENSEN, FLATEN, AND RIJKE +Corollary 4.5. The unique H-space structure on A is deloopable. +□ +Note that this gives an independent proof that it is associative (cf. Proposition 2.33). +Theorem 4.6. The type A has a unique delooping. +Proof. We must show that the type ΣB:U* (ΩB ≃∗ A) is contractible. We will use BAut1(A), with the +equivalence ψ from Proposition 4.4, as the center of contraction. Let B : U* be a pointed type with a +pointed equivalence φ : ΩB ≃∗ A. Given x : B, consider pt =B x. Since A is connected, B is simply +connected. Therefore, to give a banding on pt =B x, it suffices to do so when x is pt, in which case +we use φ. So we have defined a map f : B → BAut1(A), and it is easy to see that it is pointed. +We claim that the following triangle commutes: +ΩB +Ω BAut1(A) +A . +φ +∼ +Ωf +ψ +∼ +Let q : pt =B pt. Then (Ωf)(q) is the path associated to the equivalence +A ≃ (pt =B pt) ≃ (pt =B pt) ≃ A. +The first equivalence is φ−1 and the last is φ, as these give the pointedness of f. The middle equivalence +is the map sending p to p�q. The map ψ comes from the evaluation fibration, so to compute ψ((Ωf)(q)) +we compute what happens to the base point of A. It gets sent to refl, then q, and then φ(q). This +shows that the triangle commutes. +It follows that Ωf is an equivalence. Since B and BAut1(A) are connected, f is an equivalence as +well. So f and the commutativity of the triangle provide a path from (B, φ) to (BAut1(A), ψ) in the +type of deloopings. +□ +We conclude this section by showing how to deloop maps A →∗ A. +Definition 4.7. Given f : A →∗ A, define Bf : BAut1(A) →∗ BAut1(A) by +Bf(Xp) :≡ (X → A)(f ∗◦˜p−1), +where f ∗ :≡ f ◦id∗, and we have used that f ∗ ◦ ˜p−1 is well-defined as an element of the set-truncation. +To give a banding of (X → A)(f ∗◦˜p−1) we may induct on p and use Proposition 3.12. +The same +argument shows that Bf is a pointed map. +Note that f(a∗) = f(a)∗ for any a : A, since f is an H-space map by Proposition 2.31, so there’s +no choice involved in this definition. +Let g : BAut1(A) →∗ BAut1(A). Given a loop q : pt = pt, the loop (Ωg)(q) is the composite +pt = g(pt) = g(pt) = pt, +which uses pointedness of g and apg(q). We will identify (pt = pt) with A and then write +Ω′g : A ≃∗ (pt = pt) +Ωg +−−→∗ (pt = pt) ≃∗ A. +Proposition 4.8. We have that Ω′Bf = f for any f : A →∗ A. +Proof. The following diagram describes how Bf acts on a loop p : pt =BAut1(A) pt: +Arefl +(A → A)(f ∗) +A +Arefl +(A → A)(f ∗) +A +p +g�→g◦˜p−1 +∼ +∼ + +CENTRAL H-SPACES AND BANDED TYPES +15 +Since ˜p is in the component of the identity, we have ˜p(a) = x · a for all a : A, where x = ˜p(pt). So +˜p−1(a) = x\a. Then the composite A ≃ A on the right is seen to be +a �→ evf ∗ +�� +a · f ∗(−) +� +◦ ˜p−1 +� += evf ∗ +� +a · f ∗� +x\(−) +�� += a · f(x∗∗) = a · f(x). +The domain Arefl = Arefl is identified with A by sending a path p to ˜p(pt), which in this case is the x +above. The codomain (A ≃ A)(id) is identified with A using evid, which sends the displayed function +to pt · f(x), which equals f(x). So we have that ΩBf = f. By Lemma 2.6, they are equal as pointed +maps. +□ +Proposition 4.9. We have that BΩ′g = g for any g : BAut1(A) →∗ BAut1(A). +Proof. Given an A-band Xp, we need to show that g(Xp) = (X → A)((Ω′g)∗◦˜p−1). First we construct +a map of the underlying types from left to right. For y : g(Xp), define the map +Gy : X +∼ +−→ (pt = Xp) ≃ (Xp = pt) +apg +−−→ (g(Xp) = g(pt)) ≃ (pt = pt) → A, +where the second map is path inversion, and the fourth map uses the trivialization of g(Xp) associated +to y and pointedness of g. The identification pt = g(pt) corresponds to a unique point y0 : g(pt). +To check that Gy lies in the right component, we may induct on p and assume y ≡ y0 since g(pt) is +connected. We then get the map +Gy0 : A +id∗ +−−→ A ≃ (pt = pt) +Ωg +−−→ (pt = pt) → A, +since path inversion on (pt = pt) corresponds to inversion on A, and y0 corresponds to the pointing +of g. This map is precisely the definition of (Ω′g)∗, so G lands in the desired component. +To check that G defines an equivalence of bands we may again induct on p. Write �y0 : pt ≃ g(pt) +for the equivalence associated to the point y0 : g(pt), which is a lift of the (equivalence associated to +the) banding of g(pt). It then suffices to check that the diagram +g(pt) +(A → A)((Ω′g)∗) +pt +� +y0 +−1 +G +ev(Ω′g)∗ +commutes. Let y : g(pt), which we identify with a trivialization y′ : pt = g(pt). Chasing through the +definition of G and using that apg(refl) = refl, we see that +Gy(pt) = ev(y′ � y0) = �y0 +−1(y′(pt)) ≡ �y0 +−1(y), +where ev : (pt = pt) → A is the last map in the definition of Gy, which transports pt along a path. +Thus we see that the triangle above commutes, whence G is an equivalence of bands, as required. +□ +Theorem 4.10. We have inverse equivalences +Ω′ : (BAut1(A) →∗ BAut1(A)) ≃ (A →∗ A) : B. +In particular, the type BAut1(A) →∗ BAut1(A) is a set. +Proof. Combine Propositions 4.8 and 4.9. +□ +4.2. Tensoring bands. In this section, we will construct an H-space structure on BAut1(A), where +we continue to assume that A is a central type. This H-space structure is interesting in its own right, +and also implies that BAut1(A) is itself central. It that follows that A is an infinite loop space. +This elementary lemma will come up frequently. +Lemma 4.11. Let P : BAut1(A) → U be a set-valued type family. Then � +Xp P(Xp) is equivalent to +P(pt). + +16 +BUCHHOLTZ, CHRISTENSEN, FLATEN, AND RIJKE +Proof. Since each P(Xp) is a set, � +Xp P(Xp) is equivalent to � +X:U +� +p:A=X P(X|p|0). By path in- +duction, this is equivalent to P(A|refl|0), i.e., P(pt). +□ +A consequence of the following result is that any pointed A-band is trivial. +Proposition 4.12. Let Xp be an A-band. Then there is an equivalence (pt =BAut1(A) Xp) → X. +Proof. By Lemma 4.3, there is an equivalence (pt =BAut1(A) Xp) ≃ (A ≃ X)(˜p). We will show that +evp : (A ≃ X)(˜p) → X is an equivalence. By Lemma 4.11, it’s enough to prove this when Xp ≡ pt, +and this holds because A is central. +□ +We now show that path types between A-bands are themselves banded. This underlies the main +results of this section. +Proposition 4.13. Let Xp and Yq be A-bands. The type Xp =BAut1(A) Yq is banded by A. +Proof. We need to construct a band ∥A = (Xp = Yq)∥0. Since the goal is a set, we may induct on p +and q, thus reducing the goal to ∥A = (pt =BAut1(A) pt)∥0. Using that (pt =BAut1(A) pt) ≃ (A ≃ A)(id) +and that A is central, we may give the set truncation of the inverse of the evaluation fibration at +idA. +□ +The following is an immediate corollary of Proposition 4.12. +Corollary 4.14. For any A-band Xp, the A-band (Xp = Xp) is trivial. +□ +We next define a tensor product of banded types, using the notion of duals of bands. +Write +refl∗ : A = A for the self-identification of A associated to the inversion map id∗ (Definition 3.14) via +univalence. +Definition 4.15. Let Xp be an A-band. The band p∗ :≡ |refl∗| � p is the dual of p, and the A-band +X∗ +p :≡ Xp∗ is the dual of Xp. +Since id∗ is an involution, it follows that taking duals defines an involution on BAut1(A), meaning +that X∗∗ +p += Xp. +Lemma 4.16. We have pt = pt∗ in BAut1(A). +Proof. The underlying type of pt∗ is A, which has a base point, so this follows from Proposition 4.12. +□ +We now show how to tensor types banded by A. +Definition 4.17. For Xp, Yq : BAut1(A), define Xp ⊗ Yq :≡ (X∗ +p = Yq), with the banding from +Proposition 4.13. +It follows from Lemma 4.3 that the type Xp ⊗Yq is equivalent to (X = Y )(p∗ � q). Since taking duals +is an involution, we also have equivalences Xp ⊗ Yq ≡ (X∗ +p = Yq) ≃ (Xp = Y ∗ +q ) ≃ (X = Y )(p � q∗). +Moreover, from Corollary 4.14, we see that X∗ +p ⊗ Xp = pt. +Tensoring defines a binary operation on BAut1(A), and we now show that this operation is sym- +metric. +Proposition 4.18. For any Xp, Yq : BAut1(A), there is a path σ(Xp,Yq) : Xp ⊗ Yq =BAut1(A) Yq ⊗ Xp +such that σpt,pt = 1. +Proof. By univalence and the characterization of paths between bands, we begin by giving an equiv- +alence between the underlying types. The equivalence will be path-inversion, as a map +(X = Y )(p � q∗) −→ (Y = X)(q � p∗). +To see that this is valid it suffices to show that the inversion of p � q∗ is q � p∗. We have: +p � q∗ ≡ p � refl∗ �q = refl∗ �q � p = q � refl∗ � p = q � refl∗ �p ≡ q � p∗, + +CENTRAL H-SPACES AND BANDED TYPES +17 +where we have used associativity of path composition, and that refl∗ = refl∗ by Proposition 3.15. +To prove the transport condition, we may path induct on both p and q which then yields the +following triangle: +(A = A)(refl∗) +(A = A)(refl∗) +A . +evrefl∗ +p�→p +evrefl∗ +Here we are writing evrefl∗ for the composite (A = A)(refl∗) ≃ (A ≃ A)(id∗) +evid∗ +−−−→ A. The horizontal +map is given by path-inversion, which is homotopic to the identity by Proposition 3.15, hence the +triangle commutes. +Paths between paths between banded types correspond to homotopies between the underlying +equivalences. Thus σpt,pt = 1 since path-inversion on (A = A)(refl∗) is homotopic to the identity. +□ +We now use Lemma 2.4 to make BAut1(A) into an H-space. +Theorem 4.19. The binary operation ⊗ makes BAut1(A) into an abelian H-space. +Proof. We start by showing the left unit law. Since pt∗ = pt, we instead consider the goal (pt = Xp) = +Xp. An equivalence between the underlying types is given by Proposition 4.12, which after inducting +on p clearly respects the bands. Using Proposition 4.18 and Lemma 2.4, we obtain the desired H-space +structure. +□ +Corollary 4.20. For a central type A, the type BAut1(A) is again central. Therefore, A is an infinite +loop space, in a unique way. Moreover, every pointed map A →∗ A is infinitely deloopable, in a unique +way. +Proof. That BAut1(A) is central follows from condition (2) of Proposition 3.6, using Theorems 4.10 +and 4.19 as inputs. +That A is a infinite loop space then follows from Proposition 4.4: writing +BAut0 +1(A) :≡ A and BAutn+1 +1 +(A) :≡ BAut1(BAutn +1(A)), we see that BAutn +1(A) is an n-fold delooping +of A. The uniqueness follows from Theorem 4.6. That every pointed self-map is infinitely deloopable +in a unique way follows by iterating Theorem 4.10. +□ +Note that BAut1(A) is essentially small (Remark 4.2), so these deloopings can be taken to be in +the same universe as A. +From Theorem 4.19 we deduce another characterization of central types: +Proposition 4.21. A pointed, connected type A is central if and only if ΣX:BAut1(A) X is contractible. +Proof. If A is central, then by the left unit law of Theorem 4.19, we have +ΣX:BAut1(A) X ≃ ΣX:BAut1(A) (pt∗ =BAut1(A) X) ≃ 1. +Conversely, if ΣX:BAut1(A) X is contractible, then so is its loop space. But the loop space is equivalent +to Σf:A→∗A ∥f = id∥, i.e., the fibre of evid over the base point. Thus evid is an equivalence, since A +is connected. +□ +4.3. Bands and torsors. Let A be a central type. We define a notion of A-torsor which turns out +to be equivalent to the notion of A-band from the previous section. Under our centrality assumption, +it follows that the resulting type of A-torsors is a delooping of A. An equivalent type of A-torsors has +been independently studied by David W¨arn, who has also shown that it gives a delooping of A under +the weaker hypothesis that A has a unique H-space structure. +Definition 4.22. An action of A on a type X is a map α : A × X → X such that α(pt, x) = x for +all x : X. If X has an A-action, we say that it is an A-torsor if it is merely inhabited and α(−, x) is +an equivalence for every x : X. The type of A-torsor structures on a type X is +TA(X) :≡ +� +α:A×X→X +(α(pt, −) = idX) × ∥X∥−1 × +� +x:X +IsEquiv α(−, x), + +18 +BUCHHOLTZ, CHRISTENSEN, FLATEN, AND RIJKE +and the type of A-torsors is � +X:U TA(X). +Since A is connected, an A-action on X is the same as a pointed map A →∗ (X ≃ X)(id). Normally +one would require at a minimum that this map sends multiplication in A to composition. We explain +in Remark 4.28 why our definition suffices. +The condition that α(−, x) is an equivalence for all x is equivalent to requiring that for every +x0, x1 : X, there exists a unique a : A with α(a, x0) = x1. +It is also equivalent to saying that +(α, pr2) : A × X → X × X is an equivalence. +For any type X, write ev≃ : (A ≃ X) → X for the evaluation fibration which sends an equivalence +e to e(pt). For a map f, write Sect(f) for the type of (unpointed) sections of f. +Lemma 4.23. For any X, we have an equivalence +TA(X) ≃ ∥X∥−1 × Sect(ev≃). +Proof. This is simply a reshuffling of the data. The map from left to right sends a torsor structure +with action α : A × X → X to the map X → (A → X) sending x to α(−, x). By assumption, this +lands in the type of equivalences, and the condition α(pt, −) = idX says that it is a section. We leave +the remaining details to the reader. +□ +Lemma 4.24. Let X be an A-torsor. Then X is connected. +Proof. Since X is merely inhabited and our goal is a proposition, we may assume that we have x0 : X. +Then we have an equivalence α(−, x0) : A → X. A is connected by Proposition 3.3, so it follows that +X is. +□ +Proposition 4.25. Let X be an A-torsor. Then X is banded by A. +Proof. Associated to the torsor structure on X is a section X → (A ≃ X) of ev≃. +Since X is +0-connected, it lands in a component of A ≃ X. By univalence, this determines a banding of X. +□ +Theorem 4.26. Let X be a type. There is an equivalence TA(X) ≃ ∥A = X∥0. Therefore, there is +an equivalence between the type of A-torsors and BAut1(A). +Proof. Proposition 4.25 gives a map f. We check that the fibres are contractible. Let p : ∥A = X∥0 be +a banding of X. An A-torsor structure t on X with f(t) = p consists of a section s of ev≃ that lands in +the component (A ≃ X)(˜p), where ˜p denotes the equivalence associated to p. But by Proposition 4.12, +the evaluation fibration (A ≃ X)(˜p) → X is an equivalence, so it has a unique section. +□ +Remark 4.27. It follows that TA(X) is a set. One can also show this using Corollary 4.14 and Propo- +sition 3.6. +Remark 4.28. Let X be an A-torsor, or equivalently, an A-band. By Corollary 4.14, we have an +equivalence e : A ≃ (X ≃ X)(id). Since A has a unique H-space structure, this equivalence is an +equivalence of H-spaces, where the codomain has the H-space structure coming from composition. +Since A is connected, the A-action on X gives a map α′ : A →∗ (X ≃ X)(id). (In fact, α′ = e, but +we won’t use this fact.) Using the equivalence e, it follows from Theorem 4.10 that any map with the +same type as α′ is deloopable in a unique way. That is, it has the structure of a group homomorphism +in the sense of higher groups (see [BvDR18]). This explains why our naive definition of an A-action +is correct in this situation. +5. Examples and non-examples +We show that the Eilenberg–Mac Lane spaces K(G, n) are central whenever G is abelian and n > 0. +In addition, we produce examples of products of Eilenberg–Mac Lane spaces which are central and +examples which are not central. At present, we do not know whether there exist central types which are +not products of Eilenberg–Mac Lane spaces. Along the way, we use our results to give a self-contained, +independent construction of Eilenberg–Mac Lane spaces. To this end, we begin by discussing the base +case K(G, 1). + +CENTRAL H-SPACES AND BANDED TYPES +19 +5.1. The H-space of G-torsors. Given a group G, we construct the type TG of G-torsors and show +that it is a K(G, 1). Specifically, a pointed type X is a K(G, 1) if it is connected and comes equipped +with a pointed equivalence ΩX ≃∗ G which sends composition of loops to multiplication in G. (We +always point ΩX at refl.) +When G is abelian, we can tensor G-torsors to obtain an H-space structure on TG which is analogous +to the tensor product of bands of Theorem 4.19. These constructions are all classical and we therefore +omit some details. +Definition 5.1. Let G be a group. A G-set is a set X with a group homomorphism α : G → Aut(X). +If the set X is merely inhabited and the map α(−, x) : G → X is an equivalence for every x : X, then +(X, α) is a G-torsor. We write TG for the type of G-torsors. Given two G-sets X and Y , we write +X →G Y for the set of G-equivariant maps from X to Y , defined in the usual way. +We may write g · x instead of α(g, x) when no confusion can arise. The following is straightforward +to check: +Lemma 5.2. Let X and Y be G-torsors. There is a natural equivalence (X =T G Y ) ≃ (X →G Y ). +In particular, a G-equivariant map between G-torsors is automatically an equivalence. +□ +Any group G acts on itself by left translation, making G into a G-torsor which constitutes the base +point pt of both TG and the type of G-sets. Since a G-equivariant map pt →G X is determined by +where it sends 1 : G, the map (pt →G X) → X that evaluates at 1 is an equivalence. It is clear that +the type TG is a 1-type, which implies that its loop space is a group. +Proposition 5.3. We have a group isomorphism ΩTG ≃ G. +We only sketch a proof since this is a classical result. +Proof. Since paths between G-torsors correspond to G-equivariant maps, we have equivalences of sets +(pt =T G pt) ≃ (pt →G pt) ≃ G, +where the second equivalence is given by evaluation at 1. The first equivalence sends path compo- +sition to composition of maps, which reverses the order—i.e., it’s an anti-isomorphism. The second +equivalence evaluates a map at 1 : G. Thus, for φ, ψ : pt →G pt we have +φ(ψ(1)) = φ(ψ(1) · 1) = ψ(1) · φ(1), +where · denotes the multiplication in G. +In other words, evaluation at 1 is an anti-isomorphism, +meaning the composite (pt =T G pt) ≃ G is an isomorphism of groups. +□ +The following proposition says that the G-torsors are precisely those G-sets which lie in the com- +ponent of the base point. +Proposition 5.4. A G-set (X, α) is a G-torsor if and only if there merely exists a G-equivariant +equivalence from pt to X. +Proof. Suppose X is a G-torsor. To produce a mere G-equivariant equivalence pt ≃G X we may +assume we have some x : X, since X is merely inhabited. Then (−) · x : G → X yields an equivalence +which is clearly G-equivariant, as required. +Conversely, assume that there merely exists a G-equivariant equivalence from pt to X. Since being +a G-torsor is a proposition, we may assume we have an actual G-equivariant equivalence. But then +we are done since pt is a G-torsor. +□ +It follows that TG is connected. Thus by Proposition 5.3 we deduce: +Corollary 5.5. The type TG is a K(G, 1). +For the remainder of this section, let G be an abelian group. +Proposition 5.6. For any two G-torsors S and T, the path type S =T G T is again a G-torsor. + +20 +BUCHHOLTZ, CHRISTENSEN, FLATEN, AND RIJKE +Proof. First we make S =T G T into a G-set. This path type is equivalent to the type S →G T. Using +that G is abelian, it’s easy to check that the map +(g, φ) �−→ +� +s �→ g · φ(s) +� +: G × (S →G T) −→ (S →G T) +is well-defined and makes S →G T into a G-set. +To check that the above yields a G-torsor, we may assume that S ≡ pt ≡ T, by the previous lemma. +One can check that Proposition 5.3 gives an equivalence of G-sets, where pt →G pt is equipped with +the G-action just described. Thus pt →G pt is a G-torsor, as required. +□ +In order to describe the tensor product of G-torsors, we first need to define duals. +Definition 5.7. Let (X, α) be a G-torsor. The dual X∗ of X is the G-torsor X with action +α∗(g, x) :≡ α(g−1, x). +The tensor product of G-torsors is now defined as X ⊗ Y :≡ (X∗ =T G Y ). +Proposition 5.8. The tensor product of G-torsors makes TG into an H-space. +Proof. We verify the hypotheses of Lemma 2.4. Thus our first goal is to construct a symmetry +σX,Y : (X∗ =T G Y ) =T G (Y ∗ =T G X). +After identifying paths of G-torsors with G-equivariant equivalences, we may consider the map which +inverts such an equivalence. A short calculation shows that if φ : X∗ →G Y is G-equivariant, then +φ−1 : Y ∗ →G X is again G-equivariant. We need to check that the map sending φ to φ−1 is itself +G-equivariant, so let φ : X∗ →G Y and let g : G. Since the inverse of g · (−) is g−1 · (−), we have: +(g · φ)−1 = φ−1(g−1 · (−)) = g · φ−1(−), +using that φ−1 : Y ∗ →G X is G-equivariant. Thus inversion is G-equivariant, yielding the required +symmetry σ. +Now we argue that σpt,pt = refl, or, equivalently, that maps pt∗ →G pt are their own inverses. Such +a map is uniquely determined by where it sends 1 : G, so it suffices to show that φ(φ(1)) = 1 for every +φ : pt∗ →G pt. Fortunately, we have +φ(φ(1)) = φ(φ(1) · 1) = φ(1)−1 · φ(1) = 1. +Lastly, it is straightforward to check that the map (pt∗ →G X) → X which evaluates at 1 : G is +G-equivariant, for any G-torsor X. This yields the left unit law for the tensor product ⊗. As such we +have fulfilled the hypotheses of Lemma 2.4, giving us the desired H-space structure. +□ +Using Proposition 3.6, one can check that TG is a central H-space. (See Proposition 5.9.) +5.2. Eilenberg–Mac Lane spaces. We now use our results to give a new construction of Eilenberg– +Mac Lane spaces. For an abelian group G, recall that a pointed type X is a K(G, 1) if it is connected +and there is a pointed equivalence ΩX ≃∗ G which sends composition of paths to multiplication in +G. For n > 1, a pointed type X is a K(G, n + 1) if it is connected and ΩX is a K(G, n). It follows +that such an X is an n-connected (n + 1)-type with Ωn+1X ≃∗ G as groups. +In the previous section we saw that the type TG of G-torsors is a K(G, 1) and is central whenever +G is abelian. The following proposition may be seen as a higher analog of this fact. +Proposition 5.9. Let G be an abelian group and let n > 0. If a type A is a K(G, n) and an H-space, +then A is central and BAut1(A) is a K(G, n + 1) and an H-space. +The fact that BAut1(A) is a K(G, n + 1) also follows from [Shu], using the fact that BAut1(A) is +the 1-connected cover of BAut(A). +Proof. Suppose that A is a K(G, n) and an H-space. +Then A →∗ ΩA is contractible, since it is +equivalent to ∥A∥n−1 →∗ ΩA, and ∥A∥n−1 is contractible. +So Proposition 3.6 implies that A is +central. +By Proposition 4.4, Ω BAut1(A) ≃ A, so BAut1(A) is a K(G, n + 1). +By Theorem 4.19, +BAut1(A) is also an H-space. +□ + +REFERENCES +21 +We can use the previous proposition to define K(G, n) for all n > 0 by induction. For the base case +n ≡ 1 we let K(G, 1) :≡ TG, the type of G-torsors from the previous section. When G is abelian, we +saw that TG is an H-space, which lets us apply the previous proposition. By induction, we obtain a +K(G, n) for all n. Note that this construction produces a K(G, n) which lives n − 1 universes above +the given K(G, 1), but that it is essentially small by the join construction [Rij17]. +5.3. Products of Eilenberg–Mac Lane spaces. Here is our first example of a central type that is +not an Eilenberg–Mac Lane space. +Example 5.10. Let K = K(Z/2, 1) = RP ∞ and L = K(Z, 2) = CP ∞, and consider A = K × L. +This is a connected H-space, and +� +K × L →∗ Ω(K × L) +� +≃ +� +K →∗ Ω(K × L) +� +since K = ∥K × L∥1 +≃ +� +K →∗ ΩL +� +since K is connected +≃ +� +Z/2 →Ab Z) +by [BvDR18, Theorem 5.1] +≃ 1. +So it follows from Proposition 3.6(4) that A is central. +On the other hand, not every product of Eilenberg–Mac Lane spaces is central. +Example 5.11. Let K = K(Z/2, 1) = RP ∞ and L′ = K(Z/2, 2). A calculation like the above shows +that K × L′ →∗ Ω(K × L′) is not contractible, so K × L′ is not central. +As another example, [Cur68, Proposition Ia] shows that K(Z, 1) × K(Z, 2)) (i.e., S1 × CP ∞) has +infinitely many distinct H-space structures classically. So it is not central, by Proposition 3.3. +Clearly both of these examples can be generalized to other groups and shifted to higher dimensions. +By Proposition 3.3, centrality of a type implies that it has a unique H-space structure. +The +converse fails, as we now demonstrate. We are grateful to David W¨arn for bringing our attention to +this example. +Example 5.12. The type A :≡ K(Z, 2) × K(Z, 3) is not central, by a computation similar to the one +in the previous example. However, we note that it admits a unique H-space structure. Since A is a +loop space it admits an H-space structure, and the type of H-space structures is given by A ∧ A →∗ A +according to Theorem 2.27. Since A is 1-connected, by [CS20, Corollary 2.32] the smash product +A ∧ A is 3-connected. It follows that A ∧ A →∗ A is contractible, since A is 3-truncated. In other +words, the space of H-space structures on A is contractible. +References +[AC63] +M. Arkowitz and C. R. Curjel. “On the number of multiplications of an H–space”. In: +Topology 2 (1963), pp. 205–207. +[BR18] +Ulrik Buchholtz and Egbert Rijke. “The Cayley-Dickson construction in homotopy type +theory”. In: High. Struct. 2.1 (2018), pp. 30–41. doi: https://doi.org/10.21136/HS. +2018.02. +[Bru16] +Guillaume Brunerie. “On the homotopy groups of spheres in homotopy type theory”. +PhD thesis. Laboratoire J.A. Dieudonn´e, 2016. arXiv: 1606.05916. +[Buc19] +Ulrik Buchholtz. Non-abelian cohomology (Groups, Torsors, Gerbes, Bands & all that). +Invited talk at the workshop Geometry in Modal Homotopy Type Theory, Carnegie Mellon +University. 2019. url: https://youtu.be/eB6HwGLASJI. +[BvDR18] +U. Buchholtz, F. van Doorn, and E. Rijke. “Higher Groups in Homotopy Type Theory”. +In: Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science. +LICS ’18. Oxford, United Kingdom: ACM, 2018, pp. 205–214. isbn: 978-1-4503-5583-4. +doi: 10.1145/3209108.3209150. + +22 +REFERENCES +[Cav21] +Evan Cavallo. Pointed functions into a homogeneous type are equal as soon as they are +equal as unpointed functions. Agda formalization, part of the cubical library. 2021. url: +https://agda.github.io/cubical/Cubical.Foundations.Pointed.Homogeneous. +html#1616. +[Cop59] +A. H. Copeland. “Binary operations on sets of mapping classes.” In: Michigan Mathemat- +ical Journal 6 (1959), pp. 7–23. +[CS20] +J. Daniel Christensen and Luis Scoccola. The Hurewicz theorem in homotopy type theory. +To appear in Algebraic & Geometric Topology. 2020. arXiv: 2007.05833v2. +[Cur68] +C. R. Curjel. “On the H-space structures of finite complexes”. In: Comment. Math. Helv. +43 (1968), pp. 1–17. doi: 10.1007/BF02564376. +[Han74] +Vagn Lundsgaard Hansen. “The homotopy problem for the components in the space of +maps on the n-sphere”. In: Q. J. Math. 25.1 (Jan. 1974), pp. 313–321. eprint: https: +//academic.oup.com/qjmath/article-pdf/25/1/313/4366416/25-1-313.pdf. +[Jam55] +I. M. James. “Reduced product spaces”. In: Ann. of Math. (2) 62 (1955), pp. 170–197. +doi: 10.2307/2007107. +[Rij17] +E. Rijke. The join construction. 2017. arXiv: 1701.07538. +[Sco20] +Luis Scoccola. “Nilpotent types and fracture squares in homotopy type theory”. In: +Mathematical Structures in Computer Science 30.5 (2020), pp. 511–544. doi: 10.1017/ +s0960129520000146. +[Shu] +Mike Shulman. Fibrations with fiber an Eilenberg-MacLane space. Blog post at homotopy- +typetheory.org. url: https://homotopytypetheory.org/2014/06/30/fibrations- +with-em-fiber/. +[Uni13] +Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Math- +ematics. Institute for Advanced Study: http://homotopytypetheory.org/book/, 2013. +[vDoo18] +Floris van Doorn. “On the Formalization of Higher Inductive Types and Synthetic Ho- +motopy Theory”. PhD thesis. Carnegie Mellon University, 2018. arXiv: 1808.10690. +[Whi46] +George W. Whitehead. “On products in homotopy groups”. In: Annals of Mathematics +47 (1946), pp. 460–475. +University of Nottingham, Nottingham, United Kingdom +Email address: ulrik.buchholtz@nottingham.ac.uk +University of Western Ontario, London, Ontario, Canada +Email address: jdc@uwo.ca +University of Western Ontario, London, Ontario, Canada +Email address: jtaxers@uwo.ca +University of Ljubljana, Ljubljana, Slovenia +Email address: e.m.rijke@gmail.com + diff --git a/9dE0T4oBgHgl3EQfwwE1/content/tmp_files/load_file.txt b/9dE0T4oBgHgl3EQfwwE1/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8a021c4906211c6098c3def9c1216f5bd94b7925 --- /dev/null +++ b/9dE0T4oBgHgl3EQfwwE1/content/tmp_files/load_file.txt @@ -0,0 +1,1268 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf,len=1267 +page_content='CENTRAL H-SPACES AND BANDED TYPES ULRIK BUCHHOLTZ, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' DANIEL CHRISTENSEN, JARL G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' TAXER˚AS FLATEN, AND EGBERT RIJKE Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' We introduce and study central types, which are generalizations of Eilenberg–Mac Lane spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' A type is central when it is equivalent to the component of the identity among its own self- equivalences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' From centrality alone we construct an infinite delooping in terms of a tensor product of banded types, which are the appropriate notion of torsor for a central type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Our constructions are carried out in homotopy type theory, and therefore hold in any ∞-topos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Even when interpreted into the ∞-topos of spaces, our main results and constructions are new.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' In particular, we give a description of the moduli space of H-space structures on an H-space which generalizes a formula of Arkowitz–Curjel and Copeland which counts the number of path components of this moduli space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Contents 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Introduction 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' H-spaces and evaluation fibrations 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' H-space structures 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Evaluation fibrations 7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Unique H-space structures 9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Central types 10 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Bands and torsors 13 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Types banded by a central type 13 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Tensoring bands 15 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Bands and torsors 17 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Examples and non-examples 18 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' The H-space of G-torsors 19 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Eilenberg–Mac Lane spaces 20 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Products of Eilenberg–Mac Lane spaces 21 References 21 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Introduction In this paper we study H-spaces and their deloopings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' We work in homotopy type theory, so our results apply to any ∞-topos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Many of our results are new, even for the ∞-topos of spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' A key concept is that of a central type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' A pointed type A is central if the map (A → A)(id) → A sending a function f to f(pt) is an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Here (A → A)(id) denotes the identity component of the type of all self-maps of A, and pt denotes the base point of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Every central type is a connected H-space, and a connected H-space is central precisely when the type A →∗ A of pointed self-maps is a set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' We prove this and other characterizations of central types in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' It follows, for example, that every Eilenberg–Mac Lane space K(G, n), with G abelian and n ≥ 1, is central.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' We show in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='3 that some, but not all, products of Eilenberg–Mac Lane spaces are central.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' We don’t know whether every central type is a product of Eilenberg–Mac Lane spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Our first result is: Date: January 6, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='02636v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='AT] 6 Jan 2023 2 BUCHHOLTZ, CHRISTENSEN, FLATEN, AND RIJKE Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Let A be a central type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Then A has a unique delooping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' The key ingredient of this result and much of the paper is that we have a concrete description of the delooping of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' It is given by the type BAut1(A) :≡ ΣX:U∥A = X∥0 of types banded by A, which is the 1-connected cover of BAut(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' As an example, since K(G, n) is central for G abelian and n ≥ 1, this gives an alternative way to define K(G, n + 1) in terms of K(G, n), as previously indicated by the first author [Buc19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' We also show: Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Let A be a central type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Then every pointed map f : A →∗ A is uniquely deloopable to a map Bf : BAut1(A) →∗ BAut1(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' It follows that the type of pointed self-maps of BAut1(A) is a set, since it is equivalent to A →∗ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' One of the motivations for studying BAut1(A) is that one can define a tensoring operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Given two banded types X and Y in BAut1(A), the type X∗ = Y has a natural banding, where X∗ is a certain dual of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' We write X ⊗ Y for this banded type, and show in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='19 that it makes BAut1(A) into an abelian H-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Combined with Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='6, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='10, and the characterization of central types mentioned earlier, we therefore deduce: Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' For a central type A, the type BAut1(A) is again central.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Therefore, A is an infinite loop space, in a unique way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Moreover, every pointed map A →∗ A is infinitely deloopable, in a unique way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Our tensoring operation gives a new description of the H-space structure on K(G, n), which will be helpful for calculations of Euler classes in work in progress and is what originally motivated this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' We also give an alternate description of the delooping of a central type A as a certain type of A-torsors, and give an analogous description of K(G, 1) for any group G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' To prove the above results, we first need to further develop the theory of H-spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' One difference between our work and classical work in topology is that we emphasize the moduli space HSpace(A) of H-space structures on a pointed type A, rather than just the set of components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' For example, we prove: Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Let A be an H-space such that for all a : A, the map a · − is an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Then the type HSpace(A) of H-space structures on A is equivalent to the type A ∧ A →∗ A of pointed maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' This generalizes a classical formula of Arkowitz–Curjel and Copeland, which plays a key role in classical results on the number of H-space structures on various spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' The classical formula only de- termines the path components of the type of H-space structures, while our formula gives an equivalence of types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' From our formula it immediately follows, for example, that the type of H-space structures on the 3-sphere is Ω6S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' The proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='27 uses evaluation fibrations, which generalize the map appearing in the definition of “central.” In fact, these evaluation fibrations play an important role in much of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' For example, we include results relating the existence of sections of an evaluation fibration to the vanishing of Whitehead products, and use this to show that no even spheres besides S0 admit H-space structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' In Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='3 we show that every central type has a unique H-space structure, in the strong sense that the type HSpace(A) is contractible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' We prove several results about types with unique H-space structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' For example, we show that such H-space structures are associative and coherently abelian, and that every pointed self-map is an H-space map, a weaker version of the delooping above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' We also give an example showing that not every type with a unique H-space structure is central.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' We note that these results rely on us defining “H-space” to include a coherence between the two unit laws (see Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Outline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' In Section 2, we give results about H-spaces which do not depend on centrality, including a description of the moduli space of H-space structures, results about Whitehead products and H-space CENTRAL H-SPACES AND BANDED TYPES 3 structures on spheres, and results about unique H-space structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' In Section 3, we define central types, show that central types have a unique H-space structure, give a characterization of which H- spaces are central, and prove other results needed in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Section 4 is the heart of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' It defines the type BAut1(A) of bands for a central type A, shows that it is a unique delooping of A, proves that it is an H-space under a tensoring operation, and shows that central types and their self-maps are uniquely infinitely deloopable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' We also give the alternate description of the delooping in terms of A-torsors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Finally, Section 5 gives examples and non-examples of central types, mostly related to Eilenberg–Mac Lane spaces and their products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Notation and conventions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' In general, we follow the notation used in [Uni13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' For example, we write path composition in diagrammatic order: given paths p : x = y and q : y = z, their composite is p � q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' The reflexivity path is written refl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Given a type A and an element a : A, we write (A, a) for the type A pointed at a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' If A is already a pointed type with unspecified base point, then we write pt for the base point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' If A and B are pointed types, and f, g : A →∗ B are pointed maps, then f =∗ g is the type of pointed homotopies between f and g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' If A is an H-space, then we write x · y for the product of two elements x, y : A (unless another notation for the multiplication is given).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' For a pointed type A, we write HSpace(A) for the type of H-space structures on A with the basepoint as the identity element (Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' We write Sn for the n-sphere, and U for a fixed universe of types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' We would like to thank David Jaz Myers for many lively discussions on the content of this paper, especially related to bands and torsors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' We also thank David W¨arn for fruitful discussions and for sharing drafts of his forthcoming work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Egbert Rijke gratefully acknowledges the support by the Air Force Office of Scientific Research through grant FA9550-21-1-0024, and support by the Slovenian Research Agency research programme P1-0294.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Dan Christensen and Jarl Flaten both acknowledge the support of the Natural Sciences and Engineering Research Council of Canada (NSERC), RGPIN-2022-04739.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' H-spaces and evaluation fibrations In Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='1, we begin by recalling the notion of a (coherent) H-space structure on a pointed type A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' We discuss the type of pointed extensions of a map B ∨ C →∗ A to B × C, and show that the type of H-space structures on A is equivalent to the type of pointed extensions of the fold map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' We relate the existence of extensions to the vanishing of Whitehead products, and use this to show that there are no H-space structures on even spheres except S0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' In addition, we show that for any n-connected H-space A, the Freudenthal map π2n+1(A) → π2n+2(ΣA) is an isomorphism, not just a surjection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' In Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='2, we study evaluation fibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' We show that the type of H-space structures is equivalent to a type of sections of an evaluation fibration, and use this to show that the type of H-space structures on a left-invertible H-space A is equivalent to A ∧ A →∗ A, generalizing a classical formula of Arkowitz–Curjel and Copeland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' It immediately follows, for example, that the type of H- space structures on the 3-sphere is Ω6S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' We end with a result relating the existence of sections of an evaluation fibration to the vanishing of Whitehead products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='3 is a short section which studies the case when the type of H-space structures is con- tractible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' We stress that this is not the same as HSpace(A) having a single component, which is what is classically meant by “A has a unique H-space structure.” This situation is interesting in its own right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' We show that such H-space structures are associative and coherently abelian, and we prove that all pointed self-maps of A are automatically H-space maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' H-space structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' We begin by giving the notion of H-space structure that we will consider in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Let A be a pointed type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' (1) A non-coherent H-space structure on A consists of a binary operation µ : A → A → A, along with two homotopies µl : µ(pt, −) = idA and µr : µ(−, pt) = idA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' 4 BUCHHOLTZ, CHRISTENSEN, FLATEN, AND RIJKE (2) A (coherent) H-space structure on A consists of a non-coherent H-space structure µ on A along with a coherence µlr : µl(pt) =µ(pt,pt)=pt µr(pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' (3) We write HSpace(A) for the type of (coherent) H-space structures on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' When the H-space structure is clear from the context we may write x · y :≡ µ(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Any H-space structure yields a non-coherent H-space structure by forgetting the coherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Suppose A has a (non)coherent H-space structure µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' (4) If µ(a, −) : A → A is an equivalence for all a : A, then µ is left-invertible, and we write x\\y :≡ µ(x, −)−1(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Right-invertible is defined dually, and we write x/y :≡ µ(−, y)−1(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' (5) The twist µT of µ is the natural (non)coherent H-space structure with operation µT (a0, a1) :≡ µ(a1, a0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' When we say “H-space” we mean the coherent notion—we will only say “coherent” for emphasis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' The notion of H-space structure considered in [Uni13, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='4] corresponds to our non-coherent H- space structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' While many constructions can be carried out for non-coherent H-spaces (such as the Hopf construction), the coherent case is more natural for our purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Moreover, any non-coherent H-space can be made coherent by simply changing one of the unit laws: Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Any non-coherent H-space structure on a pointed type A gives rise to a coherent H-space structure with the same underlying binary operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Let (A, µ, µl, µr) be a non-coherent H-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' We define a new homotopy µ′ r : µ(−, pt) = id as the concatenation of paths µ(x, pt) µ(x, µ(pt, pt)) µ(x, pt) x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' apµ(x)(µr(pt))−1 apµ(x)(µl(pt)) µr(x) We claim that µl(pt) = µ′ r(pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' To see this, it suffices to show that the square µ(pt, µ(pt, pt)) µ(pt, pt) µ(pt, pt) pt apµ(pt)(µl(pt)) apµ(pt)(µr(pt)) µr(pt) µl(pt) commutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' We will show that the top path is equal to µl(µ(pt, pt)), and this turns the square into a naturality square for the homotopy µl, which always commutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' To see that apµ(pt)(µl(pt)) = µl(µ(pt, pt)), observe that µl is a homotopy µ(pt) = id, and for any homotopy H : f = id we have apf Hx = Hf(x) for all x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' □ The proposition implies that the types of non-coherent and coherent H-space structures on a pointed type are logically equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' However, they are not generally equivalent as types (see Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' We’ll be interested in abelian and associative H-spaces later on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Let A be an H-space with multiplication µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' (1) If there is a homotopy h : Πa,bµ(a, b) = µ(b, a) then µ is abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' (2) If µ = µT in HSpace(A) then µ is coherently abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' (3) If there is a homotopy α : Πa,b,c:Aµ(µ(a, b), c) = µ(a, µ(b, c)) then µ is associative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' The following lemma gives a convenient way of constructing abelian H-space structures, and will be used in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Let A be a pointed type with a binary operation µ, a symmetry σa,b : µ(a, b) = µ(b, a) for every a, b : A such that σpt,pt = refl, and a left unit law µl : µ(pt, −) = idA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Then A becomes an abelian H-space with the right unit law induced by symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' CENTRAL H-SPACES AND BANDED TYPES 5 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' For b : A, the right unit law is given by the path σb,pt � µl(b) of type µ(b, pt) = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' For coherence we need to show that the following triangle commutes: µ(pt, pt) µ(pt, pt) pt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' µl σpt,pt µl By our assumption that σpt,pt = refl, the triangle is filled reflµl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' □ For any right-invertible H-space A, for b : A one can define the two operations (−)/b and (−)·(pt/b) of type A → A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' If A is associative, then these coincide: Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Let A be an associative H-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' For any a, b : A, we have that a/b = a · (pt/b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' For all a, b : A we have (a · (pt/b)) · b = a · ((pt/b) · b) = a · pt = a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Thus by dividing by b on the right, we deduce a · (pt/b) = a/b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' □ We collect a few basic facts about H-spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' The following lemma generalizes a result of Evan Cavallo, who formalized the fact that unpointed homotopies between pointed maps into a homogeneous type A can be upgraded to pointed homotopies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Being a homogeneous type is logically equivalent to being a left-invertible H-space [Cav21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Here we do not need to assume left-invertibility, and we factor this observation through a further generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Let A be a pointed type, and consider the following three conditions: (1) A is an H-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' (2) The evaluation map (idA = idA) → (pt = pt) has a section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' (3) For every pointed type B and pointed maps f, g : B →∗ A, there is a map (f = g) → (f =∗ g) which upgrades unpointed homotopies to pointed homotopies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Then (1) implies (2) and (2) implies (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' To show that (1) implies (2), suppose that A is an H-space, and let p : pt = pt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' For any x : A we define the path px : x = x to be the concatenation x x · pt x · pt x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' µ−1 r apµ(x)(p) µr This defines a map s : (pt = pt) → (idA = idA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' To see that this map is a section of the evaluation map, it suffices to show that the square pt · pt pt · pt pt pt apµ(pt)(p) µr µr p commutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' To see this, note that µr = µl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' If we replace µr by µl in the above square, we obtain a naturality square of homotopies, which always commutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' We next show that (2) implies (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Let f, g : B →∗ A be pointed maps and let H : f = g be an unpointed homotopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' By path induction on H, we can assume we have a single function f : B → A with two pointings, fpt and f ′ pt : f(pt) = pt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Our goal is to define a homotopy K : f = f such that Kpt = r, where r :≡ fpt · f ′pt : f(pt) = f(pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' By path induction on fpt, we can assume that the basepoint of A is f(pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' By (2), we have s : (f(pt) = f(pt)) → (idA = idA) such that s(p, f(pt)) = p for all p : f(pt) = f(pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' For b : B, define Kb to be s(r, f(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Then Kpt = r, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' □ The following result is straightforward and has been formalized, so we do not include a proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' 6 BUCHHOLTZ, CHRISTENSEN, FLATEN, AND RIJKE Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Suppose A is a (left-invertible) H-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' For any pointed type B, the mapping type B →∗ A based at the constant map is naturally a (left-invertible) H-space under pointwise mul- tiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Similarly, for any type B, the mapping type B → A based at the constant map is a (left-invertible) H-space under pointwise multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' □ In particular, if A is left-invertible then for any f : B →∗ A there is a self-equivalence of B →∗ A which sends the constant map to f—namely, the pointwise multiplication by f on the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Our next goal is to rule out H-space structures on even spheres using Brunerie’s computation of Whitehead products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' (See [Bru16, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='3] for their definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=') To do so, we prove some results about Whitehead products from [Whi46] which relate to H-spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Let α : B →∗ A and β : C →∗ A be pointed maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' An (α, β)-extension is a pointed map f : B × C →∗ A equipped with a pointed homotopy filling the following diagram: B ∨ C A B × C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' α∨β f Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' It is equivalent to consider the type of unpointed (α, β)-extensions consisting of unpointed maps f : B × C → A and unpointed fillers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' The additional data in a pointed extension is a path fpt : f(pt, pt) = pt and a 2-path that determines fpt in terms of the other data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' These form a contractible pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' When α and β are maps between spheres, Whitehead instead says that f is “of type (α, β)” but we prefer to stress that we work with a structure and not a property, as the following lemma illustrates: Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' H-space structures on a pointed type A correspond to (idA, idA)-extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' □ The proof consists of straightforward reshuffling of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' If A is an H-space, then there is an (α, β)-extension for every pair α : B →∗ A and β : C →∗ A of pointed maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Using naturality of the left and right unit laws and coherence, one can show that the map (b, c) �→ α(b)·β(c) : B ×C → A is an (α, β)-extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Alternatively, observe that the (α, β)-extension problem factors through the (idA, idA)-extension problem via the map α × β : B × C → A × A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' □ The lemmas explain the relation between H-space structures and (α, β)-extensions, which are in turn related to Whitehead products via the next two results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='12 ([Whi46, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Let m, n > 0 be natural numbers and consider two pointed maps α : Sm →∗ A and β : Sn →∗ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' The type of (α, β)-extensions is equivalent to the type of witnesses that the map [α, β] : Sm+n−1 →∗ A is constant (as a pointed map).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Consider the diagram of pointed maps below, where the composite of the top two maps is [α, β] and the left diamond is a pushout of pointed types: Sm ∨ Sn Sm+n−1 Sm × Sn A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' 1 α∨β f An (α, β)-extension is the same as a pointed map f along with a pointed homotopy filling the top-right triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Since the bottom-right triangle is filled by a unique pointed homotopy, an (α, β)-extension thus corresponds exactly to the data of a filler in the outer diagram, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=', a homotopy witnessing that [α, β] is constant as a pointed map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' □ CENTRAL H-SPACES AND BANDED TYPES 7 With the notation of the previous proposition, we have the following: Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='13 ([Whi46, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Suppose A is an H-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Then [α, β] is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' The follows from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='11 and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' □ Using the above results, we can rule out H-space structures on even spheres in positive dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' The n-sphere merely admits an H-space structure if and only if [ιn, ιn] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' In particular, there are no H-space structures on the n-sphere when n > 0 is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' The implication (→) is immediate by Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Conversely, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='12 implies that [ιn, ιn] = 0 if and only if an (idSn, idSn)-extension merely exists, which by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='10 happens if and only if Sn merely admits an H-space structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Finally, Brunerie showed that [ιn, ιn] = 2 in π2n−1(Sn) for even n > 0 [Bru16, Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='4], which by the above implies that Sn cannot admit an H-space structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' □ We also record the following result and a corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Let A be a left-invertible H-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' The unit η : A →∗ ΩΣA has a pointed retrac- tion, given by the connecting map δ : ΩΣA →∗ A associated to the Hopf fibration of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Let δ : ΩΣA →∗ A be the connecting map associated to the Hopf fibration of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Recall that for a loop p : N = N, we have δ(p) :≡ p∗(pt) where p∗ : A → A denotes transport and A is the fibre above N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' By definition of the Hopf fibration, a path merid(a) : N =ΣA S sends an element x of the fibre A to a · x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Now define a homotopy δ ◦ η = id by δ(η(a)) ≡ δ(merid(a) � merid(pt)−1) = merid(pt)−1 ∗ (merid(a)∗(pt)) ≡ pt\\(a · pt) = a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Finally, we promote this to a pointed homotopy using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' □ It follows that for any n-connected H-space A, the Freudenthal map π2n+1(A) → π2n+2(ΣA) is an isomorphism, not just a surjection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' In particular, we have: Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' The natural map π5(S3) → π6(S4) is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' □ The fact that the unit η : A →∗ ΩΣA has a retraction when A is a left-invertible H-space also follows from James’ reduced product construction, as shown in [Jam55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Using [Bru16], one can see that this goes through in homotopy type theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' However, the above argument is much more elementary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' We don’t know if this argument had been observed before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Evaluation fibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' We now begin our study of evaluation fibrations and their relation to H-space structures and (α, β)-extensions from the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Given a pointed map f : B →∗ A, we will simply write ev : (B → A, f) →∗ A for the map which evaluates at pt : B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' This map is pointed since f is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' If no map f is specified, then we mean that f ≡ id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' In a moment we will define evaluation fibrations to be the restriction of ev to a component, but first we make a useful observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Let e : X →∗ A and g : B →∗ A be pointed maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' A pointed lift of g through e consists of a pointed map s : B →∗ X along with a pointed homotopy e ◦ s =∗ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' If g ≡ id, then s is more specifically a pointed section of e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Let f : B →∗ A and g : C →∗ A be pointed maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' The type of (f, g)-extensions is equivalent to the type of pointed lifts of g through ev : (B → A, f) →∗ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' □ We stress that the domain of ev is the type of unpointed maps B → A, pointed by (the underlying map of) f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' The proof of the statement is a straightforward reshuffling of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Diagrammatically, it gives a correspondence between the dashed arrows below, with pointed homotopies filling the triangles: B ∨ C A (B → A, f) B × C C A f∨g ev g 8 BUCHHOLTZ, CHRISTENSEN, FLATEN, AND RIJKE Combining Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='10 with the previous proposition, we deduce: Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Let A be a pointed type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' The type of H-space structures on A is equivalent to the type of pointed sections of ev : (A → A, id) →∗ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' □ Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Phrased another way, an H-space structure on a pointed type A is equivalent to a family µ : Π(a:A)(A, pt) →∗ (A, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' If A is a higher inductive type with a point pt, one can define µ(pt) :≡ id to simplify the task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Let A be a type and a : ∥A∥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' The path component of a in A is A(a) :≡ Σa′:A(|a′|0 = a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' If a : A then we abuse notation and write A(a) for A(|a|0), and in this case A(a) is pointed at (a, refl).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' For any pointed map α : B →∗ A, the evaluation fibration (at α) is the pointed map evα : (B → A)(α) →∗ A induced by evaluating at the base point of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Observe that the component (A → A)(id) is equivalent to (A ≃ A)(id), since being an equivalence is a property of a map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' We permit ourselves to pass freely between the two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Since pointed maps out of connected types land in the component of the base point of the codomain, we have the following consequence of Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Let A be a pointed, connected type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' The type of H-space structures on A is equivalent to the type of pointed sections of evid : (A ≃ A)(id) →∗ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' □ For certain H-spaces, various evaluation fibrations become trivial: Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Suppose A is a left-invertible H-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' We have a pointed equivalence over A (A → A) (A →∗ A) × A A , ev ∼ pr2 where the mapping spaces are both pointed at their identity maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' This pointed equivalence restricts to pointed equivalences (A ≃ A) ≃∗ (A ≃∗ A) × A over A, and (A → A)(id) ≃∗ (A →∗ A)(id) × A(pt) over A(pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Define e : (A → A) → (A →∗ A) × A by e(f) :≡ (a �→ f(pt)\\f(a), f(pt)) where the first component is a pointed map in the obvious way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Clearly e is a map over A, and moreover e is pointed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' It is straightforward to check that the triangle above is filled by a pointed homotopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' (One could also apply Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='6, but a direct inspection suffices in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=') Finally, it’s straightforward to check that e has an inverse given by (g, a) �→ (x �→ a · g(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Hence e is an equivalence, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' The restrictions to equivalences and path components follow by functoriality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' □ The hypotheses of the proposition are satisfied, for example, by connected H-spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' We obtain three pointed equivalences for any abelian group A and n ≥ 1: � K(A, n) → K(A, n) � ≃∗ Ab(A, A) × K(A, n), � K(A, n) ≃ K(A, n) � ≃∗ AutAb(A) × K(A, n), and � K(A, n) → K(A, n) � (id) ≃∗ K(A, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' CENTRAL H-SPACES AND BANDED TYPES 9 Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Taking A :≡ S3 in the previous proposition, by virtue of the H-space structure on the 3-sphere constructed in [BR18], we get three pointed equivalences: (S3 → S3) ≃∗ Ω3S3 × S3, (S3 ≃ S3) ≃∗ Ω3 ±1S3 × S3, and (S3 ≃ S3)(id) ≃∗ (S3 ≃∗ S3)(id) × S3, where Ω3 ±1S3 :≡ (Ω3S3)(1)⊔(Ω3S3)(−1) and 1 and −1 refer to the corresponding elements of π3(S3) = Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' By combining our results thus far, we obtain the following equivalence which generalizes a classical formula of [Cop59, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='5A], independently shown by [AC63], for counting homotopy classes of H-space structures on certain spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Let A be a left-invertible H-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' The type HSpace(A) of H-space structures on A is equivalent to A ∧ A →∗ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' By Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='19, the type of H-space structures on A is equivalent to the type of pointed sections of ev : (A → A) → A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='24, this type is equivalent to the type of pointed sections of pr2 : (A →∗ A) × A → A, which are simply pointed maps A →∗ (A →∗ A, id), where the codomain is pointed at the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' The latter type is equivalent to A →∗ (A →∗ A), where the codomain is pointed at the constant map, by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Finally, this type is equivalent to A ∧ A →∗ A by the smash–hom adjunction for pointed types [vDoo18, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' □ Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' It follows from the proposition that HSpace(S1) ≃ 1 and HSpace(S3) ≃ Ω6S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' We record the following result which relates Whitehead products and evaluation fibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='29 ([Han74, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Let n, m ≥ 2 and let α : πm(Sn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' The evaluation fibration evα : (Sm → Sn)(α) → Sn merely has a section if and only if the Whitehead product [α, ιn] : πn+m−1(Sn) vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' As we are proving a proposition, we may pick a representative α : Sm →∗ Sn throughout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Using Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='18 and that Sn is connected, we see that [α, ιn] vanishes if and only if there merely exists a pointed section of evα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' The fibre of the forgetful map from pointed sections of evα to unpointed sections of evα over some section (s, h) is equivalent to � k:s(pt,−)=α h(pt) =s(pt,pt)=pt k(pt) � αpt, where αpt : α(pt) = pt is the pointing of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' This fibre is (−1)-connected since s lands in the component of α and the inner part of the Σ-type is a double path space of Sn with n ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' In other words, this forgetful map is an epimorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' A pointed section of evα therefore merely exists if and only if an unpointed section merely exists, completing the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Unique H-space structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' We collect results about H-space structures which are unique, in the sense that the type of H-space structures is contractible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' In particular, we give elementary proofs that such H-space structures are automatically coherently abelian and associative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Moreover, pointed self-maps of such are automatically H-space self-maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Let A be a pointed type and suppose HSpace(A) is contractible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Then the unique H-space structure µ on A is coherently abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Since HSpace(A) is contractible, there is an identification µ = µT of H-space structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' (Here, µT is the twist, defined in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=') □ For the next result, we use the definition of the smash product from [vDoo18, Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='6] (see also [CS20, Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='29]) which avoids higher paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' For pointed types (X, x0) and (Y, y0), the smash product X ∧ Y is the higher inductive type with point constructors sm : X × Y → X ∧ Y and auxl, auxr : X ∧ Y , and path constructors gluel : � y:Y sm(x0, y) = auxl and gluer : � x:X sm(x, y0) = auxr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' It is pointed by auxl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' The smash product was shown to be associative in [vDoo18, Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' 10 BUCHHOLTZ, CHRISTENSEN, FLATEN, AND RIJKE Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Suppose A is a pointed type with a unique H-space structure, and suppose moreover that this H-space structure is left-invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Then any pointed map f : A →∗ A is an H-space map, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=', we have f(a · b) = f(a) · f(b) for all a, b : A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Let f : A →∗ A be a pointed map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' We will define an associated map ν : A ∧ A →∗ A, which records how f deviates from being an H-space map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' We define ν(sm(a, b)) :≡ � f(a · b)/f(b) � /f(a), ν(auxl) :≡ pt, and ν(auxr) :≡ pt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' For b : A, we have a path ν(sm(pt, b)) ≡ � f(pt · b)/f(b) � /f(pt) = � f(b)/f(b) � /pt = pt/pt = pt, and similarly for the other path constructor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Since A admits a unique H-space structure, the type A∧A →∗ A is contractible by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Consequently, ν is constant, whence for all a, b : A we have � f(a · b)/f(b) � /f(a) = pt, and therefore f(a · b) = f(a) · f(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' □ Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Note that when A and B are two pointed types, each with unique H-space structures, it is not necessarily the case that every pointed map f : A →∗ B is an H-space map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' For example, the squaring operation gives a natural transformation H2(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Z) → H4(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Z) which is represented by a map K(Z, 2) →∗ K(Z, 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' But since squaring isn’t a homomorphism, this map isn’t an H-space map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Suppose A is a pointed type with a unique H-space structure which is left-invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Then the H-space structure is necessarily associative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Let a : A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Define a map ν : A ∧ A →∗ A as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' We let ν(sm(b, c)) :≡ ((a · b) · c)/(a · (b · c)), ν(auxl) :≡ pt, and ν(auxr) :≡ pt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' For c : A, we have a path ν(sm(pt, c)) ≡ ((a · pt) · c)/(a · (pt · c)) = (a · c)/(a · c) = pt, and similarly for the other path constructor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Since A admits a unique H-space structure, the type A ∧ A →∗ A is contractible by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Consequently, for each a, ν is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' It follows that for all a, b, c : A we have ((a · b) · c)/(a · (b · c)) = pt, and therefore (a · b) · c = a · (b · c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' □ Note that if A ∧ A →∗ A is contractible, then it follows from the smash-hom adjunction that A∧n →∗ A is contractible for each n ≥ 2, where A∧n denotes the smash power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Central types In this and the next section we focus on pointed types which we call central.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Centrality is an elementary property with remarkable consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' For example, in the next section we will see that every central type is an infinite loop space (Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' To show this, we require a certain amount of theory about central types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' We first show that every central type has a unique H-space structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' When A is already known to be an H-space, we give several conditions which are equivalent to A being central.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' From this, it follows that every Eilenberg–Mac Lane space K(G, n), with G abelian and n ≥ 1, is central.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' We also prove several other results which we will need in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Let A be a pointed type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' The center of A is the type ZA :≡ (A → A)(id), which comes with a natural map evid : ZA →∗ A (see Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' If the map evid is an equivalence, then A is central.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' The terminology “central” comes from higher group theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Suppose A :≡ BG is the delooping of an ∞-group G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' The center of G is the ∞-group ZG :≡ Πx:G(x = x) with delooping BZG :≡ (BG ≃ BG)(id), which is our ZA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Central types and H-spaces are connected through evaluation fibrations: Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Suppose that A is central.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Then A admits a unique H-space structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' In addition, A is connected, so this H-space structure is both left- and right-invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Since evid is an equivalence, it has a unique section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' By Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='23, we deduce that A has a unique H-space structure µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' It follows from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='30 that it is coherently abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Finally, the equivalence evid : (A → A)(id) ≃ A implies that A is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Then, since µ(pt, −) and µ(−, pt) are both equal to the identity, it follows that µ is left- and right-invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' □ CENTRAL H-SPACES AND BANDED TYPES 11 It follows from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='33 and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='30 that the unique H-space structure on a central type is associative and coherently abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' In contrast, the type of non-coherent H-space structures on a central type A is rarely contractible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' We’ll show here that it is equivalent to the loop space ΩA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' First consider the type of binary operations µ : A → (A → A) which merely satisfy the left unit law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' This is equivalent to the type of maps A → (A → A)(id), since A is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Such a map µ satisfies the right unit law if and only if the composite evid ◦µ : A → A is the identity map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' In other words, µ must be a section of the equivalence evid, so there is a contractible type of such µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' The left unit law says that µ sends pt to id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' After post-composing with evid, it therefore says that it sends pt to id(pt), which equals pt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' So the type of left unit laws is pt = pt, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=', the loop space ΩA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Note that we imposed the left unit law both merely and purely, but that doesn’t change the type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' So it follows that the type of all non-coherent H-space structures on a central type A is ΩA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' We give conditions for an H-space to be central, in which case the H-space structure is the unique one coming from centrality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' For the next two results, write F :≡ Σf:A→∗A∥f = id∥ for the fibre of evid : (A → A)(id) →∗ A over pt : A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Note that the equality f = id is in the type of unpointed maps A → A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Suppose that A is a connected H-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Then F ≃ (A →∗ A)(id).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' By our assumptions, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='24 gives a trivialization of evid over A: t : (A → A)(id) ≃∗ (A →∗ A)(id) × A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Passing to the fibres of evid and pr2 over pt : A gives the desired equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' □ The lemma can also be shown using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Let A be a pointed type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Then the following are logically equivalent: (1) A is central;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' (2) A is a connected H-space and A →∗ A is a set;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' (3) A is a connected H-space and A ≃∗ A is a set;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' (4) A is a connected H-space and A →∗ ΩA is contractible;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' (5) A is a connected H-space and ΣA →∗ A is contractible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' (1) =⇒ (2): Assume that A is central.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Then Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='3 implies that A is a connected H- space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Since A is a left-invertible H-space, so is A →∗ A, by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Therefore all components of A →∗ A are equivalent to (A →∗ A)(id), and thus to F by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Now, F is contractible since evid is an equivalence, and consequently A →∗ A is a set since all of its components are contractible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' (2) =⇒ (3): This follows from the fact that A ≃∗ A embeds into A →∗ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' (3) =⇒ (1): If (A ≃∗ A) is a set, then its component (A →∗ A)(id) is contractible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Therefore F is contractible, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' It follows that evid is an equivalence, since A is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Hence A is central.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' (3) ⇐⇒ (4): Since A is a left-invertible H-space, so is A →∗ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' The latter is therefore a set if and only if the component of the constant map is contractible, which is true if and only if the loop space Ω(A →∗ A) is contractible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Finally, the equivalence Ω(A →∗ A) ≃ (A →∗ ΩA) shows that this is true if and only if A →∗ ΩA is contractible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' (4) ⇐⇒ (5): This follows from the equivalence (A →∗ ΩA) ≃ (ΣA →∗ A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' □ Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Consider the Eilenberg–Mac Lane space K(G, n) for n ≥ 1 and G an abelian group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' It is a pointed, connected type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Since K(G, n) ≃ Ω K(G, n + 1), it is an H-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' By [BvDR18, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='1], K(G, n) ≃∗ K(G, n) is equivalent to the set of automorphisms of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' It therefore follows from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='6 that K(G, n) is central.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' We will see in Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='9 a more self-contained proof of this result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' 12 BUCHHOLTZ, CHRISTENSEN, FLATEN, AND RIJKE Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Brunerie showed that π4(S3) ≃ Z/2 [Bru16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Therefore, S4 →∗ S3 is not contractible, and so S3 is not central, by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='6(5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Since this is in the stable range, it follows that Sn is not central for n ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' For a pointed type A, we have seen that A being central is logically equivalent to A being a connected H-space such that A ≃∗ A is a set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' It is natural to ask whether the reverse implication holds without the assumption that A is an H-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' However, this is not the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Consider, for example, the pointed, connected type K(G, 1) for a non-abelian group G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Then K(G, 1) ≃∗ K(G, 1) is equivalent to the set of group automorphisms of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' If K(G, 1) were central, then G would be twice deloopable, which would contradict G being non-abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' By the previous proposition, the type A →∗ A is a set whenever A is central.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Presently we observe that it is in fact a ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' For any central type A, the set A →∗ A is a ring under pointwise multiplication and function composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' It follows from A being a commutative and associative H-space that the set A →∗ A is an abelian group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' The only nontrivial thing we need to show is that function composition is linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Let f, g, φ : A →∗ A, and consider a : A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='31, φ is an H-space map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Consequently, � φ ◦ (f · g) � (a) ≡ φ(f(a) · g(a)) = φ(f(a)) · φ(g(a)) ≡ � (φ ◦ f) · (φ ◦ g) � (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' □ The following remark gives some insight into the nature of the ring A →∗ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' If BG is an ∞-group and X is a pointed type, recall that a bundle over X is G-principal if it is classified by a map X →∗ BG (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' [Sco20, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='23] for a formal definition which easily generalizes to arbitrary ∞-groups).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' In particular, it is not hard to see that the Hopf fibration of G (as the loop space of BG) is a G-principal bundle, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=', classified by a map ΣG →∗ BG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' In Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='4 we will see that any central type A has a delooping BAut1(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' This means we have equivalences (A →∗ A) ≃ � A →∗ (A ≃ A)(id) � ≃ (ΣA →∗ BAut1(A)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Thus we see that A →∗ A is the ring of principal A-bundles over ΣA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' The equivalence above maps the identity id : A →∗ A to the Hopf fibration of A (as a principal A-bundle), meaning the Hopf fibration is the multiplicative unit from this perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' In the remainder of this section we collect various results which are needed later on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' The first result is that “all” of the evaluation fibrations of a central type A are equivalences: Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Let A be a central type and let f : A →∗ A be a pointed map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' The evaluation fibration evf : (A → A)(f) →∗ A is an equivalence, with inverse given by a �→ a · f(−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' The type A → A is a left-invertible H-space via pointwise multiplication, by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' So there is an equivalence (A → A)(id) → (A → A)(f) sending g to f · g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Since f is pointed, we have evf(f · g) ≡ (f · g)(pt) ≡ f(pt) · g(pt) = pt · g(pt) = g(pt) = evid(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' In other words, evf ◦(f · −) = evid, which shows that evf is an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Since f is pointed, the stated map is a section of evf, hence is an inverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' □ Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Let A be a central type, let f : A →∗ A, and let g : (A → A)(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Then for all a : A, we have g(a) = g(pt) · f(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' □ Any central type has an inversion map, which plays a key role in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Suppose that A is central.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' The inversion map id∗ : A → A sends a to a∗ :≡ pt/a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' CENTRAL H-SPACES AND BANDED TYPES 13 The defining property of a∗ is that a∗·a = pt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Since A is abelian, we also have a·a∗ = pt, so it would have been equivalent to define the inversion to be a �→ a\\pt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' From associativity of a central H-space it follows that pt∗ = pt and a∗∗ = a for all a, so the inversion map id∗ is a pointed self-equivalence of A and an involution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' A curious property is that on the component of id∗, inversion of equivalences is homotopic to the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' This comes up in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' The map φ �→ φ−1 : (A ≃ A)(id∗) → (A ≃ A)(id∗) is homotopic to the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Let φ : (A ≃ A)(id∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' We need to show that φ = φ−1, or equivalently that φ(φ(pt)) = pt, since evid is an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' (Note that φ ◦ φ : (A ≃ A)(id).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=') Using Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='13, we have that φ(φ(pt)) = φ(pt) · φ(pt)∗ = pt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Bands and torsors We begin in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='1 by defining and studying types banded by a central type A, also called A-bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' We show that the type BAut1(A) of banded types is a delooping of A, that A has a unique delooping, and that every pointed self-map A →∗ A has a unique delooping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='2, we show that BAut1(A) is itself an H-space under a tensoring operation, from which it follows that it is again a central type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Thus we may iteratively consider banded types to obtain an infinite loop space structure on A, which is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' As a special case, taking A to be K(G, n) for some abelian group G produces a novel description of the infinite loop space structure on K(G, n), as described in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='3, we define the type of A-torsors, which we show is equivalent to the type of A-bands when A is central, thus providing an alternate description of the delooping of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' The type of A-torsors has been independently studied by David W¨arn, who has shown that it is a delooping of A under the weaker assumption that A has a unique H-space structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Types banded by a central type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' We now study types banded by a central type A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' On this type we will construct an H-space structure, which will be seen to be central.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' For a type A, let BAut1(A) :≡ ΣX:U∥A = X∥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' The elements of BAut1(A) are types which are banded by A or A-bands, for short.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' We denote A-bands by Xp, where p : ∥A = X∥0 is the band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' The type BAut1(A) is pointed by A|refl|0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Given a band p : ∥A = X∥0, we will write ˜p : ∥X ≃ A∥0 for the associated equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' It’s not hard to see that BAut1(A) is a connected, locally small type—hence essentially small, by the join construction [Rij17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' The characterization of paths in Σ-types tells us what paths between banded types are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Consider two A-bands Xp and Yq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' A path Xp = Yq of A-bands corresponds to a path e : X = Y between the underlying types making the following triangle of truncated paths commute: A X Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' p q |e|0 In other words, there is an equivalence (Xp = Yq) ≃ (X = Y )(¯p � q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' □ For the remainder of this section, let A be a central type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' We begin by showing that the type of A-bands is a delooping of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' We have that Ω BAut1(A) ≃ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' We have Ω BAut1(A) ≃ (A = A)(refl) ≃ (A ≃ A)(id) ≃ A, where the first equivalence uses Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='3 and the last equivalence is by centrality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' □ 14 BUCHHOLTZ, CHRISTENSEN, FLATEN, AND RIJKE Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' The unique H-space structure on A is deloopable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' □ Note that this gives an independent proof that it is associative (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' The type A has a unique delooping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' We must show that the type ΣB:U* (ΩB ≃∗ A) is contractible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' We will use BAut1(A), with the equivalence ψ from Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='4, as the center of contraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Let B : U* be a pointed type with a pointed equivalence φ : ΩB ≃∗ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Given x : B, consider pt =B x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Since A is connected, B is simply connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Therefore, to give a banding on pt =B x, it suffices to do so when x is pt, in which case we use φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' So we have defined a map f : B → BAut1(A), and it is easy to see that it is pointed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' We claim that the following triangle commutes: ΩB Ω BAut1(A) A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' φ ∼ Ωf ψ ∼ Let q : pt =B pt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Then (Ωf)(q) is the path associated to the equivalence A ≃ (pt =B pt) ≃ (pt =B pt) ≃ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' The first equivalence is φ−1 and the last is φ, as these give the pointedness of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' The middle equivalence is the map sending p to p�q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' The map ψ comes from the evaluation fibration, so to compute ψ((Ωf)(q)) we compute what happens to the base point of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' It gets sent to refl, then q, and then φ(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' This shows that the triangle commutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' It follows that Ωf is an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Since B and BAut1(A) are connected, f is an equivalence as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' So f and the commutativity of the triangle provide a path from (B, φ) to (BAut1(A), ψ) in the type of deloopings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' □ We conclude this section by showing how to deloop maps A →∗ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Given f : A →∗ A, define Bf : BAut1(A) →∗ BAut1(A) by Bf(Xp) :≡ (X → A)(f ∗◦˜p−1), where f ∗ :≡ f ◦id∗, and we have used that f ∗ ◦ ˜p−1 is well-defined as an element of the set-truncation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' To give a banding of (X → A)(f ∗◦˜p−1) we may induct on p and use Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' The same argument shows that Bf is a pointed map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Note that f(a∗) = f(a)∗ for any a : A, since f is an H-space map by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='31, so there’s no choice involved in this definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Let g : BAut1(A) →∗ BAut1(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Given a loop q : pt = pt, the loop (Ωg)(q) is the composite pt = g(pt) = g(pt) = pt, which uses pointedness of g and apg(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' We will identify (pt = pt) with A and then write Ω′g : A ≃∗ (pt = pt) Ωg −−→∗ (pt = pt) ≃∗ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' We have that Ω′Bf = f for any f : A →∗ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' The following diagram describes how Bf acts on a loop p : pt =BAut1(A) pt: Arefl (A → A)(f ∗) A Arefl (A → A)(f ∗) A p g�→g◦˜p−1 ∼ ∼ CENTRAL H-SPACES AND BANDED TYPES 15 Since ˜p is in the component of the identity, we have ˜p(a) = x · a for all a : A, where x = ˜p(pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' So ˜p−1(a) = x\\a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Then the composite A ≃ A on the right is seen to be a �→ evf ∗ �� a · f ∗(−) � ˜p−1 � = evf ∗ � a · f ∗� x\\(−) �� = a · f(x∗∗) = a · f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' The domain Arefl = Arefl is identified with A by sending a path p to ˜p(pt), which in this case is the x above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' The codomain (A ≃ A)(id) is identified with A using evid, which sends the displayed function to pt · f(x), which equals f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' So we have that ΩBf = f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='6, they are equal as pointed maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' □ Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' We have that BΩ′g = g for any g : BAut1(A) →∗ BAut1(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Given an A-band Xp, we need to show that g(Xp) = (X → A)((Ω′g)∗◦˜p−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' First we construct a map of the underlying types from left to right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' For y : g(Xp), define the map Gy : X ∼ −→ (pt = Xp) ≃ (Xp = pt) apg −−→ (g(Xp) = g(pt)) ≃ (pt = pt) → A, where the second map is path inversion, and the fourth map uses the trivialization of g(Xp) associated to y and pointedness of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' The identification pt = g(pt) corresponds to a unique point y0 : g(pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' To check that Gy lies in the right component, we may induct on p and assume y ≡ y0 since g(pt) is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' We then get the map Gy0 : A id∗ −−→ A ≃ (pt = pt) Ωg −−→ (pt = pt) → A, since path inversion on (pt = pt) corresponds to inversion on A, and y0 corresponds to the pointing of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' This map is precisely the definition of (Ω′g)∗, so G lands in the desired component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' To check that G defines an equivalence of bands we may again induct on p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Write �y0 : pt ≃ g(pt) for the equivalence associated to the point y0 : g(pt), which is a lift of the (equivalence associated to the) banding of g(pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' It then suffices to check that the diagram g(pt) (A → A)((Ω′g)∗) pt � y0 −1 G ev(Ω′g)∗ commutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Let y : g(pt), which we identify with a trivialization y′ : pt = g(pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Chasing through the definition of G and using that apg(refl) = refl, we see that Gy(pt) = ev(y′ � y0) = �y0 −1(y′(pt)) ≡ �y0 −1(y), where ev : (pt = pt) → A is the last map in the definition of Gy, which transports pt along a path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Thus we see that the triangle above commutes, whence G is an equivalence of bands, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' □ Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' We have inverse equivalences Ω′ : (BAut1(A) →∗ BAut1(A)) ≃ (A →∗ A) : B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' In particular, the type BAut1(A) →∗ BAut1(A) is a set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Combine Propositions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='8 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Tensoring bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' In this section, we will construct an H-space structure on BAut1(A), where we continue to assume that A is a central type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' This H-space structure is interesting in its own right, and also implies that BAut1(A) is itself central.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' It that follows that A is an infinite loop space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' This elementary lemma will come up frequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Let P : BAut1(A) → U be a set-valued type family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Then � Xp P(Xp) is equivalent to P(pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' 16 BUCHHOLTZ, CHRISTENSEN, FLATEN, AND RIJKE Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Since each P(Xp) is a set, � Xp P(Xp) is equivalent to � X:U � p:A=X P(X|p|0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' By path in- duction, this is equivalent to P(A|refl|0), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=', P(pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' □ A consequence of the following result is that any pointed A-band is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Let Xp be an A-band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Then there is an equivalence (pt =BAut1(A) Xp) → X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='3, there is an equivalence (pt =BAut1(A) Xp) ≃ (A ≃ X)(˜p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' We will show that evp : (A ≃ X)(˜p) → X is an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='11, it’s enough to prove this when Xp ≡ pt, and this holds because A is central.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' □ We now show that path types between A-bands are themselves banded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' This underlies the main results of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Let Xp and Yq be A-bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' The type Xp =BAut1(A) Yq is banded by A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' We need to construct a band ∥A = (Xp = Yq)∥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Since the goal is a set, we may induct on p and q, thus reducing the goal to ∥A = (pt =BAut1(A) pt)∥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Using that (pt =BAut1(A) pt) ≃ (A ≃ A)(id) and that A is central, we may give the set truncation of the inverse of the evaluation fibration at idA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' □ The following is an immediate corollary of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' For any A-band Xp, the A-band (Xp = Xp) is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' □ We next define a tensor product of banded types, using the notion of duals of bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Write refl∗ : A = A for the self-identification of A associated to the inversion map id∗ (Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='14) via univalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Let Xp be an A-band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' The band p∗ :≡ |refl∗| � p is the dual of p, and the A-band X∗ p :≡ Xp∗ is the dual of Xp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Since id∗ is an involution, it follows that taking duals defines an involution on BAut1(A), meaning that X∗∗ p = Xp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' We have pt = pt∗ in BAut1(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' The underlying type of pt∗ is A, which has a base point, so this follows from Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' □ We now show how to tensor types banded by A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' For Xp, Yq : BAut1(A), define Xp ⊗ Yq :≡ (X∗ p = Yq), with the banding from Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' It follows from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='3 that the type Xp ⊗Yq is equivalent to (X = Y )(p∗ � q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Since taking duals is an involution, we also have equivalences Xp ⊗ Yq ≡ (X∗ p = Yq) ≃ (Xp = Y ∗ q ) ≃ (X = Y )(p � q∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Moreover, from Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='14, we see that X∗ p ⊗ Xp = pt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Tensoring defines a binary operation on BAut1(A), and we now show that this operation is sym- metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' For any Xp, Yq : BAut1(A), there is a path σ(Xp,Yq) : Xp ⊗ Yq =BAut1(A) Yq ⊗ Xp such that σpt,pt = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' By univalence and the characterization of paths between bands, we begin by giving an equiv- alence between the underlying types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' The equivalence will be path-inversion, as a map (X = Y )(p � q∗) −→ (Y = X)(q � p∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' To see that this is valid it suffices to show that the inversion of p � q∗ is q � p∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' We have: p � q∗ ≡ p � refl∗ �q = refl∗ �q � p = q � refl∗ � p = q � refl∗ �p ≡ q � p∗, CENTRAL H-SPACES AND BANDED TYPES 17 where we have used associativity of path composition, and that refl∗ = refl∗ by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' To prove the transport condition, we may path induct on both p and q which then yields the following triangle: (A = A)(refl∗) (A = A)(refl∗) A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' evrefl∗ p�→p evrefl∗ Here we are writing evrefl∗ for the composite (A = A)(refl∗) ≃ (A ≃ A)(id∗) evid∗ −−−→ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' The horizontal map is given by path-inversion, which is homotopic to the identity by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='15, hence the triangle commutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Paths between paths between banded types correspond to homotopies between the underlying equivalences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Thus σpt,pt = 1 since path-inversion on (A = A)(refl∗) is homotopic to the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' □ We now use Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='4 to make BAut1(A) into an H-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' The binary operation ⊗ makes BAut1(A) into an abelian H-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' We start by showing the left unit law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Since pt∗ = pt, we instead consider the goal (pt = Xp) = Xp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' An equivalence between the underlying types is given by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='12, which after inducting on p clearly respects the bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Using Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='18 and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='4, we obtain the desired H-space structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' □ Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' For a central type A, the type BAut1(A) is again central.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Therefore, A is an infinite loop space, in a unique way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Moreover, every pointed map A →∗ A is infinitely deloopable, in a unique way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' That BAut1(A) is central follows from condition (2) of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='6, using Theorems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='10 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='19 as inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' That A is a infinite loop space then follows from Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='4: writing BAut0 1(A) :≡ A and BAutn+1 1 (A) :≡ BAut1(BAutn 1(A)), we see that BAutn 1(A) is an n-fold delooping of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' The uniqueness follows from Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' That every pointed self-map is infinitely deloopable in a unique way follows by iterating Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' □ Note that BAut1(A) is essentially small (Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='2), so these deloopings can be taken to be in the same universe as A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' From Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='19 we deduce another characterization of central types: Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' A pointed, connected type A is central if and only if ΣX:BAut1(A) X is contractible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' If A is central, then by the left unit law of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='19, we have ΣX:BAut1(A) X ≃ ΣX:BAut1(A) (pt∗ =BAut1(A) X) ≃ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Conversely, if ΣX:BAut1(A) X is contractible, then so is its loop space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' But the loop space is equivalent to Σf:A→∗A ∥f = id∥, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=', the fibre of evid over the base point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Thus evid is an equivalence, since A is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Bands and torsors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Let A be a central type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' We define a notion of A-torsor which turns out to be equivalent to the notion of A-band from the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Under our centrality assumption, it follows that the resulting type of A-torsors is a delooping of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' An equivalent type of A-torsors has been independently studied by David W¨arn, who has also shown that it gives a delooping of A under the weaker hypothesis that A has a unique H-space structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' An action of A on a type X is a map α : A × X → X such that α(pt, x) = x for all x : X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' If X has an A-action, we say that it is an A-torsor if it is merely inhabited and α(−, x) is an equivalence for every x : X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' The type of A-torsor structures on a type X is TA(X) :≡ � α:A×X→X (α(pt, −) = idX) × ∥X∥−1 × � x:X IsEquiv α(−, x), 18 BUCHHOLTZ, CHRISTENSEN, FLATEN, AND RIJKE and the type of A-torsors is � X:U TA(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Since A is connected, an A-action on X is the same as a pointed map A →∗ (X ≃ X)(id).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Normally one would require at a minimum that this map sends multiplication in A to composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' We explain in Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='28 why our definition suffices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' The condition that α(−, x) is an equivalence for all x is equivalent to requiring that for every x0, x1 : X, there exists a unique a : A with α(a, x0) = x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' It is also equivalent to saying that (α, pr2) : A × X → X × X is an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' For any type X, write ev≃ : (A ≃ X) → X for the evaluation fibration which sends an equivalence e to e(pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' For a map f, write Sect(f) for the type of (unpointed) sections of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' For any X, we have an equivalence TA(X) ≃ ∥X∥−1 × Sect(ev≃).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' This is simply a reshuffling of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' The map from left to right sends a torsor structure with action α : A × X → X to the map X → (A → X) sending x to α(−, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' By assumption, this lands in the type of equivalences, and the condition α(pt, −) = idX says that it is a section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' We leave the remaining details to the reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' □ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Let X be an A-torsor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Then X is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Since X is merely inhabited and our goal is a proposition, we may assume that we have x0 : X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Then we have an equivalence α(−, x0) : A → X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' A is connected by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='3, so it follows that X is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' □ Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Let X be an A-torsor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Then X is banded by A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Associated to the torsor structure on X is a section X → (A ≃ X) of ev≃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Since X is 0-connected, it lands in a component of A ≃ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' By univalence, this determines a banding of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' □ Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Let X be a type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' There is an equivalence TA(X) ≃ ∥A = X∥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Therefore, there is an equivalence between the type of A-torsors and BAut1(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='25 gives a map f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' We check that the fibres are contractible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Let p : ∥A = X∥0 be a banding of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' An A-torsor structure t on X with f(t) = p consists of a section s of ev≃ that lands in the component (A ≃ X)(˜p), where ˜p denotes the equivalence associated to p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' But by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='12, the evaluation fibration (A ≃ X)(˜p) → X is an equivalence, so it has a unique section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' □ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' It follows that TA(X) is a set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' One can also show this using Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='14 and Propo- sition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Let X be an A-torsor, or equivalently, an A-band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' By Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='14, we have an equivalence e : A ≃ (X ≃ X)(id).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Since A has a unique H-space structure, this equivalence is an equivalence of H-spaces, where the codomain has the H-space structure coming from composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Since A is connected, the A-action on X gives a map α′ : A →∗ (X ≃ X)(id).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' (In fact, α′ = e, but we won’t use this fact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=') Using the equivalence e, it follows from Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='10 that any map with the same type as α′ is deloopable in a unique way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' That is, it has the structure of a group homomorphism in the sense of higher groups (see [BvDR18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' This explains why our naive definition of an A-action is correct in this situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Examples and non-examples We show that the Eilenberg–Mac Lane spaces K(G, n) are central whenever G is abelian and n > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' In addition, we produce examples of products of Eilenberg–Mac Lane spaces which are central and examples which are not central.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' At present, we do not know whether there exist central types which are not products of Eilenberg–Mac Lane spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Along the way, we use our results to give a self-contained, independent construction of Eilenberg–Mac Lane spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' To this end, we begin by discussing the base case K(G, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' CENTRAL H-SPACES AND BANDED TYPES 19 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' The H-space of G-torsors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Given a group G, we construct the type TG of G-torsors and show that it is a K(G, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Specifically, a pointed type X is a K(G, 1) if it is connected and comes equipped with a pointed equivalence ΩX ≃∗ G which sends composition of loops to multiplication in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' (We always point ΩX at refl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=') When G is abelian, we can tensor G-torsors to obtain an H-space structure on TG which is analogous to the tensor product of bands of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' These constructions are all classical and we therefore omit some details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Let G be a group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' A G-set is a set X with a group homomorphism α : G → Aut(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' If the set X is merely inhabited and the map α(−, x) : G → X is an equivalence for every x : X, then (X, α) is a G-torsor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' We write TG for the type of G-torsors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Given two G-sets X and Y , we write X →G Y for the set of G-equivariant maps from X to Y , defined in the usual way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' We may write g · x instead of α(g, x) when no confusion can arise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' The following is straightforward to check: Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Let X and Y be G-torsors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' There is a natural equivalence (X =T G Y ) ≃ (X →G Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' In particular, a G-equivariant map between G-torsors is automatically an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' □ Any group G acts on itself by left translation, making G into a G-torsor which constitutes the base point pt of both TG and the type of G-sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Since a G-equivariant map pt →G X is determined by where it sends 1 : G, the map (pt →G X) → X that evaluates at 1 is an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' It is clear that the type TG is a 1-type, which implies that its loop space is a group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' We have a group isomorphism ΩTG ≃ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' We only sketch a proof since this is a classical result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Since paths between G-torsors correspond to G-equivariant maps, we have equivalences of sets (pt =T G pt) ≃ (pt →G pt) ≃ G, where the second equivalence is given by evaluation at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' The first equivalence sends path compo- sition to composition of maps, which reverses the order—i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=', it’s an anti-isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' The second equivalence evaluates a map at 1 : G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Thus, for φ, ψ : pt →G pt we have φ(ψ(1)) = φ(ψ(1) · 1) = ψ(1) · φ(1), where · denotes the multiplication in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' In other words, evaluation at 1 is an anti-isomorphism, meaning the composite (pt =T G pt) ≃ G is an isomorphism of groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' □ The following proposition says that the G-torsors are precisely those G-sets which lie in the com- ponent of the base point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' A G-set (X, α) is a G-torsor if and only if there merely exists a G-equivariant equivalence from pt to X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Suppose X is a G-torsor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' To produce a mere G-equivariant equivalence pt ≃G X we may assume we have some x : X, since X is merely inhabited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Then (−) · x : G → X yields an equivalence which is clearly G-equivariant, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Conversely, assume that there merely exists a G-equivariant equivalence from pt to X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Since being a G-torsor is a proposition, we may assume we have an actual G-equivariant equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' But then we are done since pt is a G-torsor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' □ It follows that TG is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Thus by Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='3 we deduce: Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' The type TG is a K(G, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' For the remainder of this section, let G be an abelian group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' For any two G-torsors S and T, the path type S =T G T is again a G-torsor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' 20 BUCHHOLTZ, CHRISTENSEN, FLATEN, AND RIJKE Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' First we make S =T G T into a G-set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' This path type is equivalent to the type S →G T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Using that G is abelian, it’s easy to check that the map (g, φ) �−→ � s �→ g · φ(s) � : G × (S →G T) −→ (S →G T) is well-defined and makes S →G T into a G-set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' To check that the above yields a G-torsor, we may assume that S ≡ pt ≡ T, by the previous lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' One can check that Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='3 gives an equivalence of G-sets, where pt →G pt is equipped with the G-action just described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Thus pt →G pt is a G-torsor, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' □ In order to describe the tensor product of G-torsors, we first need to define duals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Let (X, α) be a G-torsor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' The dual X∗ of X is the G-torsor X with action α∗(g, x) :≡ α(g−1, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' The tensor product of G-torsors is now defined as X ⊗ Y :≡ (X∗ =T G Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' The tensor product of G-torsors makes TG into an H-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' We verify the hypotheses of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Thus our first goal is to construct a symmetry σX,Y : (X∗ =T G Y ) =T G (Y ∗ =T G X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' After identifying paths of G-torsors with G-equivariant equivalences, we may consider the map which inverts such an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' A short calculation shows that if φ : X∗ →G Y is G-equivariant, then φ−1 : Y ∗ →G X is again G-equivariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' We need to check that the map sending φ to φ−1 is itself G-equivariant, so let φ : X∗ →G Y and let g : G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Since the inverse of g · (−) is g−1 · (−), we have: (g · φ)−1 = φ−1(g−1 · (−)) = g · φ−1(−), using that φ−1 : Y ∗ →G X is G-equivariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Thus inversion is G-equivariant, yielding the required symmetry σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Now we argue that σpt,pt = refl, or, equivalently, that maps pt∗ →G pt are their own inverses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Such a map is uniquely determined by where it sends 1 : G, so it suffices to show that φ(φ(1)) = 1 for every φ : pt∗ →G pt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Fortunately, we have φ(φ(1)) = φ(φ(1) · 1) = φ(1)−1 · φ(1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Lastly, it is straightforward to check that the map (pt∗ →G X) → X which evaluates at 1 : G is G-equivariant, for any G-torsor X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' This yields the left unit law for the tensor product ⊗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' As such we have fulfilled the hypotheses of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='4, giving us the desired H-space structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' □ Using Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='6, one can check that TG is a central H-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' (See Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=') 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Eilenberg–Mac Lane spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' We now use our results to give a new construction of Eilenberg– Mac Lane spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' For an abelian group G, recall that a pointed type X is a K(G, 1) if it is connected and there is a pointed equivalence ΩX ≃∗ G which sends composition of paths to multiplication in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' For n > 1, a pointed type X is a K(G, n + 1) if it is connected and ΩX is a K(G, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' It follows that such an X is an n-connected (n + 1)-type with Ωn+1X ≃∗ G as groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' In the previous section we saw that the type TG of G-torsors is a K(G, 1) and is central whenever G is abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' The following proposition may be seen as a higher analog of this fact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Let G be an abelian group and let n > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' If a type A is a K(G, n) and an H-space, then A is central and BAut1(A) is a K(G, n + 1) and an H-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' The fact that BAut1(A) is a K(G, n + 1) also follows from [Shu], using the fact that BAut1(A) is the 1-connected cover of BAut(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Suppose that A is a K(G, n) and an H-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Then A →∗ ΩA is contractible, since it is equivalent to ∥A∥n−1 →∗ ΩA, and ∥A∥n−1 is contractible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' So Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='6 implies that A is central.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='4, Ω BAut1(A) ≃ A, so BAut1(A) is a K(G, n + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' By Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='19, BAut1(A) is also an H-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' □ REFERENCES 21 We can use the previous proposition to define K(G, n) for all n > 0 by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' For the base case n ≡ 1 we let K(G, 1) :≡ TG, the type of G-torsors from the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' When G is abelian, we saw that TG is an H-space, which lets us apply the previous proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' By induction, we obtain a K(G, n) for all n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Note that this construction produces a K(G, n) which lives n − 1 universes above the given K(G, 1), but that it is essentially small by the join construction [Rij17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Products of Eilenberg–Mac Lane spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Here is our first example of a central type that is not an Eilenberg–Mac Lane space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Let K = K(Z/2, 1) = RP ∞ and L = K(Z, 2) = CP ∞, and consider A = K × L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' This is a connected H-space, and � K × L →∗ Ω(K × L) � ≃ � K →∗ Ω(K × L) � since K = ∥K × L∥1 ≃ � K →∗ ΩL � since K is connected ≃ � Z/2 →Ab Z) by [BvDR18, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='1] ≃ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' So it follows from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='6(4) that A is central.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' On the other hand, not every product of Eilenberg–Mac Lane spaces is central.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Let K = K(Z/2, 1) = RP ∞ and L′ = K(Z/2, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' A calculation like the above shows that K × L′ →∗ Ω(K × L′) is not contractible, so K × L′ is not central.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' As another example, [Cur68, Proposition Ia] shows that K(Z, 1) × K(Z, 2)) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=', S1 × CP ∞) has infinitely many distinct H-space structures classically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' So it is not central, by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Clearly both of these examples can be generalized to other groups and shifted to higher dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='3, centrality of a type implies that it has a unique H-space structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' The converse fails, as we now demonstrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' We are grateful to David W¨arn for bringing our attention to this example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' The type A :≡ K(Z, 2) × K(Z, 3) is not central, by a computation similar to the one in the previous example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' However, we note that it admits a unique H-space structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Since A is a loop space it admits an H-space structure, and the type of H-space structures is given by A ∧ A →∗ A according to Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Since A is 1-connected, by [CS20, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='32] the smash product A ∧ A is 3-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' It follows that A ∧ A →∗ A is contractible, since A is 3-truncated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' In other words, the space of H-space structures on A is contractible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' References [AC63] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Arkowitz and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Curjel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' “On the number of multiplications of an H–space”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' In: Topology 2 (1963), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' 205–207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' [BR18] Ulrik Buchholtz and Egbert Rijke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' “The Cayley-Dickson construction in homotopy type theory”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' In: High.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Struct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='1 (2018), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' 30–41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' doi: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='21136/HS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' [Bru16] Guillaume Brunerie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' “On the homotopy groups of spheres in homotopy type theory”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' PhD thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Laboratoire J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Dieudonn´e, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' arXiv: 1606.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='05916.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' [Buc19] Ulrik Buchholtz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Non-abelian cohomology (Groups, Torsors, Gerbes, Bands & all that).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Invited talk at the workshop Geometry in Modal Homotopy Type Theory, Carnegie Mellon University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' url: https://youtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='be/eB6HwGLASJI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' [BvDR18] U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Buchholtz, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' van Doorn, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Rijke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' “Higher Groups in Homotopy Type Theory”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' In: Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' LICS ’18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Oxford, United Kingdom: ACM, 2018, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' 205–214.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' isbn: 978-1-4503-5583-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='1145/3209108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='3209150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' 22 REFERENCES [Cav21] Evan Cavallo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Pointed functions into a homogeneous type are equal as soon as they are equal as unpointed functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Agda formalization, part of 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Blog post at homotopy- typetheory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' url: https://homotopytypetheory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='org/2014/06/30/fibrations- with-em-fiber/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' [Uni13] Univalent Foundations Program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Homotopy Type Theory: Univalent Foundations of Math- ematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Institute for Advanced Study: http://homotopytypetheory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='org/book/, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' [vDoo18] Floris van Doorn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' “On the Formalization of Higher Inductive Types and Synthetic Ho- motopy Theory”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' PhD thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Carnegie Mellon University, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' arXiv: 1808.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='10690.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' [Whi46] George W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' Whitehead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' “On products in homotopy groups”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' In: Annals of Mathematics 47 (1946), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' 460–475.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content=' University of Nottingham, Nottingham, United Kingdom Email address: ulrik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='buchholtz@nottingham.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='uk University of Western Ontario, London, Ontario, Canada Email address: jdc@uwo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='ca University of Western Ontario, London, Ontario, Canada Email address: jtaxers@uwo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='ca University of Ljubljana, Ljubljana, Slovenia Email address: e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='rijke@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} +page_content='com' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQfwwE1/content/2301.02636v1.pdf'} diff --git a/ENE4T4oBgHgl3EQffQ0N/content/tmp_files/2301.05105v1.pdf.txt b/ENE4T4oBgHgl3EQffQ0N/content/tmp_files/2301.05105v1.pdf.txt new file mode 100644 index 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См. https://creativecommons.org/licenses/by-nc-sa/4.0/legalcode.ru +https://doi.org/10.52605/16059921_2022_06_59 +Цифровая экономика +ИНСТИТУЦИОНАЛИЗАЦИЯ ЦИФРОВОЙ ТОРГОВЛИ В РОССИЙСКОЙ +ФЕДЕРАЦИИ: ОБРАТНЫЙ ОТСЧЕТ +Статья рекомендована к публикации главным редактором Т.В. Ершовой 31.07.2022. +Калужский Михаил Леонидович +Кандидат философских наук, доцент +МОФ «Фонд региональной стратегии развития», исполнительный директор +Омский государственный технический университет, каф. «Организация и управление наукоемкими +производствами», доцент +Омск, Российская Федерация +frsr@inbox.ru +Аннотация +Институционализация цифровой торговли является одним из важнейших направлений формирования +информационного общества в Российской Федерации. Исследования отражают наметившееся отставание +российского индекса готовности экономики поддерживать онлайн-покупки. Автор анализирует причины +отставания в контексте институциональных особенностей развития цифровой торговли. В качестве +основного препятствия, снижающего экономическую эффективность и конкурентоспособность цифровой +торговли, выделяется недостаточное внимание государства формированию инновационных институций +цифрового рынка. +Ключевые слова +сетевая экономика; цифровая торговля; цифровой рынок; электронная коммерция; институциональная +политика; контрактное производство; сетевое предпринимательство; маркетплейсы; логистический +провайдинг +Введение +Цифровизация не просто определяет ключевое направление развития российской экономики, но +служит источником институционального роста цифровой торговли. Указом Президента РФ № 203 +от 09.05.2017 г. определены национальные интересы, затрагивающие сферу цифровой торговли, +среди которых следует выделить: +1) формирование виртуальных рынков и обеспечение лидерства на них за счет развития +российской экосистемы цифровой экономики; +2) обеспечение недискриминационного доступа к товарам и услугам российских +поставщиков; +3) поддержка отраслей, использующих преимущества информационных технологий; +4) увеличение экспорта за рубеж несырьевых товаров и услуг; +5) создание платежной и логистической инфраструктуры интернет-торговли. +Перед Правительством РФ поставлена задача формирования технологической основы +цифровой экономики, в том числе через повышение доступности электронных форм коммерческих +отношений предприятиям малого и среднего бизнеса [1]. Для ее решения приняты и реализуются +«Стратегия развития информационного общества в Российской Федерации на 2017-2030 годы», +национальная программа «Цифровая экономика Российской Федерации» и федеральные проекты +«Информационная инфраструктура», «Цифровые технологии», «Цифровые услуги и сервисы +онлайн» и др. + +KyPHAA +HOOPMALOHHOE +6ECTB0ИНФОРМАЦИОННОЕ ОБЩЕСТВО | 2022 | № 6 +WWW.INFOSOC.IIS.RU +60 + +1 Российская цифровая торговля в мировых рейтингах +Рейтинговые оценки цифровой торговли в России свидетельствуют о том, что пока рано говорить +о значительных успехах. Согласно аналитическому отчету «Интернет-торговля в России: 2021» +компании Data Insight институциональное и инфраструктурное развитие цифровой торговли в +Россия пока далеко от совершенства (см. табл. 1) [2, с. 26]. +Таблица 1. Рейтинг Российской Федерации в мировой системе цифровой торговли +Рейтинг +Место +Best Countries For Investment In E-commerce And Digital Sector (Ceoworld) +– индекс привлекательности страны для инвестирования в электронную +коммерцию и цифровой сектор +15 +The Inclusive Internet Index +– индекс доступности цифровой инфраструктуры, цен, локального контента, +вовлеченности пользователей и культурных факторов +25 +The Ease of Doing Business Index +– индекс благоприятности условий предпринимательской деятельности +28 +UNCTAD B2C E-commerce Index Ranking (UNCTAD) +– индекс готовности экономики поддерживать онлайн-покупки +41 + +В целом приведенный рейтинг довольно наглядно отражает сложившееся положение в +цифровой торговле. Индекс Ceoworld показывает лучший результат за счет доминирования на +рынке крупных торговых сетей, интернет-магазинов и маркетплейсов. Inclusive Internet Index +демонстрирует вовлеченность пользователей в среду цифровой торговли. Ease of Doing Business Index +отражает отставание предложения отечественных продавцов от покупательского спроса. +Хуже всех выглядит индекс UNCTAD (ООН), согласно которому в 2020 г. готовность +экономических институтов поддерживать онлайн-покупки в России находилась на 41 месте из 152 +стран мира [3, с. 14]. Именно этот индекс отражает недостаточную эффективность +институциональной политики государства в сфере цифровой торговли. +2 Институциональные процессы в цифровой торговле +Цифровая торговля являет собой типичный пример технологической инновации, выступающей +следствием очередного институционального цикла [4, с. 25]. Институциональный цикл проходит в +своем развитии те же этапы, что и любой жизненный цикл в экономике, менеджменте или +маркетинге: выход на рынок, рост, зрелость и упадок. На первых этапах институционального цикла +доминируют институции (поведенческие шаблоны и традиции), спонтанно возникающие в +рыночной среде за пределами влияния государства. На последних этапах государство +регламентирует и ставит под свой контроль экономическую активность. Этот процесс и называется +институционализацией. +Развитие институций всегда опережает развитие институтов, поскольку они возникают +вследствие экономической активности рыночных субъектов, пытающихся выжить под гнетом +рыночных доминантов и государства. Тогда как институты представляют собой результат +реактивной деятельности государства на сокращение налогооблагаемой базы вследствие +вытеснения традиционных субъектов рынка его неинституционализированными игроками. +Проще говоря, государство озаботилось институционализацией цифровой торговли после того, как +покупатели стали отворачиваться от традиционных продавцов, а объем отправлений с AliExpress +превысил объем внутренних отправлений через ФГУП «Почта России». +Поэтому не следует ожидать синхронного развития институтов и институций цифровой +торговли. Это противоречило бы самой природе институционального развития. Речь идет о +естественном несовершенстве институциональной политики государства и ошибках при +определении ее приоритетов. Внедрение экономических новаций неизбежно связано с высокой +вероятностью незапланированного поведения рыночных субъектов. Оно нуждается в мониторинге +ситуации и корректировке. + +KyPHAA +HOOPMALOHHOE +6ECTB0ИНФОРМАЦИОННОЕ ОБЩЕСТВО | 2022 | № 6 +WWW.INFOSOC.IIS.RU +61 + +3 Институциональная эволюция цифровой торговли +В основе институций рынка цифровой торговли лежат конкурентные преимущества ее субъектов, +связанные с экономией транзакционных издержек при совершении сделок [5, с. 8]. Цифровая +экономика предоставила им неограниченный доступ к аудитории, автоматизацию продаж и +сетевую инфраструктуру логистики. Причем, на различных стадиях институционального цикла +цифровой торговли указанные преимущества доминируют в определенной последовательности +(см. рис. 1). + +Рис. 1. Конкурентные преимущества на разных стадиях институционального цикла цифровой торговли +На первой стадии появилась возможность совершать сделки через электронные доски +объявлений, в социальных сетях и на интернет-форумах. Начался бум интернет-магазинов, +возникли первые интернет-аукционы, сервисы совместных покупок и дропшиппинг. Структурные +изменения происходили вне внимания государства, поскольку на потребительском рынке сделки +совершались втемную, налоги с них – не выплачивались. Государственная статистика фиксировала +лишь кратное увеличение почтовых отправлений. +На второй стадии начался взрывной рост электронной коммерции. Увеличение масштабов +интернет-продаж +привело +к +появлению +на +рынке +провайдеров +логистических +услуг, +сформировавших +альтернативную +инфраструктуру +цифровой +торговли. +Сильнее +всего +пострадали традиционные оптово-розничные посредники, кредитно-финансовые организации, а +также обеспечиваемые ими налоговые поступления. Государству пришлось приступить к +институциональному регулированию цифровой торговли. +Третья стадия знаменуется завершением структурной перестройки экономического +ландшафта, доминированием укрупняющихся ключевых игроков рынка и сменой их +стратегических приоритетов. Если прежде основными конкурентами субъектов цифровой +торговли выступали традиционные оптово-розничные продавцы, то здесь они терпят +сокрушительное поражение и вытесняются на задворки рынка. Конкурентная борьба +разворачивается +за +повышение +эффективности +и +оптимизацию +бизнес-процессов +при +возрастающей роли государственного регулирования. +Что будет происходить на четвертой, завершающей стадии институционального цикла +цифровой торговли пока трудно предсказать, как и сроки ее начала. Сформируются новые +институции и их носители, действующие за рамками институционального регулирования +государства. Их появление станет очередной попыткой участников рынка вырваться за рамки +удушающего влияния действующих доминантов цифрового рынка. Сейчас до этого еще далеко и +на повестке дня стоят совсем иные проблемы. +4 Смена институционального вектора +Российская экономика находится в самом начале третьей стадии институционального цикла +цифровой торговли, где ведущим фактором конкурентоспособности становится сравнительная +эффективность бизнес-процессов [6, с. 334]. Между участниками рынка обостряется конкуренция, +сам рынок структурируется, значение государственного регулирования возрастает [7, с. 8]. Кроме +того, доминирующая роль в вопросах ассортимента, ценообразования и сбыта переходит от +продавцов к потребителям. Они голосуют рублем, и глобализация предоставляет им для этого все +возможности. +Институциональная политика государства осложняется новизной стоящих задач. Основная +проблема состоит в определении ориентиров и приоритетов институционального строительства. +Любая ошибка неизбежно приводит к институциональному тупику, оттоку покупателей и +отставанию в развитии цифрового рынка. В результате он переходит под контроль более успешных +доступ к +аудитории +Стадия +1 +автоматизация +продаж +Стадия +2 +сетевая +логистика +Стадия +3 + +KyPHAA +HOOPMALOHHOE +6ECTB0ИНФОРМАЦИОННОЕ ОБЩЕСТВО | 2022 | № 6 +WWW.INFOSOC.IIS.RU +62 + +зарубежных конкурентов [8, с. 113-114]. С другой стороны, наличие успешных конкурентов +позволяет изучить их опыт и применить его в своей практике. +Так, к примеру, на непродовольственном рынке цифровой торговли в России доминируют +два противоборствующих течения: розничные сети (Эльдорадо, DNS, Leroy Merlin и пр.) и +маркетплейсы (Wildberries, Lamoda, Ozon, Яндекс-Маркет, Сбермаркет и др.). Их противостояние +обусловлено разной моделью ведения бизнеса: розничные сети извлекают прибыль из своих +продаж, тогда как маркетплейсы получают ее от оказания торговых услуг. Розничные сети +стремятся закрыть и защитить каналы сбыта, а маркетплейсы, наоборот, стремятся максимально +открыть их. +Первоначально пальма первенства была у розничных сетей, довольно успешно +лоббировавших свои интересы через АКИТ (Ассоциация компаний интернет-торговли). Их +главный +интерес +состоял +в +создании +институциональных +барьеров +для +неинституционализированной трансграничной торговли. Образно говоря, сделать так, чтобы +покупка телефона Xiaomi на Aliexpress обходилась покупателям столь же дорого, как в России. +Максимальным +успехом +лоббирования +торговых +сетей +стало +снижение +порога +беспошлинного ввоза товаров для личного пользования до 200 евро. Институциональный эффект +такого +решения +представляется +весьма +спорным, +поскольку +поддержку +получили +не +производители отечественной продукции, а сфера торговли. В убытке оказались частные +потребители, чья покупательская способность снизилась. При этом институциональный цикл +торговых сетей уже находится в начале четвертой стадии: на рынке цифровой торговли взрывной +рост продаж показывают не они, а маркетплейсы [9]. +Главный интерес маркетплейсов заключается в росте продаж через привлечение +максимального числа потребителей, для которых главным фактором является цена товара. +Маркетплейсы не извлекают прибыль от продажи товаров – она формируется в процессе оказания +логистических услуг продавцам. В отличие от розничных сетей, низкие цены для них не беда, а +источник финансирования и институционального роста. +Со сменой рыночного доминанта на глазах меняется и институциональная политика +государства. Так, с 28.03.2022 в ЕАЭС порог беспошлинного ввоза товаров физическими лицами +(временно) возвращен к 1000 евро. Кроме того, лидирующие позиции членов АКИТ перешли к +крупнейшим маркетплейсам (Wildberries, Ozon, Avito, Lamoda, Яндекс-Маркет), что не могло не +сказаться на смене приоритетов ее лоббистской деятельности.1 Вектор институционального +развития +цифровой +торговли +в +России +сменился +от +попыток +ограничить +свободу +неинституционализованных ее участников к формированию ориентированной на них рыночной +инфраструктуры. +5 Большой разворот +Роль маркетплейсов на рынке цифровой торговли трудно переоценить. Первоначально +источником их институционального роста был переток покупателей из традиционной торговли с +более высоким уровнем трансакционных издержек и розничных цен. Однако к концу 2010-х гг. этот +ресурс исчерпал себя. Сегодня на потребительском рынке сопротивление им оказывают розничные +сети, с разной степенью успешности осваивающие цифровую торговлю. Наиболее эффективным +оружием против них является снижение розничных цен и повышение доступности товаров для +покупателей. +2021 год ознаменовался тектоническими изменениями ценовой политики ведущих +российских маркетплейсов: они кратно снизили комиссию для продавцов. Снижение комиссии +маркетплейсов составило у Wildberries до 5-15% (было 38%), у Ozon.ru до 5-8% (было 5-25%), у Яндекс- +Маркета до 2% (было 3-20%). Правильность принятого решения подтвердилось феноменальным +приростом продаж (в 2-3 раза) (см. табл. 2) [10]. + + + +1 Стандарты качества / Бизнесу // АКИТ. URL: https://akit.ru/business/standards (дата обращения 12.06.2022). + +KyPHAA +HOOPMALOHHOE +6ECTB0ИНФОРМАЦИОННОЕ ОБЩЕСТВО | 2022 | № 6 +WWW.INFOSOC.IIS.RU +63 + +Таблица 2. Результаты лидирующих маркетплейсов России за 2021 г. +Маркетплейс +онлайн-продажи +заказы +средний чек +млрд. руб. +прирост +млн. шт. +прирост +руб. +прирост +Wildberries +805,7 ++95% +771,9 ++153% +1040 +-23% +Ozon +446,7 ++126% +221,2 ++199% +2020 +-24% +Яндекс-Маркет +132,6 ++180% +29,7 ++151% +4110 ++12% +AliExpress +106,1 ++116% +48 ++152% +2210 +-14% +Lamoda +71,2 ++34% +14,1 ++15% +5050 ++17% + +При этом прирост продаж был обратно пропорционален размеру среднего чека: чем +меньше сумма покупки, тем больше желающих ее совершить. Из общей картины несколько +выбивается Яндекс-Маркет, но только за счет широкого присутствия на нем розничных сетей, +наоборот, ориентированных на прирост среднего чека. +Особняком на этом фоне стоит маркетплейс Lamoda, демонстративно игнорирующий +институциональные тренды цифрового рынка. У него самые высокие тарифы – 35-70% от +розничной цены продаваемых товаров. Можно предположить, что его прибыль значительно +превосходит прибыль продавцов товара. В этом Lamoda похож на розничные сети. Неудивительно, +что прирост его показателей стабильно ниже прироста продаж других лидеров цифрового рынка. +Следует особо отметить наличие огромного потенциала продаж, связанного со снижением +суммы среднего чека. Все это задает тренд на совершенствование логистических технологий, +направленных на уменьшение транзакционных издержек и снижение розничных цен для +покупателей. Участники цифровой торговли, действующие в рамках указанного тренда, +добиваются наилучших результатов. +6 Новые горизонты цифровой торговли +Практика показывает, что институции цифровой торговли оказывают решающее влияние на +конкурентоспособность ее субъектов [11, с. 60-61]. Отставание E-commerce Index Ranking лишь +подтверждает необходимость корректировки институционального регулирования цифровой +торговли и смены его приоритетов. Игнорирование рыночных трендов и закономерностей резко +снижает эффективность государственной политики и конкурентоспособность российской +цифровой экономики в целом. +В качестве институциональных ориентиров следует выделить три приоритетных +направления развития сетевой экономики и покупательский спрос как движущую силу рыночного +механизма. Выделение этих ориентиров связано с наиболее успешными институциями, +определяющими вектор институционального развития цифровой торговли. +1. Контрактное производство – институция, основанная на изготовлении продукции +независимым производителем по техническому заданию заказчика с отгрузкой «под ключ». Такое +производство переходит из категории работ в категорию услуг. Оно не производит собственную +продукцию, +оказывая +предоплаченные +услуги +заказчикам, +что +обеспечивает +большую +экономическую эффективность. +Контрактное производство не нуждается в кредитовании, имеет отрицательную +оборачиваемость средств и не несет предпринимательских рисков в торговле. В этой модели +инициаторами производства выступают независимые заказчики, отслеживающие конъюнктуру +рынка, принимающие на себя предпринимательские риски и финансирующие производство за +счет +собственных +средств. +Контрактное +производство +становится +придатком +торговли, +ориентирующейся на потребительский спрос. +Пример: Биржа контрактного производства Московского инновационного кластера.2 +2. Логистический провайдинг (англ. Third Party Logistics) – институция, основанная на +делегировании нестратегических внутрифирменных функций независимым провайдерам + +2 Биржа контрактного производства // Московский инновационный кластер. URL: https://i.moscow/contract_exchange (дата +обращения: 18.06.2022). + +KyPHAA +HOOPMALOHHOE +6ECTB0ИНФОРМАЦИОННОЕ ОБЩЕСТВО | 2022 | № 6 +WWW.INFOSOC.IIS.RU +64 + +логистических услуг. Такой провайдинг также переходит из категории работ в категорию услуг. +Провайдеры делятся с заказчиками экономией на масштабе оказываемых услуг, за счет своей узкой +специализации обеспечивая более высокое качество и эффективность. +Практически любая внутрифирменная функция может быть передана независимому +провайдеру: бухгалтерский учет, обработка заказов, разработка технической документации, +организация продаж, документооборота и т.д. [12, с. 57]. Высший уровень логистического +провайдинга (5PL) предполагает делегирование как функции, так и контроля за ее реализацией по +принципу «передал и забыл». В идеальной модели заказчик сосредотачивается на стратегическом +направлении деятельности, а все сопутствующие функции делегирует внешним провайдерам. +Примеры: маркетплейсы, аутсорсинговые и фулфилментовые компании, бухгалтерские +сервисы и т.д. +3. Сетевое предпринимательство – институция, основанная на использовании преимуществ +виртуальной среды, сетевой экономики и цифровой торговли. Они позволяют сократить +затратность ведения бизнеса и снижают входной барьер для участников цифрового рынка, +сокращая временные затраты на реализацию бизнес-проектов. +Идеальная модель сетевого предпринимательства стремится к тому, что называется +«виртуальная организация», не имеющая ни офиса, ни постоянного штата сотрудников [13, с. 279- +281]. Предприниматель здесь выступает в роли организатора и координатора «цепочек создания +ценностей», потенциал которых он использует для реализации своего бизнес-проекта. В качестве +его сетевых партнеров выступают как контрактные производители, так и провайдеры +логистических услуг. +Пример: Самодеятельные продавцы маркетплейсов (Ozon, Wildberries и Яндекс-Маркет), +продающие контрактные товары под своими брендами. +В своей совокупности все институции образуют экосреду цифровой экономики, в которой +цифровая торговля инициирует не только процесс товародвижения, но и товарного производства. +Покупатель своим спросом инициирует предпринимательскую активность продавца, который на +свой страх и риск организует контрактное производство востребованных товаров и привлекает +сетевых провайдеров логистических услуг. +В корне меняются институциональные роли участников сетевого рынка: +Покупатели – получают возможность неограниченного выбора, ставя продавцов в условия +совершенной конкуренции. +Продавцы – откликаются на запросы покупателей, первичный спрос которых инициирует их +вторичную предпринимательскую активность. +Сетевые провайдеры – оказывают логистические услуги продавцам (не покупателям!), +принимая на себя отдельные функции организации товародвижения. +Производители – оказывают услуги контрактного производства продавцам, соревнуюсь +между собой в гибкости производства и скорости выполнения заказов. +Пока наибольшую эффективность показывает институция, в рамках которой покупатель +взаимодействует с маркетплейсом, принимающим на себя все заботы по организации товаропотока +(Wildberries, Ozon, AliExpress). Однако уже сегодня многие продавцы продают свои товары +одновременно на нескольких маркетплейсах, а нелояльные покупатели, сравнивая цены, покупают +там, где дешевле. Свобода потребительского выбора размоет диктат маркетплейсов, как они сегодня +размывают диктат розничных сетей. +Рано или поздно и маркетплейсы достигнут предела институционального развития и +перейдут в категорию «при прочих равных» за счет обострения внутривидовой конкуренции. Если +это +произойдет, +то +между +продавцами +и +покупателями +сформируется +логистическая +инфраструктура, в равной мере доступная всем участникам цифрового рынка. Аналогично +электричество или компьютеры были когда-то источником рыночной конкурентоспособности, а +сегодня воспринимаются как естественная часть рыночного ландшафта. +Заключение +Приоритетом институциональной политики государства может стать превращение цифровой +торговли в один из локомотивов экономического роста. Для этого необходимо сосредоточиться на + +KyPHAA +HOOPMALOHHOE +6ECTB0ИНФОРМАЦИОННОЕ ОБЩЕСТВО | 2022 | № 6 +WWW.INFOSOC.IIS.RU +65 + +снижении транзакционных издержек в сетях товародвижения. В традиционной торговле +потребительскими товарами транзакционные издержки (маржа оптово-розничных сетей) +составляли 80-100% от конечной цены товара. В маркетплейсах типа Lamoda они и сегодня +составляют 30-70% от цены продавца. +Вместе с тем, практика институционального развития цифровой торговли задает совсем иной +вектор. Более продвинутые маркетплейсы (Wildberries, Ozon, Яндекс-Маркет) еще в начале 2021 года +инициативно снизили размер своей комиссии до 3-5% и это привело к впечатляющим результатам. +Так, например, продажи самозанятых на Wildberries только в первом квартале 2022 года выросли на +410% (до 2 млрд руб.), а их численность увеличилась почти пятикратно (до 150 тыс. чел.).3 +В условиях экономического кризиса и западных санкций снижение транзакционных +издержек в цифровой торговле способно компенсировать снижение покупательной способности +населения. +Важно +сохранить +доступность +товаров +массового +спроса +и +поддержать +товаропроизводителей. Вытесняя из торговой цепочки посредническое звено за счет ускоренной +цифровизации торговли, можно не только способствовать решению социальных задач, но и +стимулировать рост предпринимательской активности в производственной сфере. Представляется, +что именно эта цель должна стать одним из приоритетов институциональной политики +государства в отношении цифровой торговли на ближайшие годы. +Литература +1. Указ Президента РФ № 203 от 09.05.2017 г. «О Стратегии развития информационного +общества в Российской Федерации на 2017-2030 годы» / Документы // Президент России. +URL: http://kremlin.ru/acts/bank/41919. +2. Интернет-торговля в России 2021: Аналитический отчет. М.: Data Insight, 2022. 156 с. +3. The UNCTAD B2C E-commerce Index 2020 / UNCTAD Technical Notes on ICT for +Development 2021 // UNCTAD ONU. 2022. № 17. 22 р. +4. Блуммарт Т. Четвертая промышленная революция и бизнес: как конкурировать и +развиваться в эпоху сингулярности. М.: Альпина Паблишер, 2019. 204 с. +5. Данные для лучшей жизни: Обзор доклада о мировом развитии. Washington: +Международный банк реконструкции и развития / Всемирный банк, 2021. 39 с. +6. Кочетков Е.П. Цифровая трансформация экономики и технологические революции: +вызовы для текущей парадигмы менеджмента и антикризисного управления // +Стратегические решения и риск-менеджмент. 2019. Т. 10. № 4. С. 330-341. DOI: +10.17747/2618-947X-2019-4-330-341. +7. Антимонопольное регулирование в цифровую эпоху: как защищать конкуренцию в +условиях глобализации и четвертой промышленной революции: монография. М.: ВШЭ, +2019. 391 с. +8. Борисова В.В., Юань Х., Тан Л. Стратегии развития электронной платформы Aliexpress в +России // Вестник Ростовского государственного экономического университета (РИНХ). +2020. № 4 (72). С. 110-115. +9. Романова Т. Маркетплейсы рвутся вверх: за счет чего выросли обороты крупнейших +ретейлеров России / Бизнес // Forbes. [Электронный ресурс]. 7 июня 2022 г. URL: +https://www.forbes.ru/biznes/467927-marketplejsy-rvutsa-vverh-za-scet-cego-vyrosli-oboroty- +krupnejsih-retejlerov-rossii (дата обращения: 15.06.2022). +10. Рейтинг ТОП-100 крупнейших российских интернет-магазинов. М.: Data Insight, 2022. URL: +https://top100.datainsight.ru (дата обращения: 17.06.2022). +11. Слонимская М.А. Сетевые формы организации экономики. Мн.: Беларуская навука, 2018. +279 с. +12. Tan A., Shukkla S. Digital transformation of the supply chain: a practical guide for. Danvers +(USA): World Scientific Publishing, 2021. 152 p. +13. Уорнер М., Витцель М. Виртуальные организации. Новые формы ведения бизнеса в XXI +веке. М.: Добрая книга, 2005. 296 с. + + + +3 Продажи самозанятых из России на Wildberries выросли на 410% с января 2022 года. 29.06.2022. / Экономика // ТАСС. URL: +https://tass.ru/ekonomika/15065193 (дата обращения 13.07.2022). + +KyPHAA +HOOPMALOHHOE +6ECTB0ИНФОРМАЦИОННОЕ ОБЩЕСТВО | 2022 | № 6 +WWW.INFOSOC.IIS.RU +66 + +INSTITUTIONALIZATION OF DIGITAL TRADE IN THE RUSSIAN +FEDERATION: COUNTDOWN +Kaluzhsky, Mikhail Leonidovich +Candidate of philosophical sciences, associate professor +Fund of Regional Development Strategy, executive director +Omsk State Technical University, department “Organization and management of science-intensive industries”, +associate professor +Omsk, Russian Federation +frsr@inbox.ru +Abstract +The institutionalization of digital trade is one of the most important directions in the formation of the information +society in the Russian Federation. The studies reflect the emerging lag in the Russian economy readiness index to +support online shopping. The author analyzes the reasons for the lag in the context of the institutional features of +the development of digital trade. As the main obstacle that reduces the economic efficiency and competitiveness of +digital trade, insufficient attention of the state to the formation of innovative institutions of the digital market is +highlighted. +Keywords +network economy; digital trade; digital market; e-commerce; institutional policy; contract manufacturing; network +entrepreneurship; marketplaces; logistics providers +References +1. Ukaz Prezidenta RF № 203 ot 09.05.2017 g. «O Strategii razvitiya informacionnogo obshchestva v +Rossijskoj Federacii na 2017-2030 gody» / Dokumenty // Prezident Rossii. URL: +http://kremlin.ru/acts/bank/41919. +2. Internet-torgovlya v Rossii 2021: Analiticheskij otchet. M.: Data Insight, 2022. 156 s. +3. The UNCTAD B2C E-commerce Index 2020 / UNCTAD Technical Notes on ICT for +Development 2021 // UNCTAD ONU. 2022. № 17. 22 r. +4. Blummart T. Chetvertaya promyshlennaya revolyuciya i biznes: kak konkurirovat' i razvivat'sya +v ehpokhu singulyarnosti. 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Marketplejsy rvutsya vverkh: za schet chego vyrosli oboroty krupnejshikh +retejlerov Rossii / Biznes // Forbes. [Ehlektronnyj resurs]. 7 iyunya 2022 g. URL: +https://www.forbes.ru/biznes/467927-marketplejsy-rvutsa-vverh-za-scet-cego-vyrosli-oboroty- +krupnejsih-retejlerov-rossii (data obrashcheniya: 15.06.2022). +10. Rejting TOP-100 krupnejshikh rossijskikh internet-magazinov. M.: Data Insight, 2022. URL: +https://top100.datainsight.ru (data obrashcheniya: 17.06.2022). +11. Slonimskaya M.A. Setevye formy organizacii ehkonomiki. Mn.: Belaruskaya navuka, 2018. 279 s. +12. Tan A., Shukkla S. Digital transformation of the supply chain: a practical guide for. Danvers +(USA): World Scientific Publishing, 2021. 152 p. +13. Uorner M., Vitcel' M. Virtual'nye organizacii. Novye formy vedeniya biznesa v XXI veke. 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content='0/legalcode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content='ru https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content='52605/16059921_2022_06_59 Цифровая экономика ИНСТИТУЦИОНАЛИЗАЦИЯ ЦИФРОВОЙ ТОРГОВЛИ В РОССИЙСКОЙ ФЕДЕРАЦИИ: ОБРАТНЫЙ ОТСЧЕТ Статья рекомендована к публикации главным редактором Т.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content='В.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Ершовой 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content='07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Калужский Михаил Леонидович Кандидат философских наук, доцент МОФ «Фонд региональной стратегии развития», исполнительный директор Омский государственный технический университет, каф.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' «Организация и управление наукоемкими производствами», доцент Омск, Российская Федерация frsr@inbox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content='ru Аннотация Институционализация цифровой торговли является одним из важнейших направлений формирования информационного общества в Российской Федерации.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Исследования отражают наметившееся отставание российского индекса готовности экономики поддерживать онлайн-покупки.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Автор анализирует причины отставания в контексте институциональных особенностей развития цифровой торговли.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' В качестве основного препятствия, снижающего экономическую эффективность и конкурентоспособность цифровой торговли, выделяется недостаточное внимание государства формированию инновационных институций цифрового рынка.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Ключевые слова сетевая экономика;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' цифровая торговля;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' цифровой рынок;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' электронная коммерция;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' институциональная политика;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' контрактное производство;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' сетевое предпринимательство;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' маркетплейсы;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' логистический провайдинг Введение Цифровизация не просто определяет ключевое направление развития российской экономики, но служит источником институционального роста цифровой торговли.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Указом Президента РФ № 203 от 09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content='2017 г.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' определены национальные интересы, затрагивающие сферу цифровой торговли, среди которых следует выделить: 1) формирование виртуальных рынков и обеспечение лидерства на них за счет развития российской экосистемы цифровой экономики;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' 2) обеспечение недискриминационного доступа к товарам и услугам российских поставщиков;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' 3) поддержка отраслей, использующих преимущества информационных технологий;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' 4) увеличение экспорта за рубеж несырьевых товаров и услуг;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' 5) создание платежной и логистической инфраструктуры интернет-торговли.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Перед Правительством РФ поставлена задача формирования технологической основы цифровой экономики, в том числе через повышение доступности электронных форм коммерческих отношений предприятиям малого и среднего бизнеса [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Для ее решения приняты и реализуются «Стратегия развития информационного общества в Российской Федерации на 2017-2030 годы», национальная программа «Цифровая экономика Российской Федерации» и федеральные проекты «Информационная инфраструктура», «Цифровые технологии», «Цифровые услуги и сервисы онлайн» и др.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' KyPHAA HOOPMALOHHOE 6ECTB0ИНФОРМАЦИОННОЕ ОБЩЕСТВО | 2022 | № 6 WWW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content='INFOSOC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content='IIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content='RU 60 1 Российская цифровая торговля в мировых рейтингах Рейтинговые оценки цифровой торговли в России свидетельствуют о том, что пока рано говорить о значительных успехах.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Согласно аналитическому отчету «Интернет-торговля в России: 2021» компании Data Insight институциональное и инфраструктурное развитие цифровой торговли в Россия пока далеко от совершенства (см.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' табл.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' 1) [2, с.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Таблица 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Рейтинг Российской Федерации в мировой системе цифровой торговли Рейтинг Место Best Countries For Investment In E-commerce And Digital Sector (Ceoworld) – индекс привлекательности страны для инвестирования в электронную коммерцию и цифровой сектор 15 The Inclusive Internet Index – индекс доступности цифровой инфраструктуры,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' цен,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' локального контента,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' вовлеченности пользователей и культурных факторов 25 The Ease of Doing Business Index – индекс благоприятности условий предпринимательской деятельности 28 UNCTAD B2C E-commerce Index Ranking (UNCTAD) – индекс готовности экономики поддерживать онлайн-покупки 41 В целом приведенный рейтинг довольно наглядно отражает сложившееся положение в цифровой торговле.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Индекс Ceoworld показывает лучший результат за счет доминирования на рынке крупных торговых сетей, интернет-магазинов и маркетплейсов.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Inclusive Internet Index демонстрирует вовлеченность пользователей в среду цифровой торговли.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Ease of Doing Business Index отражает отставание предложения отечественных продавцов от покупательского спроса.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Хуже всех выглядит индекс UNCTAD (ООН), согласно которому в 2020 г.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' готовность экономических институтов поддерживать онлайн-покупки в России находилась на 41 месте из 152 стран мира [3, с.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Именно этот индекс отражает недостаточную эффективность институциональной политики государства в сфере цифровой торговли.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' 2 Институциональные процессы в цифровой торговле Цифровая торговля являет собой типичный пример технологической инновации, выступающей следствием очередного институционального цикла [4, с.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Институциональный цикл проходит в своем развитии те же этапы, что и любой жизненный цикл в экономике, менеджменте или маркетинге: выход на рынок, рост, зрелость и упадок.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' На первых этапах институционального цикла доминируют институции (поведенческие шаблоны и традиции), спонтанно возникающие в рыночной среде за пределами влияния государства.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' На последних этапах государство регламентирует и ставит под свой контроль экономическую активность.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Этот процесс и называется институционализацией.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Развитие институций всегда опережает развитие институтов, поскольку они возникают вследствие экономической активности рыночных субъектов, пытающихся выжить под гнетом рыночных доминантов и государства.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Тогда как институты представляют собой результат реактивной деятельности государства на сокращение налогооблагаемой базы вследствие вытеснения традиционных субъектов рынка его неинституционализированными игроками.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Проще говоря, государство озаботилось институционализацией цифровой торговли после того, как покупатели стали отворачиваться от традиционных продавцов, а объем отправлений с AliExpress превысил объем внутренних отправлений через ФГУП «Почта России».' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Поэтому не следует ожидать синхронного развития институтов и институций цифровой торговли.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Это противоречило бы самой природе институционального развития.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Речь идет о естественном несовершенстве институциональной политики государства и ошибках при определении ее приоритетов.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Внедрение экономических новаций неизбежно связано с высокой вероятностью незапланированного поведения рыночных субъектов.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Оно нуждается в мониторинге ситуации и корректировке.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' KyPHAA HOOPMALOHHOE 6ECTB0ИНФОРМАЦИОННОЕ ОБЩЕСТВО | 2022 | № 6 WWW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content='INFOSOC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content='IIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content='RU 61 3 Институциональная эволюция цифровой торговли В основе институций рынка цифровой торговли лежат конкурентные преимущества ее субъектов, связанные с экономией транзакционных издержек при совершении сделок [5, с.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Цифровая экономика предоставила им неограниченный доступ к аудитории, автоматизацию продаж и сетевую инфраструктуру логистики.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Причем, на различных стадиях институционального цикла цифровой торговли указанные преимущества доминируют в определенной последовательности (см.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' рис.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Рис.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Конкурентные преимущества на разных стадиях институционального цикла цифровой торговли На первой стадии появилась возможность совершать сделки через электронные доски объявлений, в социальных сетях и на интернет-форумах.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Начался бум интернет-магазинов, возникли первые интернет-аукционы, сервисы совместных покупок и дропшиппинг.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Структурные изменения происходили вне внимания государства, поскольку на потребительском рынке сделки совершались втемную, налоги с них – не выплачивались.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Государственная статистика фиксировала лишь кратное увеличение почтовых отправлений.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' На второй стадии начался взрывной рост электронной коммерции.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Увеличение масштабов интернет-продаж привело к появлению на рынке провайдеров логистических услуг, сформировавших альтернативную инфраструктуру цифровой торговли.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Сильнее всего пострадали традиционные оптово-розничные посредники, кредитно-финансовые организации, а также обеспечиваемые ими налоговые поступления.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Государству пришлось приступить к институциональному регулированию цифровой торговли.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Третья стадия знаменуется завершением структурной перестройки экономического ландшафта, доминированием укрупняющихся ключевых игроков рынка и сменой их стратегических приоритетов.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Если прежде основными конкурентами субъектов цифровой торговли выступали традиционные оптово-розничные продавцы, то здесь они терпят сокрушительное поражение и вытесняются на задворки рынка.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Конкурентная борьба разворачивается за повышение эффективности и оптимизацию бизнес-процессов при возрастающей роли государственного регулирования.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Что будет происходить на четвертой, завершающей стадии институционального цикла цифровой торговли пока трудно предсказать, как и сроки ее начала.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Сформируются новые институции и их носители, действующие за рамками институционального регулирования государства.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Их появление станет очередной попыткой участников рынка вырваться за рамки удушающего влияния действующих доминантов цифрового рынка.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Сейчас до этого еще далеко и на повестке дня стоят совсем иные проблемы.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' 4 Смена институционального вектора Российская экономика находится в самом начале третьей стадии институционального цикла цифровой торговли, где ведущим фактором конкурентоспособности становится сравнительная эффективность бизнес-процессов [6, с.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' 334].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Между участниками рынка обостряется конкуренция, сам рынок структурируется, значение государственного регулирования возрастает [7, с.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Кроме того, доминирующая роль в вопросах ассортимента, ценообразования и сбыта переходит от продавцов к потребителям.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Они голосуют рублем, и глобализация предоставляет им для этого все возможности.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Институциональная политика государства осложняется новизной стоящих задач.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Основная проблема состоит в определении ориентиров и приоритетов институционального строительства.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Любая ошибка неизбежно приводит к институциональному тупику, оттоку покупателей и отставанию в развитии цифрового рынка.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' В результате он переходит под контроль более успешных доступ к аудитории Стадия 1 автоматизация продаж Стадия 2 сетевая логистика Стадия 3 KyPHAA HOOPMALOHHOE 6ECTB0ИНФОРМАЦИОННОЕ ОБЩЕСТВО | 2022 | № 6 WWW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content='INFOSOC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content='IIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content='RU 62 зарубежных конкурентов [8, с.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' 113-114].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' С другой стороны, наличие успешных конкурентов позволяет изучить их опыт и применить его в своей практике.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Так, к примеру, на непродовольственном рынке цифровой торговли в России доминируют два противоборствующих течения: розничные сети (Эльдорадо, DNS, Leroy Merlin и пр.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=') и маркетплейсы (Wildberries, Lamoda, Ozon, Яндекс-Маркет, Сбермаркет и др.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Их противостояние обусловлено разной моделью ведения бизнеса: розничные сети извлекают прибыль из своих продаж, тогда как маркетплейсы получают ее от оказания торговых услуг.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Розничные сети стремятся закрыть и защитить каналы сбыта, а маркетплейсы, наоборот, стремятся максимально открыть их.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Первоначально пальма первенства была у розничных сетей, довольно успешно лоббировавших свои интересы через АКИТ (Ассоциация компаний интернет-торговли).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Их главный интерес состоял в создании институциональных барьеров для неинституционализированной трансграничной торговли.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Образно говоря, сделать так, чтобы покупка телефона Xiaomi на Aliexpress обходилась покупателям столь же дорого, как в России.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Максимальным успехом лоббирования торговых сетей стало снижение порога беспошлинного ввоза товаров для личного пользования до 200 евро.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Институциональный эффект такого решения представляется весьма спорным, поскольку поддержку получили не производители отечественной продукции, а сфера торговли.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' В убытке оказались частные потребители, чья покупательская способность снизилась.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' При этом институциональный цикл торговых сетей уже находится в начале четвертой стадии: на рынке цифровой торговли взрывной рост продаж показывают не они, а маркетплейсы [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Главный интерес маркетплейсов заключается в росте продаж через привлечение максимального числа потребителей, для которых главным фактором является цена товара.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Маркетплейсы не извлекают прибыль от продажи товаров – она формируется в процессе оказания логистических услуг продавцам.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' В отличие от розничных сетей, низкие цены для них не беда, а источник финансирования и институционального роста.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Со сменой рыночного доминанта на глазах меняется и институциональная политика государства.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Так, с 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content='03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content='2022 в ЕАЭС порог беспошлинного ввоза товаров физическими лицами (временно) возвращен к 1000 евро.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Кроме того, лидирующие позиции членов АКИТ перешли к крупнейшим маркетплейсам (Wildberries, Ozon, Avito, Lamoda, Яндекс-Маркет), что не могло не сказаться на смене приоритетов ее лоббистской деятельности.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content='1 Вектор институционального развития цифровой торговли в России сменился от попыток ограничить свободу неинституционализованных ее участников к формированию ориентированной на них рыночной инфраструктуры.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' 5 Большой разворот Роль маркетплейсов на рынке цифровой торговли трудно переоценить.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Первоначально источником их институционального роста был переток покупателей из традиционной торговли с более высоким уровнем трансакционных издержек и розничных цен.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Однако к концу 2010-х гг.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' этот ресурс исчерпал себя.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Сегодня на потребительском рынке сопротивление им оказывают розничные сети, с разной степенью успешности осваивающие цифровую торговлю.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Наиболее эффективным оружием против них является снижение розничных цен и повышение доступности товаров для покупателей.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' 2021 год ознаменовался тектоническими изменениями ценовой политики ведущих российских маркетплейсов: они кратно снизили комиссию для продавцов.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Снижение комиссии маркетплейсов составило у Wildberries до 5-15% (было 38%), у Ozon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content='ru до 5-8% (было 5-25%), у Яндекс- Маркета до 2% (было 3-20%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Правильность принятого решения подтвердилось феноменальным приростом продаж (в 2-3 раза) (см.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' табл.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' 2) [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' 1 Стандарты качества / Бизнесу // АКИТ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' URL: https://akit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content='ru/business/standards (дата обращения 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content='06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content='2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' KyPHAA HOOPMALOHHOE 6ECTB0ИНФОРМАЦИОННОЕ ОБЩЕСТВО | 2022 | № 6 WWW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content='INFOSOC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content='IIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content='RU 63 Таблица 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Результаты лидирующих маркетплейсов России за 2021 г.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Маркетплейс онлайн-продажи заказы средний чек млрд.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' руб.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' прирост млн.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' шт.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' прирост руб.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' прирост Wildberries 805,7 +95% 771,9 +153% 1040 23% Ozon 446,7 +126% 221,2 +199% 2020 24% Яндекс-Маркет 132,6 +180% 29,7 +151% 4110 +12% AliExpress 106,1 +116% 48 +152% 2210 14% Lamoda 71,2 +34% 14,1 +15% 5050 +17% При этом прирост продаж был обратно пропорционален размеру среднего чека: чем меньше сумма покупки, тем больше желающих ее совершить.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Из общей картины несколько выбивается Яндекс-Маркет, но только за счет широкого присутствия на нем розничных сетей, наоборот, ориентированных на прирост среднего чека.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Особняком на этом фоне стоит маркетплейс Lamoda, демонстративно игнорирующий институциональные тренды цифрового рынка.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' У него самые высокие тарифы – 35-70% от розничной цены продаваемых товаров.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Можно предположить, что его прибыль значительно превосходит прибыль продавцов товара.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' В этом Lamoda похож на розничные сети.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Неудивительно, что прирост его показателей стабильно ниже прироста продаж других лидеров цифрового рынка.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Следует особо отметить наличие огромного потенциала продаж, связанного со снижением суммы среднего чека.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Все это задает тренд на совершенствование логистических технологий, направленных на уменьшение транзакционных издержек и снижение розничных цен для покупателей.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Участники цифровой торговли, действующие в рамках указанного тренда, добиваются наилучших результатов.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' 6 Новые горизонты цифровой торговли Практика показывает, что институции цифровой торговли оказывают решающее влияние на конкурентоспособность ее субъектов [11, с.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' 60-61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Отставание E-commerce Index Ranking лишь подтверждает необходимость корректировки институционального регулирования цифровой торговли и смены его приоритетов.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Игнорирование рыночных трендов и закономерностей резко снижает эффективность государственной политики и конкурентоспособность российской цифровой экономики в целом.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' В качестве институциональных ориентиров следует выделить три приоритетных направления развития сетевой экономики и покупательский спрос как движущую силу рыночного механизма.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Выделение этих ориентиров связано с наиболее успешными институциями, определяющими вектор институционального развития цифровой торговли.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Контрактное производство – институция, основанная на изготовлении продукции независимым производителем по техническому заданию заказчика с отгрузкой «под ключ».' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Такое производство переходит из категории работ в категорию услуг.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Оно не производит собственную продукцию, оказывая предоплаченные услуги заказчикам, что обеспечивает большую экономическую эффективность.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Контрактное производство не нуждается в кредитовании, имеет отрицательную оборачиваемость средств и не несет предпринимательских рисков в торговле.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' В этой модели инициаторами производства выступают независимые заказчики, отслеживающие конъюнктуру рынка, принимающие на себя предпринимательские риски и финансирующие производство за счет собственных средств.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Контрактное производство становится придатком торговли, ориентирующейся на потребительский спрос.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Пример: Биржа контрактного производства Московского инновационного кластера.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Логистический провайдинг (англ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Third Party Logistics) – институция, основанная на делегировании нестратегических внутрифирменных функций независимым провайдерам 2 Биржа контрактного производства // Московский инновационный кластер.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' URL: https://i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content='moscow/contract_exchange (дата обращения: 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content='06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content='2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' KyPHAA HOOPMALOHHOE 6ECTB0ИНФОРМАЦИОННОЕ ОБЩЕСТВО | 2022 | № 6 WWW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content='INFOSOC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content='IIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content='RU 64 логистических услуг.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Такой провайдинг также переходит из категории работ в категорию услуг.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Провайдеры делятся с заказчиками экономией на масштабе оказываемых услуг, за счет своей узкой специализации обеспечивая более высокое качество и эффективность.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Практически любая внутрифирменная функция может быть передана независимому провайдеру: бухгалтерский учет, обработка заказов, разработка технической документации, организация продаж, документооборота и т.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content='д.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' [12, с.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' 57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Высший уровень логистического провайдинга (5PL) предполагает делегирование как функции, так и контроля за ее реализацией по принципу «передал и забыл».' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' В идеальной модели заказчик сосредотачивается на стратегическом направлении деятельности, а все сопутствующие функции делегирует внешним провайдерам.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Примеры: маркетплейсы, аутсорсинговые и фулфилментовые компании, бухгалтерские сервисы и т.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content='д.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Сетевое предпринимательство – институция, основанная на использовании преимуществ виртуальной среды, сетевой экономики и цифровой торговли.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Они позволяют сократить затратность ведения бизнеса и снижают входной барьер для участников цифрового рынка, сокращая временные затраты на реализацию бизнес-проектов.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Идеальная модель сетевого предпринимательства стремится к тому, что называется «виртуальная организация», не имеющая ни офиса, ни постоянного штата сотрудников [13, с.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' 279- 281].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Предприниматель здесь выступает в роли организатора и координатора «цепочек создания ценностей», потенциал которых он использует для реализации своего бизнес-проекта.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' В качестве его сетевых партнеров выступают как контрактные производители, так и провайдеры логистических услуг.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Пример: Самодеятельные продавцы маркетплейсов (Ozon, Wildberries и Яндекс-Маркет), продающие контрактные товары под своими брендами.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' В своей совокупности все институции образуют экосреду цифровой экономики, в которой цифровая торговля инициирует не только процесс товародвижения, но и товарного производства.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Покупатель своим спросом инициирует предпринимательскую активность продавца, который на свой страх и риск организует контрактное производство востребованных товаров и привлекает сетевых провайдеров логистических услуг.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' В корне меняются институциональные роли участников сетевого рынка: Покупатели – получают возможность неограниченного выбора, ставя продавцов в условия совершенной конкуренции.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Продавцы – откликаются на запросы покупателей, первичный спрос которых инициирует их вторичную предпринимательскую активность.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Сетевые провайдеры – оказывают логистические услуги продавцам (не покупателям!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' ), принимая на себя отдельные функции организации товародвижения.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Производители – оказывают услуги контрактного производства продавцам, соревнуюсь между собой в гибкости производства и скорости выполнения заказов.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Пока наибольшую эффективность показывает институция, в рамках которой покупатель взаимодействует с маркетплейсом, принимающим на себя все заботы по организации товаропотока (Wildberries, Ozon, AliExpress).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Однако уже сегодня многие продавцы продают свои товары одновременно на нескольких маркетплейсах, а нелояльные покупатели, сравнивая цены, покупают там, где дешевле.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Свобода потребительского выбора размоет диктат маркетплейсов, как они сегодня размывают диктат розничных сетей.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Рано или поздно и маркетплейсы достигнут предела институционального развития и перейдут в категорию «при прочих равных» за счет обострения внутривидовой конкуренции.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Если это произойдет, то между продавцами и покупателями сформируется логистическая инфраструктура, в равной мере доступная всем участникам цифрового рынка.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Аналогично электричество или компьютеры были когда-то источником рыночной конкурентоспособности, а сегодня воспринимаются как естественная часть рыночного ландшафта.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Заключение Приоритетом институциональной политики государства может стать превращение цифровой торговли в один из локомотивов экономического роста.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Для этого необходимо сосредоточиться на KyPHAA HOOPMALOHHOE 6ECTB0ИНФОРМАЦИОННОЕ ОБЩЕСТВО | 2022 | № 6 WWW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content='INFOSOC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content='IIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content='RU 65 снижении транзакционных издержек в сетях товародвижения.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' В традиционной торговле потребительскими товарами транзакционные издержки (маржа оптово-розничных сетей) составляли 80-100% от конечной цены товара.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' В маркетплейсах типа Lamoda они и сегодня составляют 30-70% от цены продавца.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Вместе с тем, практика институционального развития цифровой торговли задает совсем иной вектор.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Более продвинутые маркетплейсы (Wildberries, Ozon, Яндекс-Маркет) еще в начале 2021 года инициативно снизили размер своей комиссии до 3-5% и это привело к впечатляющим результатам.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Так, например, продажи самозанятых на Wildberries только в первом квартале 2022 года выросли на 410% (до 2 млрд руб.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' ), а их численность увеличилась почти пятикратно (до 150 тыс.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' чел.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content='3 В условиях экономического кризиса и западных санкций снижение транзакционных издержек в цифровой торговле способно компенсировать снижение покупательной способности населения.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Важно сохранить доступность товаров массового спроса и поддержать товаропроизводителей.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Вытесняя из торговой цепочки посредническое звено за счет ускоренной цифровизации торговли, можно не только способствовать решению социальных задач, но и стимулировать рост предпринимательской активности в производственной сфере.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Представляется, что именно эта цель должна стать одним из приоритетов институциональной политики государства в отношении цифровой торговли на ближайшие годы.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Литература 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Указ Президента РФ № 203 от 09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content='2017 г.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' «О Стратегии развития информационного общества в Российской Федерации на 2017-2030 годы» / Документы // Президент России.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' URL: http://kremlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content='ru/acts/bank/41919.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Интернет-торговля в России 2021: Аналитический отчет.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' М.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=': Data Insight, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' 156 с.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' The UNCTAD B2C E-commerce Index 2020 / UNCTAD Technical Notes on ICT for Development 2021 // UNCTAD ONU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' № 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' 22 р.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Блуммарт Т.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Четвертая промышленная революция и бизнес: как конкурировать и развиваться в эпоху сингулярности.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' М.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=': Альпина Паблишер, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' 204 с.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Данные для лучшей жизни: Обзор доклада о мировом развитии.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Washington: Международный банк реконструкции и развития / Всемирный банк, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' 39 с.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Кочетков Е.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content='П.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Цифровая трансформация экономики и технологические революции: вызовы для текущей парадигмы менеджмента и антикризисного управления // Стратегические решения и риск-менеджмент.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Т.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' № 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' С.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' 330-341.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content='17747/2618-947X-2019-4-330-341.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Антимонопольное регулирование в цифровую эпоху: как защищать конкуренцию в условиях глобализации и четвертой промышленной революции: монография.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' М.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=': ВШЭ, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' 391 с.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Борисова В.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content='В.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=', Юань Х.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=', Тан Л.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Стратегии развития электронной платформы Aliexpress в России // Вестник Ростовского государственного экономического университета (РИНХ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' № 4 (72).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' С.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' 110-115.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Романова Т.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Маркетплейсы рвутся вверх: за счет чего выросли обороты крупнейших ретейлеров России / Бизнес // Forbes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' [Электронный ресурс].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' 7 июня 2022 г.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' URL: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content='forbes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content='ru/biznes/467927-marketplejsy-rvutsa-vverh-za-scet-cego-vyrosli-oboroty- krupnejsih-retejlerov-rossii (дата обращения: 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content='06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content='2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Рейтинг ТОП-100 крупнейших российских интернет-магазинов.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' М.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=': Data Insight, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' URL: https://top100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content='datainsight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content='ru (дата обращения: 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content='06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content='2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Слонимская М.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content='А.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Сетевые формы организации экономики.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Мн.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' : Беларуская навука, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' 279 с.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Tan A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=', Shukkla S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Digital transformation of the supply chain: a practical guide for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Danvers (USA): World Scientific Publishing, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' 152 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Уорнер М.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=', Витцель М.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Виртуальные организации.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Новые формы ведения бизнеса в XXI веке.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' М.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=': Добрая книга, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' 296 с.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' 3 Продажи самозанятых из России на Wildberries выросли на 410% с января 2022 года.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content='06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' / Экономика // ТАСС.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' URL: https://tass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content='ru/ekonomika/15065193 (дата обращения 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content='07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content='2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' KyPHAA HOOPMALOHHOE 6ECTB0ИНФОРМАЦИОННОЕ ОБЩЕСТВО | 2022 | № 6 WWW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content='INFOSOC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content='IIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content='RU 66 INSTITUTIONALIZATION OF DIGITAL TRADE IN THE RUSSIAN FEDERATION: COUNTDOWN Kaluzhsky, Mikhail Leonidovich Candidate of philosophical sciences, associate professor Fund of Regional Development Strategy, executive director Omsk State Technical University, department “Organization and management of science-intensive industries”, associate professor Omsk, Russian Federation frsr@inbox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content='ru Abstract The institutionalization of digital trade is one of the most important directions in the formation of the information society in the Russian Federation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' The studies reflect the emerging lag in the Russian economy readiness index to support online shopping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' The author analyzes the reasons for the lag in the context of the institutional features of the development of digital trade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' As the main obstacle that reduces the economic efficiency and competitiveness of digital trade, insufficient attention of the state to the formation of innovative institutions of the digital market is highlighted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Keywords network economy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' digital trade;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' digital market;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' e-commerce;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' institutional policy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' contract manufacturing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' network entrepreneurship;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' marketplaces;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' logistics providers References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Ukaz Prezidenta RF № 203 ot 09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content='2017 g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' «O Strategii razvitiya informacionnogo obshchestva v Rossijskoj Federacii na 2017-2030 gody» / Dokumenty // Prezident Rossii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' URL: http://kremlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content='ru/acts/bank/41919.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Internet-torgovlya v Rossii 2021: Analiticheskij otchet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=': Data Insight, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' 156 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' The UNCTAD B2C E-commerce Index 2020 / UNCTAD Technical Notes on ICT for Development 2021 // UNCTAD ONU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' № 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' 22 r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Blummart T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=" Chetvertaya promyshlennaya revolyuciya i biznes: kak konkurirovat' i razvivat'sya v ehpokhu singulyarnosti." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' M.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' 7 iyunya 2022 g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' URL: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content='forbes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content='ru/biznes/467927-marketplejsy-rvutsa-vverh-za-scet-cego-vyrosli-oboroty- krupnejsih-retejlerov-rossii (data obrashcheniya: 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content='06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content='2022).' metadata={'source': 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+page_content=' Digital transformation of the supply chain: a practical guide for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Danvers (USA): World Scientific Publishing, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' 152 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Uorner M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=", Vitcel' M." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=" Virtual'nye organizacii." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' Novye formy vedeniya biznesa v XXI veke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=': Dobraya kniga, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' 296 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} +page_content=' KyPHAA HOOPMALOHHOE 6ECTB0' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf'} diff --git a/ENFQT4oBgHgl3EQfQTaG/content/tmp_files/load_file.txt b/ENFQT4oBgHgl3EQfQTaG/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..457ec6d2d5642732b0451aaf6c46d3140e12040d --- /dev/null +++ b/ENFQT4oBgHgl3EQfQTaG/content/tmp_files/load_file.txt @@ -0,0 +1,520 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf,len=519 +page_content='Pressure Reconstruction from the Measured Pressure Gradient Using Gaussian Process Regression Zejian You∗ and Qi Wang† San Diego State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' San Diego,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' California,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' 92182,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' USA Xiaofeng Liu‡ San Diego State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' San Diego,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' California,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' 92182,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' USA Many numerical algorithms have been established to reconstruct pressure fields from mea- sured kinematic data with noise by Particle Image Velocimetry (PIV),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' such as the Pressure Pois- son solver and the Omni-Directional Integration (ODI) method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' This study adopts Gaussian Process Regression (GPR), a probabilistic framework with an intrinsic de-noising mechanism to tackle drawbacks of traditional Pressure Poisson solver and compares the performance with ODI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' To evaluate the accuracy of the algorithm, GPR and ODI are tested in detail in a canon- ical setup of forced homogeneous isotropic turbulence from the Johns Hopkins Turbulence Database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' According to the result, GPR has the same level of accuracy as ODI with optimized hyper-parameters for the isotropic turbulence flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' However, GPR has the tendency to flatten impulsive signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' Therefore, without further modifications, it is not suitable to detect flow structures with impulsive true signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' The error propagation of the proposed framework is also analyzed and discussed in both physical and spectral spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='Nomenclature ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='𝑝(𝒙) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='pressure field ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='�𝑝(𝒙) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='true pressure field ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='𝒙 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='spatial coordinates ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='𝒖 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='velocity field ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='𝜌 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='density ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='𝑡 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='time ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='𝜇 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='dynamic viscosity ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='𝑘 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='realization number ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='¯𝑝(𝒙) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='mean of the pressure field ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='GP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='Gaussian Process ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='N ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='Gaussian distribution ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='C ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='covariance matrix of the Gaussian process ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='𝑙 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='correlation length in the radial basis function kernel ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='𝑙𝑝 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='correlation length scale from the true pressure field ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='𝜎(𝒙) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='standard deviation of Gaussian Process at a spatial location 𝒙 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='𝜎𝑝 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='standard deviation of the pressure field ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='𝜎𝜖 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='standard deviation of assumed noise level in pressure gradient ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='𝜎∇𝑝 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='standard deviation of embedded noise in pressure gradient ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='𝑶 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='vector formed by observations of material derivatives ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='𝑿∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' 𝑿 = vector formed by observation locations and general spatial locations Σ𝑖 𝑗 = Covariance matrices 𝑅𝜆 = Reynolds number based on Taylor micro scale 𝑝𝐺𝑃𝑅,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' 𝑝𝑂𝐷𝐼 = pressure field reconstructed by GPR or ODI ∗Doctoral student,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' Department of Aerospace Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' San Diego State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' Student Member AIAA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' †Assistant Professor, Department of Aerospace Engineering, San Diego State University, Member AIAA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' E-mail: qwang4@sdsu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='edu ‡Associate Professor, Department of Aerospace Engineering, San Diego State University, Associate Fellow AIAA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='13282v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='flu-dyn] 30 Jan 2023 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' Introduction A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' Pressure field estimation from PIV U nderstanding the pressure field is essential in various turbulence and hydrodynamic researches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' The generation of acoustic noise [1, 2] and the onset of boundary separation [3, 4], for example, are instances where an accurate estimation of the pressure is of paramount importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' However, non-intrusive measurements of the detailed instantaneous pressure field is a leading challenge in experimental studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' By applying Particle Image Velocimetry (PIV), the gradient information of pressure can be obtained from the balance of the Navier-Stokes equation, in which the material acceleration is the dominant term, while the viscous term is negligible at regions away from the wall for high Reynolds number flow [5], ∇𝑝 = −𝜌 𝐷𝒖 𝐷𝑡 + 𝜇∇2𝒖 ≈ −𝜌 𝐷𝒖 𝐷𝑡 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' (1) Numerical tools have been established to further reconstruct the instantaneous pressure field based on the measured pressure gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' While the traditional approach solves the pressure-Poisson equation, which often utilizes an inaccurate boundary condition, the state-of-art tool, named Omni-directional Integration (ODI), integrates the pressure gradients in a collection of directions and averages the resulting field [5–8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' This has led to a robust de-noising framework that greatly mitigates the effect of measurement error [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' A recent example shows that the time-averaged pressure reconstructed from Reynolds Averaged Navier-Stokes (RANS) equation based on stereo PIV measurements even reaches spatial resolution beyond that of PIV [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' The goal of the current study is to extend the idea of denoising and establish a probabilistic framework using Gaussian Process Regression to reconstruct pressure from its gradient information with uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' Gaussian process regression Gaussian process regression has been used to solve regression problem when the underlying function is unknown and hard to evaluate analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' For example, apply Gaussian Process upper confidence bound sampling (GP-UCB) to provide movie recommendation[11];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' reconstruct missing temporal and spatial sensor data of a dynamic nonlinear response [12, 13];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' analyze motion trajectory of moving targets from sparse observations [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' This non-parametric Bayesian approach has been proven to be very powerful in exploration and exploitation scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' Furthermore, GPR can capture a wide variety of relations between inputs and outputs by the kernel-encoded prior assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' In this study, we reconstruct the pressure from pressure gradient by deliberately encoding the prior assumption with different kernel functions while previous works mainly focus on obtaining function from observations of the function itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' The rest of the paper is structured as follows: in §II we introduce the mathematical formulation for Gaussian Process Regression when applied to pressure field reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' In §III we explain how we choose the optimal hyper-parameters for GPR and how we evaluate the performance of GPR and ODI through the forced homogeneous isotropic turbulence database from Johns Hopkins Turbulence Database (JHTDB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' In §IV we show some comparison results and analyses of reconstructed pressure field by GPR and ODI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' In §V, we conclude current results and propose some prospective future works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' Mathematical formulation In the current study, we propose a new approach, adopting the idea of Gaussian Process Regression (GPR) [see 11, for a brief tutorial] This probability framework takes into account measurement noise and could help to perform field inversion from gradient information and mitigate the effect of measurement noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' In Gaussian process regression, we assume the observation of pressure gradient at any spatial location 𝒙 can be expressed by the true pressure gradient with an additional noise: ∇𝑝(𝒙) = ∇�𝑝(𝒙) + 𝜖 (2) where the noise term 𝜖 follows normal distribution 𝜖 ∼ N (0, 𝜎2 𝜖 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' (3) Furthermore, GPR regards the pressure field, 𝑝(𝒙) as a Gaussian process in infinite dimensional space, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' 𝑝(𝒙) ∼ GP( ¯𝑝(𝒙), C(𝒙, 𝒙′)), (4) 2 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' 1 (a): An example of one-dimensional GPR when observations of the values of a smooth function are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' Red dashed lines mark the location of observations while the black dashed lines are samples drawn from the posterior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' Two red solid lines mark the region within one standard deviation in the posterior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' (b): When observations of function derivatives are available, a similar approach can also be adopted to infer the values of the function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' in which ¯𝑝(𝒙) represents the mean, or the expected value of the prior distribution for the pressure, ¯𝑝(𝒙) = E[𝑝(𝒙)], and 𝒙, 𝒙′ represent two different spatial locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' The kernel of the Gaussian process, C(𝒙, 𝒙′) represents the covariance of the uncertain pressure fields at two different spatial locations, 𝑝(𝒙) and 𝑝(𝒙′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' Or equivalently, C(𝒙, 𝒙′) = E [(𝑝(𝒙) − ¯𝑝(𝒙))(𝑝(𝒙′) − ¯𝑝(𝒙′))].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' For a continuous and stationary random process, radial basis function kernel is popularly introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' The Gaussian kernel, for example, reads C(𝒙, 𝒙′) = 𝜎(𝒙)𝜎(𝒙′) exp(−1 2 ||𝒙 − 𝒙′||2 𝑙2 ), (5) in which 𝜎(𝒙) is the standard deviation and 𝑙 is the correlation length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' The standard deviation 𝜎(𝒙) will be updated during the inference and can naturally lead to uncertainty quantification (UQ) of the reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' The correlation length 𝑙 is a hyper-parameter representing the smoothness of the pressure field, which can be tuned to achieve the best performance for different scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' The corresponding correlation function of the pressure field for the kernel function C(𝒙, 𝒙′) is defined as K(𝒙, 𝒙′) = C(𝒙, 𝒙′) 𝜎(𝒙)𝜎(𝒙′) (6) where 𝜎𝑝 is the root mean square of entire pressure field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' For radial basis function kernel, the correlation function K(𝒙, 𝒙′) is only function of distance 𝑟 between two point, K(𝒙, 𝒙′) = K(𝑟) (7) in which 𝑟 = ||𝒙 −𝒙′||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' In order to better comprehend the physical meaning of pressure field, we introduce the correlation length scale 𝑙𝑝, which could be obtained by fitting the correlation function K(𝑟) of true pressure field into the correlation function of Gaussian kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' While most applications of GPR deal with observing the values of the function directly [13], as shown in Figure 1(𝑎), the formulation of GPR is general and can be applied to observations of the gradient information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' An example of such reconstruction for a one-dimensional case is shown in Figure 1(𝑏), where the dashed lines marks the observation location for the gradient of the function 𝑝(𝑥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' Given 𝑿∗ a vector collection of 𝑛 spatial locations 𝒙𝒏 where data are observed, the samples of the pressure gradient observations would follow a multivariate Gaussian distribution, 𝑶 = ∇𝑝 (𝑿∗) ∼ N � ∇ ¯𝑝 �� 𝑿∗, ∇𝒙∇𝒙′C (𝑿∗, 𝑿∗) + 𝜎2 𝜖 I∗ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' (8) Here 𝜎𝜖 is the assumed noise level of synthetic noise introduced as a mimic of experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' I∗ is the identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' Moreover, once the observations are drawn from the above distribution, the observations of pressure gradient at 𝑿∗ and unknown values of pressure field at 𝑿, a vector collection of spatial locations where reconstruction are 3 α) 2 1 1 p I 1 1 1 1 2 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='4 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='8 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='8 1 1Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' 2 (a): True pressure field from an isotropic turbulence DNS database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' (b)-(c): Pressure gradient obtained from the DNS pressure field by central finite difference method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' (d)-(e): Sample realization of 1000 error embedded pressure gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' conducted, would follow the joint Gaussian distribution, ������� 𝑶 𝑝(𝑿) ������� ∼ N ������ � ��������� ∇𝑝 (𝑿∗) ¯𝑝(𝑿) ��������� , ���������� ∇𝒙∇𝒙′C (𝑿∗, 𝑿∗) + 𝜎2 𝜖 I∗ �������������������������������������������������������� Σ11 , ∇𝒙′C (𝑿, 𝑿∗) �������������������������� Σ12 ∇𝒙C (𝑿∗, 𝑿) ������������������������ Σ21 , C (𝑿, 𝑿) �������������� Σ22 ���������� ������ � (9) The last piece of this joint Gaussian distribution is the reference pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' Since the value of reference pressure does not influence the dynamic structure of reconstructed pressure fields, we include one additional observation of pressure 𝑝(𝒙0) = 0 at location 𝒙0 in the formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' Once noisy measurements for the pressure gradient become available, the pressure field can be recovered using Bayes’ theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' And the posterior conditional distribution is given by, 𝑝 (𝑿) ∼ N � ¯𝑝 − Σ21Σ−1 11 � 𝑶 − ∇ ¯𝑝 �� 𝑿∗ � , Σ22 − Σ21Σ−1 11Σ12 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' (10) The updated mean of the posterior distribution is regarded as the reconstructed pressure field from GPR, 𝑝𝐺𝑃𝑅(𝑿) = ¯𝑝 − Σ21Σ−1 11 � 𝑶 − ∇ ¯𝑝 �� 𝑿∗ � (11) Notice that the matrix inversion in the above expression requires O(𝑁3) operations, with 𝑁 being the number of observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' The inversion would therefore be computationally intractable for large 𝑁.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' Nevertheless, for large-scale computations, we could transform the matrix inversion into an iterative algorithm using the Conjugate Gradient method [15], which is part of future efforts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' Moreover, the covariance matrix of the posterior probability distribution can be computed from, 𝐶𝐺𝑃𝑅(𝑿, 𝑿) = Σ22 − Σ21Σ−1 11Σ12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' (12) Square roots of the diagonal elements of the above matrix are the standard deviation 𝜎𝑝(𝑿), representing the uncertainty of the reconstructed pressure fields at different spatial locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' Problem Setup A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' Data acquisition As the first step, we test our algorithm in a homogeneous isotropic turbulence flow field with Reynolds number around 𝑅𝜆 ∼ 433 based on Taylor microscale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' Instead of using data from experiments, we extract data in the Johns Hopkins 4 a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='8 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='2 30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='75Turbulent database [16–18] from a direct numerical simulation (DNS) as a surrogate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' The simulated data provides access to the fully resolved velocity and pressure fields, enabling us to test our algorithm thoroughly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' Although the database is three-dimensional, the algorithm of both ODI and GPR is able to reconstruct pressure on a two-dimensional plane given observations of the in-plane components of the gradient vectors, as shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' The observations are the projection of material derivatives onto the 𝑥 − 𝑦 plane at certain observation locations extracted from the database, at four times the DNS grid spacing in 𝑥 and 𝑦 directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' A total number of 150 × 150 observations of pressure gradient are obtained on a two-dimensional plane in the turbulent field on the 𝑥 − 𝑦 plane with 𝐿𝑥 = 𝐿𝑦 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' In order to compare the performance of GPR and ODI, we adopted 1000 realizations of error-embedded pressure gradient generated by Liu and Moreto [9] and reconstruct the pressure field by ODI as well as GPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' The error- embedded pressure gradient is generated by adding random noise of uniform distribution with the magnitude as 40% of (|∇𝑝|DNS)max, the maximum magnitude of the true pressure gradient, at each point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' The true pressure field, true pressure gradient as well as one sample realization of error embedded pressure gradient are shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' Evaluation of pressure reconstruction To quantify the accuracy of pressure reconstruction when subject to noise in the measurements, as often observed in PIV, we evaluate the cumulative error over 1000 realizations of noisy measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' First, we calculate the error of the reconstructed pressure field by GPR and ODI by subtracting the true pressure field �𝑝(𝒙) at each point, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=', 𝜖𝑖 𝑗 = 𝑝𝑖 𝑗 − �𝑝𝑖 𝑗, (13) where 𝑖, 𝑗 refer to Cartesian indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' Then, the standard deviation of error 𝜖std is defined as 𝜖std = � � � � 1 𝑁𝑥 × 𝑁𝑦 − 1 𝑁𝑦 ∑︁ 𝑗=1 𝑁𝑥 ∑︁ 𝑖=1 (𝜖𝑖 𝑗 − 𝜖𝑖 𝑗)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' (14) Finally, we calculate the averaged standard deviation over 1000 realizations, the cumulative error 𝜀std, as 𝜀std = 1 𝑘 𝑘 ∑︁ 𝑛=1 � 𝜖std 𝑝std � 𝑛 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' (15) C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' Hyper-parameter Optimization The performance of both ODI and GPR methods depends on setting up proper parameters, or hyper-parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' For Omni-directional Integration, we adopt the state-of-art formulation with Rotating Parallel Rays to create different integration paths [see 7, for details].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' The parallel rays are rotated with an increment of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='2 degrees, which creates a total number of 1800 different orientations of parallel integration paths over the entire domain of calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' The optimal separation between adjacent parallel rays at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='4 times of the grid size as suggested by [9] is adopted in the ODI calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' For GPR, the prior distribution is set with mean ¯𝑝(𝒙) = 0 and variance 𝜎(𝒙) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' However, correlation length 𝑙 and assumed noise level 𝜎𝜖 still need to be optimized, which is going to be elaborated in the following sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' In order to optimize the performance of GPR, we calculate the averaged error of 10 realizations with different hyper-parameters sets and find the optimal hyper-parameters with the lowest error as the correlation length 𝑙 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='0625 and the assumed noise level 𝜎𝜖 = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' The error is slightly smaller with assumed noise level 𝜎𝜖 less than 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' However, because the influence of equivalent noise level is not very significant once it is smaller than 6, we choose 𝜎𝜖 = 6 as our optimal hyper-parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' The correlation length scale 𝑙𝑝 could be obtained by fitting the Gaussian kernel into the correlation function of the true pressure field, which is equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='0574 for the isotropic turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' So the correlation length scale 𝑙𝑝 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='0574.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' Embedded noise is a uniform distribution from -12 to 12, the standard deviation of which is 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='928.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' Thus the magnitude of embedded noise is 𝜎∇𝑝 = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='918.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' We could find out that optimal hyper-parameters correlation length 𝑙 approximates to correlation length scale 𝑙𝑝 while assumed noise level 𝜎𝜖 approximates to the magnitude of embedded noise 𝜎∇𝑝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' This phenomenon proves that the kernel of GPR depends on the correlation function of the true pressure field as well as the standard deviation of noise in the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' 5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' 3 (a): Correlation function 𝐾(𝑟) of the true pressure field and the optimal Gaussian kernel function from curve-fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' (b): The cumulative error of 10 realizations with different correlation length 𝑙 and assumed noise level 𝜎𝜖 as well as the location of optimal hyper-parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' Results A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' Error Analysis in Physical Space The cumulative error of GPR and ODI on 150 by 150 grids as well as the cumulative error of ODI on 254 by 254 grids are shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' From the result, we could observe that error of GPR converges to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='153 over 1000 realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' Furthermore, GPR has a similar level of accuracy with ODI method on 150 by 150 grids domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' However, ODI on 254 by 254 grids can reach to lower cumulative error 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='148, the same as the result of previous work by Liu and Moreto [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' But GPR requires more memory to conduct the matrix inversion in the regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' Therefore, for large-scale problems, it is necessary to convert matrix inversion into an iterative algorithm, such as the conjugate gradient iteration for better efficiency, which is part of future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' To further investigate the performance of GPR and ODI, Figure 5 shows the exact pressure field �𝑝, the pressure field reconstructed by GPR for the instance of the worst case scenario among the 1000 realizations, pressure field reconstructed by ODI for the instance of the worst case scenario among the 1000 realizations, the standard deviation of reconstructed pressure field by GPR, error of GPR result as well as error of ODI result of the worst performance case among 1000 realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' In Figure 5, we could observe that the reconstructed pressure field by GPR is significantly smoother than ODI result and both reconstructed pressure fields are fairly accurate compared to the exact pressure field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' Although the reconstructed pressure field by GPR has a smaller global error, it also has a larger local error compared to the reconstructed pressure field by ODI, indicating the tendency of GPR in flattening impulsive local pressure changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' Furthermore, the reconstructed pressure field by ODI preserves more fine structures (combined with noise and high-frequency pressure signal) while GPR seems to have a stronger denoising effect according to the smooth distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' While the mean of the posterior distribution can be used to represent the reconstructed pressure field, we could also evaluate the accuracy of reconstruction based on the standard deviation of the posterior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' Notice that adding or subtracting a constant field on the pressure does not alter the agreement with the observation of the pressure gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' This property, sometimes coined as “gauge invariance", has to be eliminated in order to obtain meaningful results for uncertainty analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' Therefore, we assume that the reconstructed pressure at a given reference point, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' 𝒙0 at the center of the domain is solely due to such freedom of adding an arbitrary constant field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' The variances of the pressure field at other locations further take into account the effect of 𝜎𝑝(𝒙0), which should be subtracted to obtain a reasonable estimation of the reconstruction uncertainty without the influence of such gauge invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' For this reason, we here plot 𝜎𝑝(𝑿) − 𝜎𝑝(𝒙0) in Figure 5(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' In order to visualize how the error is propagated throughout the entire pressure field by different methods, we add a 6 a) b) ×10-1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='0 correlation function of p ★ optimal parameters 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='6 fitted correlation function 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='2 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='4 三 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='0 + p$3 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='8 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='4 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='4 100 5 × 100 1/lpFig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' 4 Cumulative error 𝜀std and standard deviation (𝜖std)𝑘 of 1000 realizations with correlation length 𝑙 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='0625 and assumed noise level 𝜎𝜖 = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' The bottom histogram shows the standard deviation of error (𝜖std)𝑘 in the 𝑘-th realization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' The blue bar represents the standard deviation of error by GPR and red line represents the standard deviation of error by ODI in 150 × 150 grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' Three converged lines on the top show the cumulative error of GPR and ODI in 150 × 150 and 254 × 254 grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' unit impulse in 𝜕𝑝 𝜕𝑥 at the middle of the computational domain and use GPR and ODI to reconstruct the pressure field, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' Differences between the reconstructed pressure field with and without the unit impulse are then visualized in Figure 6 to quantify the domain of influence for such point-wise perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' Such approaches have been adopted in previous research to study the domain of influence or the domain of dependence to fully understand the forward and backward propagation of perturbations in a dynamical system or data assimilation algorithm [19, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' The results of this impulse response have profound implications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' First of all, the ODI method indicates a clear singularity point at the location of perturbation, manifested by very large positive and negative values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' The implication here is that although ODI averages the error across the whole computational domain, the reconstructed pressure still relies heavily on the local pressure gradient information near the point of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' In other words, most of the local error remains local, and correspondingly, the error diffusion is relatively not strong in comparison with that of GPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' On the other hand, the GPR method exhibits nearly zero influence at the point of perturbation and has a larger influence slightly farther away.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' This difference could explain the stronger de-noising effect in GPR than in ODI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' Error Analysis in Wave Number Space The different behavior of GPR and ODI in Figure 5 leads to a comparison of power spectrum density of the reconstructed pressure field by GPR and ODI, as shown in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' From Figure 7(a), we could see that both methods perform well on low wave number space: the power spectrum density of GPR and ODI coincides with the power spectrum density of true pressure field perfectly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' However, in high wave number space, the power spectrum density of different pressure field seem to diverge: GPR has a lower power spectrum density while ODI has a larger power spectrum density compared to the power spectrum density of true pressure field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' This phenomenon validates previous observations in the pressure field shown in Figure 5, as well as the impulse response in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' GPR has a stronger denoising effect but also smooths out some pressure information in high wave number space while ODI preserves more fine structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' Some of them are dynamic behavior of the pressure field, others are noise in the observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' Furthermore, Figure 7(b) clearly indicates that if the pressure gradient information is accurate, ODI can faithfully replicate the dynamic behavior of the pressure field over the entire spectral range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' However, in contrast, because 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='20 GPR (150×150) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='45 ODI (254×254) ODI (150×150) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='16 Estd p4s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='10 0 200 400 600 800 1000 Realization kFig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' 5 (a): True pressure field from isotropic turbulence DNS database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' (b): Instant of realization of recon- structed pressure field by GPR with the largest standard deviation of error 𝜖std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' (c): Instant of realization of reconstructed pressure field by ODI with the largest standard deviation of error 𝜖std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' (d): Standard deviation of reconstructed pressure field by GPR, 𝜎(𝑿) subtracted by the standard deviation of reference point 𝜎(𝒙0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' (e): Error distribution of reconstructed pressure field by GPR, obtained by subtracting (a) from (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' (f): Error distribution of reconstructed pressure field by ODI, obtained by subtracting (a) from (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' 6 Error Propagation of GPR and ODI, represented by the change of reconstructed pressure field when perturbation in the form of a unit impulse of 𝜕𝑝𝜕𝑥 is added at the center of the computational domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' 8 a) b) c) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='6 Pressure 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='4- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='0 - a d) e) X10-4 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='8 4 1 2 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='75ODI GPR ×10-4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='8 2 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='6 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='4 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='2 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='0 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content='75 a aFig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' 7 (a): Power spectrum density of true pressure field as well as reconstructed pressure field by GPR and ODI from error embedded pressure gradients of 150 by 150 grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' (b): Power spectrum density of true pressure field as well as reconstructed pressure field by GPR and ODI from true pressure gradients of 150 by 150 grids the hyperparameters used in this GPR computation were optimized with the error-embedded data (therefore are flow dependent), GPR produces a reconstructed pressure spectrum with an overall lower fluctuation amplitude over the entire spectral domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' This indicates that the optimization of GPR needs to be adjusted according to the actual flow properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' This problem might be solved by switching a proper kernel function rather than a Gaussian kernel in this case since its denoising effect is too strong to preserve necessary information, which requires future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' Conclusion and future work We adopt the framework of Gaussian Process Regression (GPR) to the problem of determining the pressure fields from measured pressure gradients, with the potential of reconstructing pressure from sparsely measured data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' The formulation naturally avoids the burden of solving Poisson equation with inaccurate boundary conditions and takes into account the effect of measurement noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' Furthermore, this framework provides more possibilities to improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' From the comparison between the reconstructed pressure field by GPR and ODI, we are able to conclude that the reconstructed pressure field by GPR achieves accuracy comparable to the state-of-art Omni-directional integration method (ODI) and has a stronger denoising effect compared to ODI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' However, pressure reconstruction by GPR might also smooth out some pressure information in high wave number space by mistake, especially dealing with accurate pressure gradient data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' This problem might be able to be solved by switching the Gaussian kernel, which requires further investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' In future directions, the following will be studied in detail: (a) improvement of computational efficiency for large-scale problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' (b) different kernel functions for the Gaussian Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' (c) effect of the sparseness of the observation in terms of reconstruction quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' Acknowledgments The support from San Diego State University is gratefully acknowledged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' Thanks to Jose Moreto for his assistance in the usage of the parallel-ray Omni-directional integration code and for providing the error-embedded pressure gradients data in previous study by Liu and Moreto [9].' metadata={'source': 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steady scalar sources from remote measurements in turbulent flow,” Journal of Fluid Mechanics, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' 870, 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' 316–352.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} +page_content=' 10' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFQT4oBgHgl3EQfQTaG/content/2301.13282v1.pdf'} diff --git a/F9AyT4oBgHgl3EQf5Pp6/content/tmp_files/2301.00801v1.pdf.txt b/F9AyT4oBgHgl3EQf5Pp6/content/tmp_files/2301.00801v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..2a35883a129a7c56fac90370d9999cf63cc202a7 --- /dev/null +++ b/F9AyT4oBgHgl3EQf5Pp6/content/tmp_files/2301.00801v1.pdf.txt @@ -0,0 +1,2919 @@ +arXiv:2301.00801v1 [stat.ML] 2 Jan 2023 +Causal Inference (C-inf) — asymmetric scenario of +typical phase transitions +Agostino Capponi ∗ Mihailo Stojnic † +Department of Industrial Engineering and Operations Research +Columbia University, New York, NY 10027, USA +Abstract +In this paper, we revisit and further explore a mathematically rigorous connection between Causal in- +ference (C-inf) and the Low-rank recovery (LRR) established in [10]. Leveraging the Random duality +- Free probability theory (RDT-FPT) connection, we obtain the exact explicit typical C-inf asymmetric +phase transitions (PT). We uncover a doubling low-rankness phenomenon, which means that exactly two +times larger low rankness is allowed in asymmetric scenarios compared to the symmetric worst case ones con- +sidered in [10]. Consequently, the final PT mathematical expressions are as elegant as those obtained in [10], +and highlight direct relations between the targeted C-inf matrix low rankness and the time of treatment. +Our results have strong implications for applications, where C-inf matrices are not necessarily symmetric. +Index Terms: Causal inference; Random duality theory; Algorithms; Matrix completion; Spar- +sity. +1 +Introduction +Causal inference (C-inf) deals with the design of estimation strategies that allow researchers to draw +causal conclusions based on data. The overarching goal is to draw a conclusion regarding the effect of a +causal variable, which is typically referred to as the “treatment” or the “intervention” on some outcome of +interest. For example, suppose we want to estimate the causal effect of a drug on deadly cancer progression +(vs no exposure to the drug). Then we want to compare metastasis in the patient’s body one month after +the drug regime has begun versus metastasis in the absence of exposure to the drug. The main challenge for +causal inference is that we are not generally able to observe both of these states: at the point in time when +we are measuring the outcomes, each individual either has had drug exposure or has not. +The problem of estimating the counterfactual, i.e., what would have been the outcome in the absence +of a treatement, is central in many disciplines, including economics, health, and social sciences (see, e.g. +[2,11,13,14,32,33,50]), machine learning and theoretical computer science (see, e.g. [25–28]). Methodological +developments to estimate causal effects have been based on experimental or observational data. Experimental +research offers the most plausibly unbiased estimates, but experiments are frequently infeasible because they +are costly or subject to moral objections. +Observational data instead are becoming increasing available +due to technological advancements in the design of sensor and hardware devices. Our focus is on causal +inference in observational studies, and specifically on the design of efficient algorithmic techniques to estimate +counterfactuals. +The C-inf approaches can be broadly classified into three categories: 1) the unconfoundedness (see, +e.g. [14, 32]); 2) the synthetic control (see, e.g. [1, 2, 11]); and 3) the matrix completion (see, e.g. [3, 4, 15]. +Matrix completion methods build upon the foundation works of [7,9,29]). Perhaps unexpectedly, all three +methods heavily rely on mathematical, statistical, and ultimately algorithmic concepts with very deep roots +∗e-mail: ac3827@columbia.edu +†e-mail: flatoyer@gmail.com +1 + +in information theory. Our work is positioned within the third line of work that mathematically resembles +the matrix completion (MC) problem. +Along the same lines, our work extends significantly the analysis developed in the companion paper [10]. +Therein, we obtained the exact explicit typical worst case C-inf phase transitions (PT), and further showed +that these phase transitions are achievable by the symmetric targeted C-inf matrices. In the present paper, +we consider a generic asymmetric context, to deal with the situation that C-inf matrices are not necessarily +always symmetric in real applications. +This allows us improving upon the results from [10] in certain +scenarios. We build further upon the RDT-FPT synergistic mechanisms considered in [10], and precisely +characterize the corresponding asymmetric PTs. We also uncover a doubling low-rankness phenomenon, +which means that exactly two times larger low rankness is allowed in asymmetric scenarios compared to the +symmetric worst case ones of [10]. +2 +Causal inference mathematical setup +In this section, we revisit the explicit causal inference (C-inf) ↔ matrix completion (MC) connection, +established in [10]. Therein, we have discussed the connection between low rank recovery (LRR), matrix +completion (MC), and the causal inference (C-inf). Mathematically speaking, one has that the MC is a +special case of the LRR and the C-inf is a special case of the MC itself. Consequently, the mathematical +models that describe the LRR problems can be used to describe the MC and ultimately the C-inf ones as +well. Below we present the C-inf mathematical setup developed through such a connection in [10]. +We start with a low rank matrix Xsol ∈ Rn×n with the singular value decomposition (SVD) +X = UΣV T , +(1) +where +σ(X) ≜ diag(Σ) +and +U T U = In×n +and +V T V = In×n, +(2) +with In×n being the n × n identity matrix and diag(·) being the operator that creates a column vector of +the diagonal elements of its matrix argument. We then define ℓ∗ +p(X) to be the so-called ℓp (quasi) norm of +σ(X) (the vector of the singular values of X), i.e. +ℓ∗ +p(X) ≜ ℓp(σ(X)), p ∈ R+. +(3) +The following limiting ℓp(·) connections are important as well +ℓ∗ +0(Xsol) ≜ ℓ0(σ(Xsol)) = ∥σ(Xsol)∥0 = lim +p−→0 ∥σ(Xsol)∥p = lim +p−→0 ℓp(σ(Xsol)) = lim +p−→0 ℓ∗ +p(Xsol). +(4) +Moreover, we also define the so-called block masking matrix M as (see Figure 1 as well) +M matrix in block causal inference (C-inf): +M ≜ M (l) ≜ 1n×11T +n×1 − I(l)(I(l))T 1n×11T +n×1I(l)(I(l))T +and +I(l) ≜ +� 0l×(n−l) +I(n−l)×(n−l) +� +. +(5) +One then has the following two optimization problems that are at the heart of the C-inf ↔ MC connection +ℓ∗ +0-minimization (C-inf – MMT) +min +X +ℓ∗ +0(X) +subject to +Y = M ◦ X. +(6) +ℓ∗ +1-minimization (C-inf – MMT) +min +X +ℓ∗ +1(X) +subject to +Y = M ◦ X, +(7) +2 + +M = +1 +Matrix M – block causal inference (C-inf) +1 +0 +1 +0 and 1 grouped in blocks +l × l block of all 1s +l × (n − l) block of all 1s +(n − l) × (n − l) block of all 0s +(n − l) × l block of all 1s +Figure 1: Matrix M ≜ M (l) – block causal inference (C-inf) setup +where ◦ stands for the component-wise multiplication. Namely, keeping in mind that ℓ∗ +0(X) effectively counts +the number of the nonzero singular values of X, the optimization problem in (6) is exactly the recovery of +the C-inf targeted low rank matrix X from the linear observations Y obtained through a masking via M. +Moreover, the problem in (6) (with a generic M) is a standard matrix completion setup which on the other +hand is a special case of the LRR problems (expressed in the “masking matrix terminology” (MMT)). On +the other hand, the optimization problem in (7) is the tightest convex relaxation heuristic typically utilized +in the matrix completion literature for solving NP-hard problem (6). For more on the origin of these two +problems and their connection within the LRR and MC context we refer to the introductory LRR/MC +papers [7,30,36]. More on their importance and different related algorithmic considerations can be found in +many papers that followed (see, e.g. [8,16–22,31]). +Here though, we would particularly like to point out reference [4] where the very same C-inf context +was considered and the very same C-inf ↔ MC connection recognized. Considerations from [4] are in +fact especially convenient to properly understand in what C-inf contexts the block structure of the matrix +M might appear. To see that one can connect it to the so-called counterfactuals and the units/treatments +terminology employed in [4]. +First we note that M can be alternatively defined as +Mi,j = +� +1, +(i, j)-th element of Xsol is observed +0, +otherwise. +(8) +It is then rather clear that ones in M allow reading out the corresponding elements of Xsol while zeros block +(mask) them. Then the context of [4] is roughly as follows. One first assumes that the matrix X contains +observations about a certain set of, say, n units (e.g. individuals, subpopulations, and geographic regions) +over a period of say, n, time instances. After that the rows of X are allocated to the units and the columns +to the time instances and one would like to estimate the effects that a certain treatment may have on the +treated units. A subset of the units (say those that correspond to the rows i > l) is then at time l exposed +to an irreversible treatment. Examples of treatments include health therapies, socio-economic policies, and +taxes. To ensure an appropriate assessment of the resulting treatment effects, in addition to having the +values of X after the treatment, one would need to have the access to the so-called counterfactuals – the +values of the treated units – had the treatment not been applied. Relating back to the matrix completion +terminology, one would basically need to estimate (a presumably low rank) X while not having access to its +3 + +portion covered by the block-mask M = M (l). In other words, one would need to solve (6) with M = M (l). +The above describes the C-inf via counterfactuals and the underlying role of matrix M. Moreover, if +one views things in the time domain, i.e. if the columns of M represent time axis, then the observations +in ceratin rows will not be available after a fixed point in time. In the block scenario this point is fixed +across the affected rows. However, it does not necessarily need to be fixed (for more in this direction we refer +to [2] (in particular, the California tobacco example), [49] (in particular, the latent factor modeling in the +context of the simultaneous/staggered treatment adoption), and to [5, 6, 34] (in particular, the health care +applications) as excellent references for understanding the need of various C-inf scenarios). As this and [10] +are the introductory papers, where we present the overall methodology, we selected the block causal inference +scenario as probably the most representative and well-known one. In some of our companion papers we will +show how the methodology that we are introducing here can be utilized to handle other C-inf scenarios as +well. +3 +ℓ∗ +0 − ℓ∗ +1 equivalence +As mentioned earlier, solving the generic LRR (and consequently the C-inf as its a special case) might be +difficult due to a highly non-convex objective function in (6). Various heuristics can be employed depend- +ing on the practical scenarios that one can face. In the mathematically most challenging so-called linear +regime, the above mentioned ℓ∗ +1-minimization relaxation heuristic (often called nuclear norm minimization) +is typically viewed as the best known provably polynomial one. We adopt the same view in what follows +and take it as a current benchmark for the algorithmic handling of the C-inf. As mentioned above, a rather +remarkable feature of this heuristic is that sometimes it can actually solve the underlying problems exactly. +When that happens we say that the following ℓ∗ +0 − ℓ∗ +1-equivalence phenomenon occurs. +ℓ∗ +0 − ℓ∗ +1-equivalence (C-inf): ℓ∗ +0 ⇐⇒ ℓ∗ +1 +Let Xsol be the solution of (6) and let ˆX be a solution of (7) and set +RMSE ≜ ∥vec( ˆX) − vec(Xsol)∥2. +If and only if ( ˆX = Xsol and RMSE = 0) +then +(ℓ∗ +0 − minimization ⇐⇒ ℓ∗ +1 − minimization). +(9) +The above basically means that when the ℓ∗ +0 − ℓ∗ +1-equivalence happens the optimization problems in (6) +and (7) are equivalent and as such replaceable by each other. We denote such a phenomenon as ℓ∗ +0 ⇐⇒ ℓ∗ +1. +That would, of course, be an ideal scenario where it would be basically possible to replace the non-convex +optimization problem with the convex one without losing anything in terms of the accuracy of the obtained +solutions. Since the mere existence of such a phenomenon is rather remarkable we will in this paper be +interested in uncovering the underlying intricacies that enable for it ro happen. Moreover, as it will turn +out that its occurrence is not an anomaly but rather a consequence of a generic property, we will then +raise the bar accordingly and attempt to provide not only the proof of its existence but also its a complete +analytical characterization. This will include a full characterization as to how often and in what scenarios it +might happen. To do so we will combine the Random Duality Theory (RDT) tools from [37–44] and several +advanced sophisticated probabilistic concepts that we will introduce along the way in the sections that follow +below. +We start with some algebraic ℓ∗ +0 − ℓ∗ +1-equivalence preliminaries which are borrowed from the RDT. The +first one is a generic LRR ℓ∗ +0−ℓ∗ +1-equivalence result (the result is basically an adaptation of the corresponding +CS equivalence condition from [39–41] (similar adaptation can also be found in [24])). +Theorem 1. ( [10] ℓ∗ +0 − ℓ∗ +1-equivalence condition (LRR) – general X) Consider a ¯U ∈ Rn×k such that +¯U T ¯U = Ik×k and a ¯V ∈ Rn×k such that ¯V T ¯V = Ik×k and a rank− k matrix Xsol = X ∈ Rn×n with all of its +columns belonging to the span of ¯U and all of its rows belonging to the span of ¯V T . Also, let the orthogonal +spans ¯U ⊥ ∈ Rn×(n−k) and ¯V ⊥ ∈ Rn×(n−k) be such that U ≜ +� ¯U +¯U ⊥� +and V ≜ +� ¯V +¯V ⊥� +and +U T U ≜ +� ¯U +¯U ⊥�T � ¯U +¯U ⊥� += In×n +and +V T V ≜ +� ¯V +¯V ⊥�T � ¯V +¯V ⊥� += In×n. +(10) +4 + +For a given matrix A ∈ Rm×n2 (m ≤ n2) assume that y = Avec(X) = Avec(Xsol) ∈ Rm and let ˆX be the +solution of (7). If +(∀W ∈ Rn×n|Avec(W) = 0m×1, W ̸= 0n×n) +− tr ( ¯U T W ¯V ) < ℓ∗ +1(( ¯U ⊥)T W ¯V ⊥), +(11) +then +ℓ∗ +0 ⇐⇒ ℓ∗ +1 +and +RMSE = ∥vec( ˆX) − vec(Xsol)∥2 = 0, +(12) +and the solutions of (6) and (7) coincide. Moreover, if +(∃W ∈ Rn×n|Avec(W) = 0m×1, W ̸= 0n×n) +− tr ( ¯U T W ¯V ) ≥ ℓ∗ +1(( ¯U ⊥)T W ¯V ⊥), +(13) +then there is an X from the above set of matrices with columns belonging to the span of ¯U and rows belonging +to the span of ¯V such that the solutions of (6) and (7) are different. +Proof. The proof is a trivial adaptation of the proof for symmetric matrices given in Appendix A. +Continuing further in the spirit of the RDT the following corollary is a matrix completion specific variant +of the above theorem. +Corollary 1. ( [10] ℓ∗ +0 − ℓ∗ +1-equivalence condition via masking matrix (MC/C-inf) – general X) +Assume the setup of Theorem 1 with Xsol being the unique solution of (6). Let the masking matrix M ∈ Rn×n +have m ones and (n2−m) zeros and let A be generated via M, i.e. let A be the matrix obtained after removing +all the zero rows from diag−1(vec(M))In2×n2. If and only if +min +W,W T W=1,M◦W=0n×n +tr ( ¯U T W ¯V ) + ℓ∗ +1(( ¯U ⊥)T W ¯V ⊥) ≥ 0, +(14) +then +ℓ∗ +0 ⇐⇒ ℓ∗ +1 +and +RMSE = ∥vec( ˆX) − vec(Xsol)∥2 = 0, +(15) +and the solutions of (6) and (7) coincide. +Finally, the following spectral oriented corollary was proven in [10] as well. +Corollary 2. ( [10] ℓ∗ +0−ℓ∗ +1-equivalence condition via mask-modified bases spectra (C-inf) – general +X) Assume the setup of Theorem 1 with k ≤ l. Let M ≜ M (l) ∈ Rn×n and I(l) ∈ Rn×(n−l) be as defined in +(5). Set +ΛV +≜ +((I(l))T ¯V ⊥)−1(I(l))T ¯V +ΛU +≜ +((I(l))T ¯U ⊥)−1(I(l))T ¯U +Q += +� +(I(l))T ¯V ⊥( ¯V ⊥)T I(l)�−1 +− I +Q⊥ +1 += +� +(I(l))T ¯U ⊥( ¯U ⊥)T I(l)�−1 +− I. +(16) +C-inf perfectly succeeds: ℓ∗ +0 ⇐⇒ ℓ∗ +1 +and +RMSE = ∥vec( ˆX) − vec(Xsol)∥2 = 0 +If and only if +λmax(ΛT +V ΛV ΛT +UΛU) ≤ 1. +(17) +Moreover, if +λmax (Q) λmax +� +Q⊥ +1 +� +≤ 1, +(18) +then again ℓ∗ +0 ⇐⇒ ℓ∗ +1 and RMSE = ∥vec( ˆX − vec(Xsol)∥2 = 0 and the C-inf perfectly succeeds as well. +5 + +Since we will be working in the mathematically most challenging large n linear regime, we find it useful +to introduce the following large dimensional scalings +β ≜ lim +n→∞ +k +n +and +η ≜ lim +n→∞ +l +n +and +α ≜ lim +n→∞ +m +n2 = lim +n→∞ +n2 − (n − l)2 +n2 += 1 − (1 − η)2. +(19) +The key highlight result of [10] is the following theorem obtained through an analysis that relied on the +above corollary and a combination of the Random duality theory (RDT) and Free probability theory (FPT). +It basically establishes the worst case phase-transition (PT) that ℓ∗ +1, tightest convex relaxation heuristic, +exhibits when used for solving C-inf in a typical statistical scenario. +Theorem 2. (ℓ∗ +1 – phase transition – C-inf (typical worst case)) Consider a rank-k matrix Xsol = +X ∈ Rn×n with the Haar distributed ( not necessarily independent) bases of its orthogonal row and column +spans ¯U ⊥ ∈ Rn×(n−k) and ¯V ⊥ ∈ Rn×(n−k) (XT +sol ¯U ⊥ = Xsol ¯V ⊥ = 0n×(n−k)). Let M ≜ M (l) ∈ Rn×n be as +defined in (5). Assume a large n linear regime with β ≜ limn→∞ k +n and η ≜ limn→∞ l +n and let βwc and η +satisfy the following +C-inf ℓ∗ +1 worst case phase transition (PT) characterization +ξ(wc) +η +(β) ≜ β − 1 +2 + +� +η − η2 = 0. +(20) +If and only if β ≤ βwc +lim +n→∞ P(ℓ∗ +0 ⇐⇒ ℓ∗ +1) = +lim +n→∞ P(RMSE = 0) = 1, +(21) +and the solutions of (6) and (7) coincide with overwhelming probability. +The results obtained based on the above theorem are shown in Figure 2, where one can clearly see that +the phase transition curve splits the entire (β, η) region into two subregions: 1) the first one (below (or to +the right of) the curve) where the ℓ∗ +0 − ℓ∗ +1-equivalence phenomenon occurs; and 2) the second one (above (or +to the left of) the curve) where the ℓ∗ +0 − ℓ∗ +1-equivalence is lacking. This means that one can recover Xsol +masked by M as in (6) via the ℓ∗ +1 heuristic from (7) with the residual mean square error (RMSE) equal to +zero. In other words, for the system parameters (β, η) that belong to the subregion below the curve one has +a perfect recovery with Xsol and ˆX (the respective solutions of (6) and (7)) being equal to each other and +consequently with RMSE = ∥vec( ˆX) − vec(Xsol)∥2 = 0. On the other hand, in the subregion above the +curve, the ℓ∗ +1 heuristic fails and one can even find an Xsol for which RMSE → ∞. +4 +Analysis of the ℓ∗ +0 −ℓ∗ +1-equivalence – typical asymmetric scenario +In this section we consider when the conditions given in Corollary 2 are met. As in [10], we will be working +in a “typical” statistical context. On the other hand, differently from [10], instead of focusing on the worst +case (symmetric) scenario we here consider a typical asymmetric scenario setup. Practically this means +two things: 1) as in [10], both ¯V and ¯U will be assumed as Haar distributed; and 2) differently from Theorem +2 and [10], ¯V and ¯U will now be assumed as independent of each other. In a way one can view the worst case +scenario from [10] as an extreme where ¯V and ¯U are “not independent at all” (or, in other words, equal to +each other). Along similar lines, one can then view the scenario that we will consider here as another extreme +where ¯V and ¯U are “not dependent at all” (or, in other words, completely independent). In situations where +no particular structure of a low rank nonsymmetric Xsol is favored over any other this one would naturally be +a most reasonable choice. In other words, it is not only an extreme case, but actually the one that typically +might most faithfully describe the performance of the underlying C-inf heuristics. +4.1 +Free probability theory (FPT) – preliminaries +Below we provide a short preview of the most basic FPT concepts needed for our analysis (we refer to our +companion paper [10] for a more detail treatment). As is by now well known, the work od Dan Voiculescu +6 + +η +0.5 +0.55 +0.6 +0.65 +0.7 +0.75 +0.8 +0.85 +0.9 +0.95 +1 +β +0 +0.05 +0.1 +0.15 +0.2 +0.25 +0.3 +0.35 +0.4 +0.45 +0.5 +(β, η) region of success/failure — C-inf ℓ∗1 PT +RMSE −→ ∞, ℓ∗1 +fails +RMSE = 0, ℓ∗1 succeeds +ℓ∗1’s PT: ξ(wc) +η +(β) = β − 1 +2 + �η − η2 = 0 +Figure 2: Causal inference (C-inf) – typical worst case ℓ∗ +1 phase transition +on group theories (see, e.g. [46–48]) established the foundations of the FPT. As the practical importance +of FPT became immediately evident a substantial interest for further studying was generated and, in the +years that followed, quite a few nice results appeared that made the whole theory more approachable and +ultimately presentable in an easily understandable way. Along the same lines, we follow into the footsteps +of [10], leave all the abstractions out and focus on the FPT’s key practically applicable components (for +further details see also, e.g. [12,23,35,45–48]). +4.1.1 +Basics of FPT – random matrix variables +We assume large n linear regime and consider two symmetric matrices A = AT ∈ Rn×n and B = BT ∈ Rn×n +with Haar distributed eigenspaces. We also assume that their individual respective spectral laws are fA(·) +and fB(·). Three different transforms of these spectral densities will be needed. We start with the first one, +the so-called Stieltjes (or G) transform +G(z) +≜ +� +If +f(x) +z − xdx, +z ∈ C \ If, +(22) +where If is the domain of f(·). The following inverse relation is also well known +f(x) = lim +ǫ→0+ +G(x − iǫ) − G(x + iǫ) +2iπ +or +f(x) = − lim +ǫ→0+ +imag(G(x + iǫ)) +π +. +(23) +For the above to hold it makes things easier to implicitly assume that f(x) is continuous. We will, however, +utilize it even in discrete (or semi-discrete) scenarios since the obvious asymptotic translation to continuity +would make it fully rigorous. A bit later though, when we see some concrete examples where things of this +nature may appear, we will say a few more words and explain more thoroughly what exactly can be discrete +and how one can deal with such a discreteness. In the meantime we proceed with general principles not +necessarily worrying about all the underlying technicalities that may appear in scenarios deviating from the +typically seen ones and potentially requiring additional separate addressing. To that end we continue by +considering the R(·)- and S(·)-transforms that satisfy the following +R(G(z)) + +1 +G(z) = z, +(24) +7 + +and +S(z) = +1 +R(zS(z)) +and +R(z) = +1 +S(zR(z)). +(25) +Let fA(·) and fB(·) be the spectral distributions of A and B and let RA(z)/SA(z) and RB(z)/SB(z) be their +associated R(·)-/S(·)-transforms. One then has the following +Key Voiculescu’s FPT concepts [46, 47]: +C += +A + B +=⇒ +RC(z) += +RA(z) + RB(z) +C += +AB +=⇒ +SC(z) += +SA(z)SB(z). +(26) +Now it is relatively easy to see that (22)-(26) are sufficient to determine the spectral distribution of the sum +or the product of two independent matrices with given spectral densities and the Haar distributed bases of +eigenspaces. The above is of course a generic principle. It can be applied pretty much always as long as +one has access to the statistics of the underlying matrices A and B. In the following section we will raise +the bar a bit higher and show that in the C-inf context one can use all of the above in such a manner that +eventually all the quantities of interest are explicitly determined. +4.1.2 +Spectral preliminaries +We start by recalling on Q from (16) and introducing Q1 +Q1 +≜ +ΛT +V ΛV +Q +≜ +� +(I(l))T ¯V ⊥( ¯V ⊥)T I(l)�−1 +− I +Sp(Q1) +⇐⇒\0 +Sp(Q), +(27) +where Sp(·) stands for the spectrum of the matrix argument and ⇐⇒\0 means the equivalence of the parts +of the spectra outside the zero eigenvalues. It is rather obvious that it will then be sufficient to handle the +spectrum of +D +≜ +(I(l))T ¯V ⊥( ¯V ⊥)T I(l). +(28) +Consider Haar distributed ¯U ⊥ +D ∈ Rn×(n−l) with ( ¯U ⊥ +D)T ¯U ⊥ +D = I(n−l)×(n−l) and let +UD += +� ¯UD +¯U ⊥ +D +� +with +U T +DUD = In×n. +(29) +Also, we assume that ¯U ⊥ +D (and UD) are independent of ¯V ⊥. After setting +¯D +≜ +(I(l))T U T +D ¯V ⊥( ¯V ⊥)T UDI(l), +(30) +we have that the spectra of D and ¯D are statistically identical, i.e. +Sp(D) ≜ Sp((I(l))T ¯V ⊥( ¯V ⊥)T I(l)) ⇐⇒P Sp((I(l))T U T +D ¯V ⊥( ¯V ⊥)T UDI(l)) ≜ Sp( ¯D), +(31) +where ⇐⇒P stands for the statistical/probabilistic equivalence. Two facts enable the above statistical iden- +tity: 1) the spectrum of the projector ¯V ⊥( ¯V ⊥)T does not change under pre- and post-unitary multiplications +on both sides; and 2) the Haar structure of ¯V ⊥ remains preserved. Modulo zero eigenvalues, we then further +have +Sp((I(l))T U T +D ¯V ⊥( ¯V ⊥)T UDI(l)) ⇐⇒P\0 Sp( ¯V ⊥( ¯V ⊥)T UDI(l)(I(l))T U T +D) ⇐⇒ Sp( ¯V ⊥( ¯V ⊥)T ¯U ⊥ +D( ¯U ⊥ +D)T ), (32) +where, similarly as above, ⇐⇒P\0 stands for the statistical/probabilistic equivalence in the part of the +spectrum outside the zero eignevalues (introduced due to the non-square underlying matrices). Clearly, the +8 + +key object of our interest below will be +˜D +≜ +¯V ⊥( ¯V ⊥)T ¯U ⊥ +D( ¯U ⊥ +D)T , +(33) +where both ¯V ⊥ and ¯U ⊥ +D are Haar distributed and independent of each other. After setting +V +≜ +¯V ⊥( ¯V ⊥)T +U +≜ +¯U ⊥ +D( ¯U ⊥ +D)T , +(34) +we easily have from (33) +˜D +≜ +VU. +(35) +The following lemma proven in [10] characterizes the G-transform of ˜D.. +Lemma 1. ( [10]) Let ¯V ⊥ ∈ Rn×(n−k) and ¯U ⊥ +D ∈ Rn×(n−k) be Haar distributed unitary bases of (n − k)- +dimensional subspaces of Rn. Let V and U be as in (34) and ˜D as in (35), i.e. +V +≜ +¯V ⊥( ¯V ⊥)T +U +≜ +¯U ⊥ +D( ¯U ⊥ +D)T +˜D +≜ +VU. +(36) +In the large n linear regime, with β ≜ limn→∞ k +n, the G-transform of the spectral density of ˜D, f ˜ +D(·), is +G± +˜ +D(z) = z − (β + η) ± +� +(z − (β + η))2 + 4βη(z − 1) +2(z2 − z) +. +(37) +4.2 +Asymmetric scenario – FPT analysis of the ℓ∗ +0 − ℓ∗ +1-equivalence +As in [10], we will again rely on the free probability theory. This time though things will be a bit more +complicated as we will be determining, so to say, the “joint spectrum” of λT +V λV λT +UλU. In other words, based +on Corollary 2 and (17), we have +ℓ∗ +0 − ℓ∗ +1 − −equivalence +⇐⇒ +λmax(λT +V λV λT +UλU) ≤ 1, +(38) +and consequently determining the upper edge of the “joint spectrum” of λT +V λV λT +UλU would be then sufficient +to establish ℓ∗ +0 − ℓ∗ +1-equivalence. We recall that in [10] we determined only the individual spectra λT +V λV and +λT +UλU (which in the worst case was sufficient to ultimately obtain corresponding C-inf ℓ∗ +1 PT). While the +calculations and supporting technicalities might on occasion be a bit heavy the overall methodology will be +fairly similar to what we presented in [10]. In fact, to make things easier to follow we will try to parallel the +presentation from [10] as much as possible. We start by recalling on Q1 and introducing Q⊥ +1 , and Q1 +Q1 +≜ +λT +V λV +Q⊥ +1 +≜ +λT +UλU +Q1 +≜ +Q1Q⊥ +1 . +(39) +We also recall on the definitions of Q and Q⊥ and introduce Q in the following way +Q +≜ +� +(I(l))T ¯V ⊥( ¯V ⊥)T I(l)�−1 +− I +Q⊥ +≜ +� +(I(l))T ¯U ⊥( ¯U ⊥)T I(l)�−1 +− I +Q +≜ +QQ⊥. +(40) +9 + +4.2.1 +The spectrum of Q1 ≜ λT +V λV λT +UλU – theoretical considerations +Since (Q1, Q) and (Q⊥ +1 , Q⊥) are statistically identical pairs, we will, for the time being, focus on only one of +them, say (Q1, Q). To that end, we first recall the statistical relations within the pairs +Q1 +≜ +ΛT +V ΛV +Q +≜ +� +(I(l))T ¯V ⊥( ¯V ⊥)T I(l)�−1 +− I = D−1 − I +Sp(Q1) +⇐⇒\0 +Sp(Q), +(41) +where ⇐⇒\0 stands for the spectral equivalence outside the zeros eigenvalues. We will also find it convenient +to work with the spectrum of Q. Later on we will make the necessary adjustments so that the results fully +fit the spectrum of Q1. To start things off we first note +GQ(z) = GD−1(z + 1). +(42) +To see that (42) indeed holds, we first observe that the spectral functions of Q and D−1, fQ(x) and fD−1(x), +can be connected in the following way +fQ(x) = fD−1(x + 1). +(43) +Then from (22) we have +GQ(z) = +� fQ(x) +z − x dx = +� fD−1(x + 1) +z − x +dx = +� +fD−1(x + 1) +z + 1 − (x + 1)dx = +� +fD−1(x) +z + 1 − xdx = GD−1(z + 1). +(44) +From (41) and (42) we also have +RQ(z) = RD−1(z) − 1. +(45) +Namely, (24) first gives +RQ(GQ(z)) = z − +1 +GQ(z), +(46) +and then +RQ(z) += +G−1 +Q (z) − 1 +z +⇐⇒ +z += +GQ +� +RQ(z) + 1 +z +� +RD−1(z) += +G−1 +D−1(z) − 1 +z +⇐⇒ +z += +GD−1 � +RD−1(z) + 1 +z +� +. +(47) +Combining (42) and (47) we obtain +GQ +� +RQ(z) + 1 +z +� += +GD−1 � +RD−1(z) + 1 +z +� +⇐⇒ +GQ +� +RQ(z) + 1 +z +� += +GQ +� +RD−1(z) + 1 +z − 1 +� +⇐⇒ +RQ(z) + 1 +z += +RD−1(z) + 1 +z − 1 +⇐⇒ +RQ(z) += +RD−1(z) − 1, +(48) +which is exactly (46). From (25) we further have +SQ(z) = +1 +RQ(zSQ(z)) = +1 +RD−1(zSQ(z)) − 1, +(49) +and +RD−1(zSQ(z)) = +1 +SQ(z) + 1. +(50) +10 + +Relying further on (25) we also have +RD−1(zSQ(z)) = +1 +SD−1(zSQ(z)RD−1(zSQ(z)) = +1 +SD−1 +� +zSQ(z) +� +1 +SQ(z) + 1 +�� = +1 +SD−1 (z + zSQ(z)). +(51) +A combination of (50) and (51) gives a way to connect the S-transforms of D−1 and Q +1 +SD−1 (z + zSQ(z)) = +1 +SQ(z) + 1. +(52) +From (40) and the key FPT principles (26) we find +SQ(z) = SQ(z)SQ⊥(z) = (SQ(z))2, +(53) +where we used the fact that Q and Q⊥ are statistically identical and as such have the same S-transform. +One can now rewrite (52) with z → zRQ(z) and utilize (25) to obtain +1 +SD−1 (zRQ(z)+zRQ(z)SQ(zRQ(z))) += +1 +SQ(zRQ(z)) + 1 +⇐⇒ +1 +SD−1 +� +zRQ(z)+zRQ(z)√ +SQ(zRQ(z)) +� += +1 +√ +SQ(zRQ(z)) + 1 +⇐⇒ +1 +SD−1 +� +zRQ(z)+z√ +RQ(z) +� += +� +RQ(z) + 1. +(54) +Replacing z → GQ(z), (54) can be further rewritten +1 +SD−1 +� +zRQ(z)+z√ +RQ(z) +� += +� +RQ(z) + 1 +⇐⇒ +1 +SD−1 +� +GQ(z)RQ(GQ(z))+GQ(z)√ +RQ(GQ(z)) +� += +� +RQ(GQ(z)) + 1. +(55) +From (24) we find +RQ(GQ(z)) + +1 +GQ(z) += +z +⇐⇒ +GQ(z)RQ(GQ(z)) += +zGQ(z) − 1. +(56) +Combining further (55) and (56) we also have +1 +SD−1 +� +GQ(z)RQ(GQ(z))+GQ(z)√ +RQ(GQ(z)) +� += +� +RQ(GQ(z)) + 1 +⇐⇒ +1 +SD−1 +� +zGQ(z)−1+√ +GQ(z)√ +zGQ(z)−1 +� += +� +zGQ(z)−1 +GQ(z) ++ 1. +(57) +As in [12] one has for the connection between the S-transforms of the matrix and its inverse +SD(z) = +1 +SD−1(−1 − z). +(58) +Keeping (58) in mind, one can rewrite (57) in the following way +1 +SD−1 +� +GQ(z)RQ(GQ(z))+GQ(z)√ +RQ(GQ(z)) +� += +� +RQ(GQ(z)) + 1 +⇐⇒ +SD +� +−zGQ(z) − +� +GQ(z) +� +zGQ(z) − 1 +� += +� +zGQ(z)−1 +GQ(z) ++ 1. +(59) +We will make a SD(z) − GD(z) connection below. however, before doing so, we will need to make certain +adjustments in the GD(z) transform itself. +i) Adjusting GD(z) for the difference between ˜D and ¯D +11 + +We now briefly recall on the connection between ˜D, ¯D, and D. First, from (32) and (19) we have +˜D += +¯V ⊥( ¯V ⊥)T ¯U ⊥ +D( ¯U ⊥ +D)T +¯D += +( ¯U ⊥ +D)T ¯V ⊥( ¯V ⊥)T ¯U ⊥ +D. +(60) +The spectra of ˜D and ¯D are modulo scalings practically identical. Since ¯D has all the eigenvalues that ˜D +has with l = ηn zero eigenvalues removed one can connect their G-transforms in the following way. +G ¯ +D(z) += +1 +1 − η +� +G ˜ +D(z) − η +z +� +. +(61) +To see that (61) is indeed true we first connect the spectral pdfs of ˜D and ¯D +f ¯ +D(x) += +1 +1 − η (f ˜ +D(x) − ηδ(x)) . +(62) +Then from (22) we have +G ¯ +D(x) = +� f ¯ +D(x) +z − x dx = +1 +1 − η +�� f ˜ +D(x) +z − x dx − η +� +δ(x) +z − xdx +� += +1 +1 − η +� +G ˜ +D(z) − η +z +� +. +(63) +Connecting beginning and end in (63) we obtain (61). +ii) Adjusting GD(z) for the difference between Q1 and Q +We recall that Q1 has the same eigenvalues as Q minus n − l − k zero eigenvalues (when n − l − k ≤ 0 that +means that Q1 has all the eigenvalues of Q plus |n−l−k| zero eigenvalues). To account for this difference we +find it useful to introduce a matrix D1 obtained by removing/adding |n − (l + k)| ones into the spectrum of +D. As these added ones are inversion invariant they remain in the spectrum after the inversion. This means +that after the inversion of D1 and subtraction of the identity matrix they become zeros and basically have +an effect on Q as if |n − (l + k)| zeros were added or removed which is exactly what we need to account for +the difference between Q1 and Q. To put everything in the right mathematical context, let D1 be a matrix +with the Haar distributed eigen-space basis and the spectral function defined int he following way +fD1 = +1 +1 − η − (1 − (β + η)) (f ˜ +D − ηδ(x) − (1 − (β + η))δ(x − 1)) , +(64) +where we have now taken into the account the above mentioned adjusting between ˜D and ¯D ( ¯D and D have +identical spectral functions). Utilizing again (22) we similarly to (63) have +GD1(z) = +1 +1 − η − (1 − β + η) +� +G ˜ +D(z) − η +z − 1 − (β + η) +z − 1 +� +. +(65) +Recalling once again on (24) we have +RD1(GD1(z)) + +1 +GD1(z) += +z +⇐⇒ +RD1(z) + 1 +z += +G−1 +D1(z). +(66) +After taking z → RD1(z) + 1 +z we can rewrite (65) as +GD1 +� +RD1(z) + 1 +z +� += 1 +β +� +G ˜ +D +� +RD1(z) + 1 +z +� +− +η +RD1(z) + 1 +z +− +1 − (β + η) +RD1(z) + 1 +z − 1 +� +, +(67) +and after utilizing (66) +z = 1 +β +� +G ˜ +D +� +RD1(z) + 1 +z +� +− +η +RD1(z) + 1 +z +− +1 − (β + η) +RD1(z) + 1 +z − 1 +� +. +(68) +12 + +After another replacement, z → zSD1(z), (68) becomes +zSD1(z) = 1 +β +� +G ˜ +D +� +RD1(zSD1(z)) + +1 +zSD1(z) +� +− +η +RD1(zSD1(z)) + +1 +zSD1 (z) +− +1 − (β + η) +RD1(zSD1(z)) + +1 +zSD1 (z) − 1 +� +. +(69) +Using (25) we from (69) further find +zSD1(z) = 1 +β +� +G ˜ +D +� +1 +SD1(z) + +1 +zSD1(z) +� +− +η +1 +SD1 (z) + +1 +zSD1 (z) +− +1 − (β + η) +1 +SD1(z) + +1 +zSD1(z) − 1 +� +, +(70) +and +zSD1(z) = 1 +β +� +G ˜ +D +� z + 1 +zSD1(z) +� +− +η +z+1 +zSD1 (z) +− 1 − (β + η) +z+1 +zSD1(z) − 1 +� +. +(71) +Taking D → D1 in (41) and correspondingly denoting Q → Q1 and Q → Q1, one can repeat all the steps +between (41) and (59) to arrive at the following +SD1 +� +−zGQ1(z) − +� +GQ1(z) +� +zGQ1(z) − 1 +� += +� +zGQ1(z) − 1 +GQ1(z) ++ 1. +(72) +Setting +z1(z) +≜ +−zGQ1(z) − +� +GQ1(z) +� +zGQ1(z) − 1 +y(z) +≜ +z1(z) + 1 +z1(z)SD1(z1(z)), +(73) +one has from (72) +SD1(z1(z)) += +� +zGQ1(z) − 1 +GQ1(z) ++ 1. +(74) +After taking z → z1 and rewriting (71) one finally obtains +z1(z) + 1 +y(z) += 1 +β +� +G ˜ +D(y(z)) − +η +y(z) − 1 − (β + η) +y(z) − 1 +� +. +(75) +We summarize the above results in the following lemma. +Lemma 2. Let Q1 be as in (39). Then its G-transform, GQ1(z), satisfies +z1(z) + 1 +y(z) += 1 +β +� +G ˜ +D(y(z)) − +η +y(z) − 1 − (β + η) +y(z) − 1 +� +. +(76) +with z1(z) and y(z) as in (73), SD1(z1(z)) as in (74), and G ¯ +D(y(z)) as in Lemma 1. +A combination of Lemma 1 (where G ˜ +D(·) is explicitly given) and (73)-(75) is then sufficient to determine +GQ1(z) . Utilizing (23) then enables one to fully determine the spectral distribution. This is a generic +procedure that in principle works. Below we will move things a step further and provide a more detailed +analysis of the edges of the spectrum as they play a critical role in the ℓ∗ +0 − ℓ∗ +1-equivalence. It will turn +out that one can provide their a sufficiently explicit characterization so that the explicit closed form for the +corresponding C-inf phase transitions can again be obtained. Later on we will return to the above described +procedure for determining the entire spectrum of Q1 and show what type of results such a procedure actually +produces. +13 + +4.2.2 +Explicit characterization of Q1’s spectral edges +As we have seen earlier, the upper edge of the spectrum of Q (or Q1), λmax(Q) = λmax(Q1) is directly +related to the success of the ℓ∗ +1-minimization heuristic in causal inference. More precisely, as Corollary 2 +states, one will have the ℓ∗ +0 − ℓ∗ +1-equivalence if and only if λmax(Q) = λmax(Q1) ≤ 1. Clearly, an explicit +characterization of λmax(Q) = λmax(Q1) will be sufficient to explicitly characterize the ℓ∗ +0 − ℓ∗ +1-equivalence. +That will then be enough to conclude when ℓ∗ +1 can be used reliable to handle the casual inference. +To provide an explicit characterization of λmax(Q) = λmax(Q1) ≤ 1 we rely on the results that we +presented in the previous section. We start by observing that the spectral function of Q1, fQ1(x), can be +obtained by utilizing (23) and the above discussed GQ1(z) transform. Moreover, at the edge of the spectrum +GQ1(z) should be real (the edge of the spectrum is actually the breaking point where the GQ1(z) becomes +complex, i.e. starts having a nonzero imaginary part). That basically means that at the edge of the spectrum +one should have (76) satisfied for a real GQ1(z). Moreover, since our targeted edge of the spectrum is one +that means that (76) needs to be satisfied for a real GQ1(1). Rewriting (73)-(75) for z = 1 gives +z1(1) += +−GQ1(1) − +� +GQ1(1) +� +GQ1(1) − 1 +y(1) +≜ +z1(1) + 1 +z1(1)SD1(z1(1)), +(77) +and +SD1(z1(1)) += +� +GQ1(1) − 1 +GQ1(1) ++ 1 = − z1(1) +GQ1(1), +(78) +and +z1(1) + 1 +y(1) += 1 +β +� +G ˜ +D(y(1)) − +η +y(1) − 1 − (β + η) +y(1) − 1 +� +. +(79) +From (77) one further finds +GQ1(1) += +− (z1(1))2 +1 + 2z1(1) +y(1) += +−(z1(1) + 1)GQ1(1) +(z1(1))2 += z1(1) + 1 +1 + 2z1(1). +(80) +The second equality then also gives +z1(1) += +y(1) − 1 +1 − 2y(1), +(81) +and +z1(1) + 1 += +y(1) +2y(1) − 1. +(82) +Plugging (82) into (79) one has +1 +2y(1) − 1 = 1 +β +� +G ˜ +D(y(1)) − +η +y(1) − 1 − (β + η) +y(1) − 1 +� +, +(83) +or +ζ1(y) ≜ − +1 +2y − 1 + 1 +β +� +G ˜ +D(y) − η +y − 1 − (β + η) +y − 1 +� += 0. +(84) +Utilizing G ˜ +D(z) (with the “−” sign as the lower edge in the bulk of the spectrum of ˜D corresponds to the +14 + +upper edge in the spectrum of Q) from Lemma 1 we further have +ζ1(y) = − +1 +2y − 1 + +2β − 1 +2β(y − 1) + +1 +2βy(y − 1) +� +−β + η − +� +(y − (β + η))2 + 4βη(y − 1) +� += 0, +(85) +and +ζ1(y) += +−2βy(y − 1) + y(2β − 1)(2y − 1) + (2y − 1) +� +−β + η − +� +(y − (β + η))2 + 4βη(y − 1) +� +2βy(y − 1)(2y − 1) += +2(β − 1)y2 + (1 − 2β + 2η)y + β − η − (2y − 1) +� +(y − (β + η))2 + 4βη(y − 1) +2βy(y − 1)(2y − 1) +. +(86) +Setting +ζ2(y) +≜ +2(β − 1)y2 + (1 − 2β + 2η)y + β − η − (2y − 1) +� +(y − (β + η))2 + 4βη(y − 1) +ζ(y) +≜ +(2(β − 1)y2 + (1 − 2β + 2η)y + β − η)2 − ((2y − 1) +� +(y − (β + η))2 + 4βη(y − 1))2, +(87) +we easily have +ζ1(y) = 0 +⇐⇒ +ζ2(y) = 0 +⇐⇒ +ζ(y) = 0. +(88) +We therefore below focus on ζ(y). After squaring and grouping the terms we have +ζ(y) = 4β(c3y4 + c2y3 + c1y2 + c0y + c00), +(89) +with +c3 += +β − 2 +c2 += +5 − 2β − 2η +c1 += +β − 4 + 3η +c0 += +1 − η +c00 += +0. +(90) +From (89) we then also have +ζ(y) = 4βy(c3y3 + c2y2 + c1y + c0). +(91) +Since we are interested in an edge or a breaking point of the spectrum ζ(y) should touch zero for certain y +which means that it should have a stationary point at such y. To find such a stationary point we take the +derivative +d +� +ζ(y) +4βy +� +dy += 3c3y2 + 2c2y + c1 = 0. +(92) +Solving over y gives +y = −c2 + +� +c2 +2 − 3c1c3 +3c3 +. +(93) +Setting +r ≜ c2 +2 − 3c1c3 = 1 + β2 + 4η2 − 2β − 2η − βη, +(94) +15 + +we have from (93) +yopt = −c2 + √r +3c3 +. +(95) +First we set +ζ3(y) ≜ c3y3 + c2y2 + c1y + c0. +(96) +Clearly, from (91) one has +ζ(y) = 4βyζ3(y). +(97) +Then we plug the value for yopt from (95) and after a bit of algebraic transformations obtain +ζ3(yopt) = −2(√r)3 − c3 +2 + 3rc2 + 27c2 +3c0. +(98) +From (90) we first have +c2 += +−2c3 − 1 + 2c0 +c1 += +c3 − 3c0 + 1, +(99) +and then from (94) +r = c2 +3 + 1 + 4c2 +0 + c3 − 4c0 + c0c3. +(100) +Combining (98)-(100) after a bit of additional algebraic transformations gives +ζ3(yopt) = −2(√r)3 + 2c3 +3 − 3c3 + 6c3c2 +0 + 3c2 +3 + 3c3c0 + 3c0c2 +3 − 2 − 24c2 +0 + 12c0 + 16c3 +0. +(101) +Below we show that +c2 +3 = −1 − 2c3 + 4c0 − 4c2 +0 +⇐⇒ +ζ3(yopt) = 0. +(102) +We first use (102) to systematically linearize ζ3(yopt) in c3 and obtain +ζ3(yopt) = −2(√r)3 + (−3c3 + 5c3c0 − 2c3c2 +0 − 1 − 8c2 +0 + 5c0 + 4c3 +0). +(103) +Transforming further we also have +ζ3(yopt) += +−2(√r)3 + (−3c3 + 5c3c0 − 2c3c2 +0 − 1 − 8c2 +0 + 5c0 + 4c3 +0) += +−2(√r)3 + (c3(c0 − 1)(3 − 2c0) + (−4c0 + 1 + 4c2 +0)(c0 − 1)) += +2(√c3 +√ +c0 − 1)3 + (c3(c0 − 1)(3 − 2c0) + (−4c0 + 1 + 4c2 +0)(c0 − 1)) += +� +2(√c3)3√ +c0 − 1 + (c3(3 − 2c0) + (−4c0 + 1 + 4c2 +0)) +� +(c0 − 1). +(104) +where the third equality follows after noting that with condition (102) in place r in 100) becomes +c2 +3 = −1 − 2c3 + 4c0 − 4c2 +0 +=⇒ +r = −c3 + c0c3. +(105) +We find it useful to rewrite (104) as +ζ3(yopt) += +ζ(1) +3 (yopt) + ζ(2) +3 (yopt), +(106) +where +ζ(1) +3 (yopt) +≜ +2(√c3)3√ +c0 − 1 +ζ(2) +3 (yopt) +≜ +c3(3 − 2c0) + (−4c0 + 1 + 4c2 +0). +(107) +16 + +We then look at the squared values of these quantities. First we start with ζ(1) +3 (yopt) +(ζ(1) +3 (yopt))2 = 4c3 +3(c0 − 1), +(108) +and utilize the condition (102) to systematically linearize in c3. First we remove the cubic c3 term to arrive +at the following +(ζ(1) +3 (yopt))2 += +4c3 +3(c0 − 1) += +4c3(−1 − 2c3 + 4c0 − 4c2 +0)(c0 − 1) += +−8c2 +3c0 + 32c3c2 +0 − 16c3c3 +0 − 8 − 12c3 + 32c0 − 32c2 +0 − 20c3c0, +(109) +and apply the same procedure again to arrive at a fully linearized form +(ζ(1) +3 (yopt))2 = 4(c3((−2c2 +0 + c0 + 1)(2c0 − 3)) − 2(1 − c0)(2c0 − 1)2). +(110) +Then we turn to ζ(2) +3 (yopt) +(ζ(2) +3 (yopt))2 = (c3(3 − 2c0) + (−4c0 + 1 + 4c2 +0))2, +(111) +and again utilize the condition (102) to linearize in c3. This time the procedure is simpler as there is only a +quadratic term in c3 and there is no need to apply the procedure from above in two steps. Instead only one +step suffices and we have +(ζ(2) +3 (yopt))2 += +(c3(3 − 2c0) + (−4c0 + 1 + 4c2 +0))2 += +c2 +3(3 − 2c0)2 + (−4c0 + 1 + 4c2 +0)2 + 2(−4c0 + 1 + 4c2 +0)c3(3 − 2c0) += +(−1 − 2c3 + 4c0 − 4c2 +0)(3 − 2c0)2 + (−4c0 + 1 + 4c2 +0)2 + 2(−4c0 + 1 + 4c2 +0)c3(3 − 2c0) += +−2c3(9 − 12c0 + 4c2 +0 − (−4c0 + 1 + 4c2 +0)(3 − 2c0)) − (−4c0 + 1 + 4c2 +0)(8 − 8c0) += +4(c3((−2c2 +0 + c0 + 1)(2c0 − 3)) − 2(1 − c0)(2c0 − 1)2). +(112) +Comparing (110) and (112) we have +(ζ(1) +3 (yopt))2 += +(ζ(2) +3 (yopt))2. +(113) +Now we will show that one also has (ζ(1) +3 (yopt))2 = −(ζ(2) +3 (yopt))2. We again look at the condition in (102) +and replace the values for c0 and c3 from (91) to obtain +c2 +3 += +−1 − 2c3 + 4c0 − 4c2 +0 +⇐⇒ +(β − 2)2 += +−1 − 2(β − 2) + 4(1 − η) − 4(1 − η)2 +⇐⇒ +(β − 2)2 + 2(β − 2) + 1 += +4(1 − η) − 4(1 − η)2 +⇐⇒ +(β − 2 + 1)2 += +4η(1 − η) +⇐⇒ +β += +1 − 2 +� +η(1 − η). +(114) +From (107) we then also have +ζ(1) +3 (yopt) +≜ +2(√c3)3√c0 − 1 = 2( +� +β − 2)3√−η = 2(2 − β) +� +η(2 − β) ≥ 0. +(115) +Similarly, we have +ζ(2) +3 (yopt) +≜ +c3(3 − 2c0) + (−4c0 + 1 + 4c2 +0) += +(β − 2)(1 + 2η) + (1 − 2η)2 += +(−1 − 2 +� +η(1 − η))(1 + 2η) + (1 − 2η)2 += +−2 +� +η(1 − η)(1 + 2η) − 6η + 4η2 +17 + +≤ +−2 +� +η(1 − η)(1 + 2η) − 6η + 4η += +−2 +� +η(1 − η)(1 + 2η) − 2η +≤ +0. +(116) +A combination of (106), (107), (113), (115), and (116) finally gives +(ζ(1) +3 (yopt))2 +(113) += +(ζ(2) +3 (yopt))2 +(115),(116) +⇐⇒ +ζ(1) +3 (yopt) += +−ζ(2) +3 (yopt) +⇐⇒ +ζ(1) +3 (yopt) + ζ(2) +3 (yopt) += +0 +(106) +⇐⇒ +ζ3(yopt) += +0. +(117) +Moreover, a combination of (102), (114), and (117) gives +β = 1 − 2 +� +η(1 − η) +⇐⇒ +c2 +3 = −1 − 2c3 + 4c0 − 4c2 +0 +⇐⇒ +ζ3(yopt) = 0. +(118) +After combining (97) and (118) one then also has +β = 1 − 2 +� +η(1 − η) +⇐⇒ +c2 +3 = −1 − 2c3 + 4c0 − 4c2 +0 +⇐⇒ +ζ3(yopt) = 0 +⇐⇒ +ζ(yopt) = 0. (119) +From (88) one then has that for yopt +ζ1(yopt) = ζ2(yopt) = 0, +(120) +which means that y = yopt is indeed a choice for y that ensures that functional equation used to determine +GQ1(z) is satisfied. Moreover, since the derivative condition is met as well, i.e. since ζ(yopt) = 0, one has +that not only is yopt a point where ζ(y) crosses zero, it is actually a point where it touches zero. That is +exactly what is needed to determine an edge of the spectrum. Since we operated using the “−” sign in the +definition of G ˜ +D(z) that means (based on the considerations from [10]) that we have determined the lower +edge in the corresponding spectrum of ˜D (or any of ¯D and D) which after the inversion means that we have +determined the upper edge in the spectrum of Q1 or Q. +One can even explicitly determine yopt. From (90), (95), (99), and (105) we obtain +yopt = −c2 + √r +3c3 += −(5 − 2β − 2η) + +� +η(2 − β) +3(β − 2) +. +(121) +In Figure 3 we show yopt as a function of η. The whole mechanism of “touching zero” as β decreases is +shown in Figure for η = 0.9. As can be seen from the figure, for β > 1−2 +� +η(1 − η) = 0.4 ζ1(y) remains below +zero one therefore can not be a part of the spectrum. On the other hand, for β ≤ 1−2 +� +η(1 − η) = 0.4 ζ1(y) +does intersect zero line which implies that one is now in the spectrum (there is y = y(11) and consequently +a real GQ1(1) such that ζ1(y) = 0). The borderline or the breaking point happens exactly when the ζ1(y) +curve touches the zero line. As figure indicates that happens for y = yopt = 0.25, exactly as the theory +predicts. +We summarize the above results in the following lemma. +Lemma 3. Assume the setup of Lemmas 1 and 2 with Q1 as in (39). Then we have for the upper edge of +the Q1’s spectrum +β = 1 − 2 +� +η(1 − η) +⇐⇒ +λmax(Q1) = 1. +(122) +Moreover, +β ≤ 1 − 2 +� +η(1 − η) +⇐⇒ +λmax(Q1) ≤ 1. +(123) +Proof. Follows from the above discussion. +18 + +η +0.5 +0.55 +0.6 +0.65 +0.7 +0.75 +0.8 +0.85 +0.9 +0.95 +1 +yopt +0 +0.05 +0.1 +0.15 +0.2 +0.25 +0.3 +0.35 +0.4 +0.45 +0.5 +yopt as a function of η +yopt = +√1−η +√η−√1−η +η = 0.9 +=⇒ +yopt = +√1−η +√η+√1−η = 0.25 +Figure 3: yopt as a function of η +4.2.3 +The spectrum of Q1 ≜ λT +V λV λT +UλU – practical evaluations +Now that we have fully characterized the upper edge of the Q1’s spectrum we can return to the consideration +of the entire spectrum. Relying on the above presented machinery we can establish the following lemma. +Lemma 4. Assume the setup of Lemmas 1 and 2 with Q1 as in (39). Let GQ1(z) be the solution of the +following system of equations: +y(z) += +� +zGQ1(z) − 1 +� +zGQ1(z) − 1 + z +� +GQ1(z) +G ˜ +D(y(z)) += +y(z) − (β + η) ± +� +(y(z) − (β + η))2 + 4βη(y(z) − 1) +2((y(z))2 − y(z)) +1 +β +� +G ˜ +D(y(z)) − +η +y(z) − 1 − (β + η) +y(z) − 1 +� += +−( +� +zGQ1(z) − 1 + +� +GQ1(z))( +� +zGQ1(z) − 1 + z +� +GQ1(z)). +(124) +Then the spectral function of Q1, fQ1(x), is obtained as +fQ1(x) = − lim +ǫ→0+ +imag(GQ1(x + iǫ)) +π +. +(125) +Proof. Follows from Lemma 2 through a combination of the results of Lemma 1 (where G ˜ +D(·) is explicitly +given) and (73)-(75). The following two sequences of identities are then sufficient to prove the lemma +y(z) += +z1(z) + 1 +z1(z)SD(z1(z)) += +((−zGQ1(z) − +� +GQ1(z) +� +zGQ1(z) − 1) + 1) +� +GQ1(z) +(−zGQ1(z) − +� +GQ1(z) +� +zGQ1(z) − 1)( +� +zGQ1(z) − 1 + +� +GQ1(z)) += +� +zGQ1(z) − 1 +� +zGQ1(z) − 1 + z +� +GQ1(z) +, +(126) +19 + +y +0 +0.05 +0.1 +0.15 +0.2 +0.25 +ζ1(y) +-0.25 +-0.2 +-0.15 +-0.1 +-0.05 +0 +0.05 +ζ1(y) as a function of y; η = 0.9 +β = 0.41 +β = 0.4 +β = 0.39 +β = 0.4, η = 0.9 +=⇒ +yopt = +√1−η +√η+√1−η = 0.25 +Figure 4: ζ1y as a function of y +and +z1(z) + 1 +y(z) += +SD(z1(z)) +z1(z) += +(−zGQ1(z) − +� +GQ1(z) +� +zGQ1(z) − 1)( +� +zGQ1(z) − 1 + +� +GQ1(z)) +� +GQ1(z) += +( +� +zGQ1(z) − 1 + z +� +GQ1(z))( +� +zGQ1(z) − 1 + +� +GQ1(z)). +(127) +In Figure 5 we show the entire spectrum of fQ1(x). We chose β = 0.4 and η = 0.9 and ran the experiments +with n = 4000. As can be seen from the figure, the obtained numerical results are in a strong agreement +with what the theory predicts. +4.2.4 +ℓ∗ +0 − ℓ∗ +1-equivalence via the spectral limit – asymmetric scenario +From Corollary 2, (17), and (38) one has in the asymmetric scenario +ℓ∗ +0 − ℓ∗ +1 − equivalence +⇐⇒ +λmax(λT +V λV λT +UλU) ≤ 1 +⇐⇒ +λmax(Q1) ≤ 1. +(128) +From (123) and (128) we finally have +ℓ∗ +0 − ℓ∗ +1 − equivalence +⇐⇒ +β ≤ 1 − 2 +� +η − η2. +(129) +Analogously to Theorem 2 we can now establish a precise asymmetric scenario location of the phase transition +in a typical statistical context. +Theorem 3. (ℓ∗ +1 – phase transition – C-inf (typical asymmetric scenario)) Assume the setup of +Theorem 2 with rank-k matrix Xsol = X ∈ Rn×n that now has Haar distributed independent bases of its +orthogonal row and column spans ¯U ⊥ ∈ Rn×(n−k) and ¯V ⊥ ∈ Rn×(n−k) (XT +sol ¯U ⊥ = Xsol ¯V ⊥ = 0n×(n−k)). +Let M ≜ M (l) ∈ Rn×n be as defined in (5). Let βac and η satisfy the following +20 + +x +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +fQ1(x) +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +1.6 +1.8 +2 Spectral distribution fQ1(x); η = .4; β = .9; n = 4000 +simulated +theory +fQ1(x) +Bulk +Figure 5: fQ1(x) – spectral function of Q1; β = 0.4 and η = 0.9 +C-inf ℓ∗ +1 asymmetric scenario phase transition (PT) characterization +ξ(ac) +η +(β) ≜ β − 1 + 2 +� +η − η2 = 0. +(130) +If and only if β ≤ βac +lim +n→∞ P(ℓ∗ +0 ⇐⇒ ℓ∗ +1) = +lim +n→∞ P(RMSE = 0) = 1, +(131) +and the solutions of (6) and (7) coincide with overwhelming probability. +Proof. Follows from Lemma 3 and the above discussion. +The results related to the use of the ℓ∗ +1-minimization heuristic for solving the causal inference problems +obtained based on the above theorem are shown in Figure 6. +As in the worst case scenario, the phase +transition (PT) curve splits the (β, η) region into two separate subregions where the ℓ∗ +0 − ℓ∗ +1-equivalence +phenomenon either occurs or fails to occur. Basically, below the curve one has a perfect recovery with the +residual RMSE = ∥vec( ˆX) − vec(Xsol)∥2 = 0. Contrary to that, above the curve though, there is an Xsol +for which RMSE → ∞ and ℓ∗ +1 fails. +The following corollary adapts the above results so that they fit the standard (α, β) representation +typically used in the compressed sensing (CS), low rank recovery (LRR), and matrix completion (MC) +literature. +Corollary 3. (ℓ∗ +1 – phase transition – C-inf (typical asymmetric scenario; standard (α, β) rep- +resentation)) Assume the setup of Theorem 3. Let m be the total number of ones in matrix M and let +α ≜ limn→∞ m +n2 . Let β and αw satisfy the +C-inf ℓ∗ +1 asymmetric scenario PT (standard (α, β) representation) +ξ(wc,s) +β +(α) ≜ β − 1 + 2 +�√ +1 − α − 1 + α = 0. +(132) +21 + +η +0.5 +0.55 +0.6 +0.65 +0.7 +0.75 +0.8 +0.85 +0.9 +0.95 +1 +β +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +(η, β) region of success/failure — C-inf ℓ∗1 PT; asymmetric scenario +worst case +average case +RMSE −→ ∞, ℓ∗1 +fails +ℓ∗1’s PT: ξ(ac) +η +(β) = β − 1 + 2�η − η2 = 0 +RMSE = 0, ℓ∗1 succeeds +Doubling low rankness: +ξ(ac) +η +(2β) = 2ξ(wc) +η +(β) +Figure 6: Causal inference (C-inf) – typical asymmetric scenario ℓ∗ +1 phase transition +If and only if α ≥ αw +lim +n→∞ P(ℓ∗ +0 ⇐⇒ ℓ∗ +1) = +lim +n→∞ P(RMSE = 0) = 1, +(133) +and the solutions of (6) and (7) coincide with overwhelming probability. +Proof. Follows as a direct consequence of Theorem 3 after noting that m = n2 − (n − l)2 and consequently +α = 1 − (1 − η)2. +Figure 7 shows the results obtained based on the above corollary in the standard (α, β) region format. +As usual in the PT considerations, the entire (α, β) region is split in the part below the curve where +RMSE = ∥vec( ˆX) − vec(Xsol)∥2 = 0 and the part above the curve where even RMSE → ∞ is achievable. +We should point out an interesting similarity between what we observed here in the above corollary and +in Figure 7 on the one side and what is known to hold in generic LRR. Namely, as Corollary 3 states (and as +is emphasized in Figure 7), for the same value of α one achieves exactly two times larger β in the asymmetric +case than in the worst case. As the worst case is basically symmetric, one has that the PTs of the symmetric +and the nonsymmetric scenarios are distinguished by a factor of two. Similar observation was in place when +it comes to the comparison between the LRR of the symmetric and the general (nonsymmetric) matrices. +However, one should keep in mind a fundamental difference as well. In LRR the underlying symmetry is a +priori known and can be utilized in the algorithms design whereas here it is just the choice of the worst case +problem instance and is not assumed to be known to the algorithm itself. Of course, given the properties of +the LRR, such a choice is not necessarily very surprising. +4.3 +Numerical results +To complement the above theoretical findings and see how successful in characterizing the utilization of the +ℓ∗ +1-minimization in C-inf problems they indeed are, we conducted a set of numerical experiments and show +the obtained results in Figure 8. As in [10], we again observe both the PT’s existence and a solid agreement +between its theoretical prediction and the results obtained through the simulations. +In the conducted numerical experiments we chose n = 80 and η in the range [0.6, 0.95]. Clearly, such fairly +small matrix sizes correspond to the settings quite opposite from the ones that we used in the theoretical +analysis. Still, even though the theory is predicated on the large n assumption, it is not impossible that +its conclusions remain valid for smaller values of n as well. The results form Figure 8 confirm that this is +22 + +α +0.75 +0.8 +0.85 +0.9 +0.95 +1 +β/α +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +(α, β) region of success/failure — C-inf ℓ∗1 PT; asymmetric scenario +worst case +average case +RMSE = 0, ℓ∗1 succeeds +RMSE −→ ∞, ℓ∗1 +fails +ℓ∗1’s PT: ξ(ac,s) +β +(α) = β − 1 + 2 +�√1 − α − 1 + α = 0 +ξ(ac,s) +2β +(α) = 2ξ(ac,s) +β +(α) +Figure 7: Causal inference (C-inf) – typical asymmetric scenario ℓ∗ +1 phase transition ((α, β) region) +indeed the case. Moreover, one can then say that the large n regime, needed for the theoretical consideration, +practically may start ro kick in already for rather small (of order of a few tens!) values of n. This ultimately +means that the presented results, although theoretical in nature, have in themselves a strong practical +component as well. Finally, we should also add that for larger values of n an even better agreement between +the theoretical and the simulated results is to be expected. +A few additional points regarding the simulations setup might be useful. First, one should emphasize, +that in order to be in an agreement with the theoretical considerations, we, in all numerical experiments, +considered the so-called typical behavior. Following further into the footsteps of the theoretical consider- +ations, the presented simulations results were obtained for the square matrices. As was the case in [10], +all theoretical considerations can be repeated assuming the non-square scenarios as well. We, however, (as +in [10]) prioritized the clarity of the presentations over simple generalizations and opted for the square sce- +narios which are substantially easier to present. Also, all the simulations needed for Figure 8 were done with +the singular values of the unknown targeted matrices equal to one. While we refer to [10] for a bit more +complete discussion regarding such a choice, we here briefly mention that choices of this type are known +to serve as the worst case examples in establishing the reversal ℓ0 − ℓ1-equivalence conditions. As in [10], +we also ran the simulations where the singular values were randomly chosen with results either identically +matching or improving on the ones shown in Figure 8. +5 +Conclusion +In this paper, we have built on the mathematical Causal inference (C-inf) ↔ low-rank recovery +(LRR) connection established in [10] to deal with asymmetric PTs phenomena. The results of [10] proved +that the nuclear norm (ℓ∗ +1) minimization heuristic, when used for solving the low rank recovery C-inf problems, +exhibits the so-called phase transition (PT) phenomenon. Moreover, in a typical statistical scenario, [10] +characterized the exact location of the worst case PT. This effectively meant that there are problem in- +stances where the ℓ∗ +1 predicated behavior might be improved upon. Here we showed that this is indeed true. +Considering an asymmetric scenario (in contrast with the symmetric worst case one from [10]) we deter- +mined the underlying exact phase transitions locations. Moreover, we uncovered a doubling low rankness +phenomenon, which means that, throughout the entire region of allowed system parameters, matrices of +exactly two times larger rank can be recovered when compared to the worst case scenario from [10]. Such a +phenomenon also ensures that the simplicity of the worst case PTs from [10] is preserved in the asymmetric +23 + +η +0.5 +0.6 +0.7 +0.8 +0.9 +1 +β +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +(η, β) region of success/failure — ℓ∗1’s PT; simulated/theory +ℓ∗ +1’s PT – simulated +ℓ∗ +1’s PT – theory +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +Success +Failure +Figure 8: C-inf ℓ∗ +1’s asymmetric scenario phase transition (PT) +scenarios as well. Consequently, one is again able to elegantly pin down the relation between the low rankness +of the target C-inf matrix and the time when the treatment is applied. +Throughout the process of creating the theoretical phase transitions characterizations we also established +several mathematical results that are of independent interest. All of our theoretical findings we supplemented +with the results obtained from the corresponding numerical experiments. 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Generalized synthetic control method: Causal inference with interactive fixed effects models. +Political Analysis, 25:57–76, 2017. +A +Proof of Theorem 1 +As mentioned earlier, the proof of Theorem 1 is conceptually identical to the corresponding proof when +matrix X is symmetric. A detailed proof for the symmetric matrices is given below. Before being able to +present the proof we need a couple of technical lemmas. +Lemma 5. Let C = CT ∈ Rn×n. Also let all eigenvalues of C belong to the interval [−1, 1]. Finally, let the +first k entries on the main diagonal, Ci,i, 1 ≤ i ≤ k, be larger than or equal to 1. Then the upper k × k left +block of C, C1:k,1:k, is an identity matrix, i.e. +C1:k,1:k += +Ik×k. +(134) +Proof. Let λmax(C) be the maximum eigenvalue of C. Then +λmax(C) +≜ +max +∥c∥2=1 cT Cc. +(135) +Since by assumptions 1 ≤ Ci,i, 1 ≤ i ≤ k and λmax(C) ≤ 1 we also have for any 1 ≤ i ≤ k +1 ≤ Ci,i ≤ max +∥c∥2=1 cT Cc ≜ λmax(C) ≤ 1, +(136) +which implies C(i, i) = 1, 1 ≤ i ≤ k. The proof that all other elements of C1:k,1:k are equal to zero proceeds +inductively. +1) Induction move from l = 1 to l = 2: First we look at the upper block of size 2 × 2, i.e. at C1:2,1:2. +We then have +1 ≥ max +∥c∥2=1 cT Cc ≥ +max +∥c1:2∥2=1 cT +1:2C1:2,1:2c1:2 +≥ +max +∥c1:2∥2=1 (∥c1:2∥2 + 2|c1c2C1,2|) +≥ +max +∥c1:2∥2=1 (1 + 2|c1c2C1,2|) ≥ 1, +(137) +which implies C1,2 = 0. +2) Induction move from l to l + 1: Now we look at the upper block of size (l + 1) × (l + 1), i.e. at +C1:l+1,1:l+1 while assuming that C1:l,1:l = Il×l. We then have +1 +≥ +max +∥c∥2=1 cT Cc +≥ +max +∥c1:l+1∥2=1 cT +1:l+1C1:l+1,1:l+1c1:l+1 +≥ +max +∥c1:l+1∥2=1 +� +∥c1:l+1∥2 + 2|cT +1:lC1:l,l+1cl+1| +� +≥ +max +∥c1:l+1∥2=1 +� +1 + 2|cT +1:lC1:l,l+1cl+1| +� +≥ +1, +(138) +which implies C1:l,l+1 = 0l×1 and completes the proof. +27 + +Lemma 6. Assume the setup of Lemma 5. Then the upper k × k left block of C, C1:k,1:k, is an identity +matrix and the upper k × (n − k) right block of C, C1:k,n−k+1:n is a zero matrix, i.e. +C1:k,1:k += +Ik×k +C1:k,n−k+1:n += +0k×(n−k). +(139) +Proof. The first part follows by Lemma 5. We now focus on the second part. Consider the following partition +of matrix C +C += +� +C1:k,1:k +C1:k,n−k+1:n +Cn−k+1:n,1:k +Cn−k+1:n,n−k+1:n +� += +� +Ik×k +C1:k,n−k+1:n +Cn−k+1:n,1:k +Cn−k+1:n,n−k+1:n +� +. +(140) +Then assuming that the largest nonzero singular value of C1:k,n−k+1:n is equal to b > 0, we have +1 +≥ +max +∥c∥2=1 cT Cc +≥ +max +∥c1:k∥2=a,cn−k+1:n +� +cT +1:kC1:k,1:kc1:k + 2|cT +1:kC1:k,n−k+1:ncn−k+1:n| + cT +n−k+1:nCn−k+1:n,n−k+1:ncn−k+1:n +� +≥ +max +∥c1:k∥2=a,cn−k+1:n +� +a2 + 2|cT +1:kC1:k,n−k+1:ncn−k+1:n| + cT +n−k+1:nCn−k+1:n,n−k+1:ncn−k+1:n +� +≥ +max +∥c1:k∥2=a,cn−k+1:n +� +a2 + 2|cT +1:kC1:k,n−k+1:ncn−k+1:n| − cT +n−k+1:ncn−k+1:n +� +≥ +max +a∈[0,1] +� +a2 + 2ba +� +1 − a2 − (1 − a2) +� += +max +a∈[0,1] +� +2a2 − 1 + 2ba +� +1 − a2 +� +, +(141) +where the fourth inequality follows since the minimum eigenvalue of Cn−k+1:n,n−k+1:n is larger than or equal +to the minimum eigenvalue of C which is by the lemma’s assumption larger than or equal to -1. Now, we +further have +c ≜ 2a +� +1 − a2 +and +2a2 − 1 + 2ba +� +1 − a2 = +� +1 − c2 + bc, +(142) +and +d( +√ +1 − c2 + bc) +dc += +−c +√ +1 − c2 + b = 0. +(143) +From (143) we then easily obtain +c = +b +√ +1 + b2 . +(144) +A combination of (141), (142), and (144) gives +1 ≥ max +∥c∥2=1 cT Cc ≥ max +a∈[0,1] +� +2a2 − 1 + 2ba +� +1 − a2 +� += +√ +1 + b2, +(145) +which implies b = 0 and automatically C1:k,n−k+1:n = 0k×1. This completes the proof. +Now we can consider the above mentioned theorem that adapts the general ℓ1 equivalence condition +result from [39–41] to the corresponding one for the ℓ1 norm of the singular/eigenvalues (similar adaptation +can also be found in [24]). +Theorem 4. (ℓ∗ +0 − ℓ∗ +1-equivalence condition (LRR) – symmetric X) Consider a ¯U ∈ Rn×k such that +¯U T ¯U = Ik×k and a rank − k a priori known to be symmetric matrix Xsol = X ∈ Rn×n with all of its +columns belonging to the span of ¯U. For concreteness, and without loss of generality, assume that X has only +28 + +positive nonzero eigenvalues. For a given matrix A ∈ Rm×n2 (m ≤ n2) assume that y = Avec(X) ∈ Rm. If +(∀W ∈ Rn×n|Avec(W) = 0m×1, W = W T ̸= 0n×n) +− tr ( ¯U T W ¯U) < ℓ∗ +1(( ¯U ⊥)T W ¯U ⊥), +(146) +then the solutions of (6) and (7) coincide. Moreover, if +(∃W ∈ Rn×n|Avec(W) = 0m×1, W = W T ̸= 0n×n) +− tr ( ¯U T W ¯U) ≥ ℓ∗ +1(( ¯U ⊥)T W ¯U ⊥), +(147) +then there is an X from the above set of the symmetric matrices with columns belonging to the span of ¯U +such that the solutions of (6) and (7) are different. +Proof. The proof follows literally step-by-step the proof of the corresponding theorem in [39–41] and adapts +it to matrices or their singular/eigenvalues. For experts in the field this adaptation is highly likely to be +viewed as trivial and certainly doesn’t need to be as detailed as we will make it to be. Nonetheless, to ensure +a perfect clarity of all arguments we provide a step-by-step instructional derivation. For concreteness and +without loss of generality we also assume that the eigen-decomposition of X is +X = UΛU T = +� ¯U +¯U ⊥� � +¯ΛX +0k×(n−k) +0(n−k)×k +¯Λ⊥ +X +� � ¯U +¯U ⊥�T . +(148) +(i) =⇒ (the if part): Following step-by-step the proof of Theorem 2 in [41], we start by assuming that +ˆX is the solution of (7). Then we want to show that if (146) holds then ˆX = X. As usual, we instead of that, +assume opposite, i.e. we assume that (146) holds but ˆX ̸= X. Then since y = Avec( ˆ +X) and y = Avec(X) +must hold simultaneously there must exist W such that ˆX = X + W with W ̸= 0, Avec(W) = 0. Moreover, +since ˆX is the solution of (7) one must also have +ℓ∗ +1(X + W) = ℓ∗ +1( ˆX) +≤ +ℓ∗ +1(X) +⇐⇒ +ℓ∗ +1( +� ¯U +¯U ⊥�T (X + W) +� ¯U +¯U ⊥� +) +≤ +ℓ∗ +1(X) +=⇒ +ℓ∗ +1( ¯U T (X + W) ¯U) + ℓ∗ +1(( ¯U ⊥)T (X + W) ¯U ⊥) +≤ +ℓ∗ +1(X). +(149) +The last implication follows after one trivially notes +ℓ∗ +1( +� ¯U +¯U ⊥�T (X + W) +� ¯U +¯U ⊥� +) += +max +Λ∗=ΛT +∗ ∈L∗ +tr (Λ∗ +� ¯U +¯U ⊥�T (X + W) +� ¯U +¯U ⊥� +) +≥ +max +Λ∗=ΛT +∗ ∈L0∗ +tr (Λ∗ +� ¯U +¯U ⊥�T (X + W) +� ¯U +¯U ⊥� +) += +ℓ∗ +1( ¯U T (X + W) ¯U) + ℓ∗ +1(( ¯U ⊥)T (X + W) ¯U ⊥), +(150) +where +L0 +∗ +≜ +� +Λ∗ ∈ Rn×n|Λ∗ = ΛT +∗ , Λ∗ΛT +∗ ≤ I, Λ∗ = +� +Λ∗,1 +0k×(n−k) +0(n−k)×k +Λ∗,2 +�� +⊆ +� +Λ∗ ∈ Rn×n|Λ∗ = ΛT +∗ , Λ∗ΛT +∗ ≤ I +� +≜ L∗. +(151) +The key observation – “Removing the absolute values”: +Now, the key observation made in [41] comes into play. Namely, one notes that the absolute values can +be removed in the nonzero part and that the ℓ∗ +1(·) can be “replaced” by tr (·). Such a simple observation +is the most fundamental reason for all the success of the RDT when used for the exact performance +characterization of the structured objects’ recovery. From (149) we then have +ℓ∗ +1( ¯U T (X + W) ¯U) + ℓ∗ +1(( ¯U ⊥)T (X + W) ¯U ⊥) +≤ +ℓ∗ +1(X) +=⇒ +tr ( ¯U T (X + W) ¯U) + ℓ∗ +1(( ¯U ⊥)T (W) ¯U ⊥) +≤ +ℓ∗ +1(X) +⇐⇒ +tr ( ¯U T W ¯U) + ℓ∗ +1(( ¯U ⊥)T W ¯U ⊥) +≤ +0. +(152) +29 + +We have arrived at a contradiction as the last inequality in (152) is exactly the opposite of (146). This +implies that our initial assumption ˆX ̸= X cannot hold and we therefore must have ˆX = X. This is precisely +the claim of the first part of the theorem. +(ii) ⇐= (the only if part): We now assume that (147) holds, i.e. +(∃W ∈ Rn×n|Avec(W) = 0m×1, W ̸= 0n×n) +− tr (( ¯U)T W ¯U) ≥ ℓ∗ +1(( ¯U ⊥)T W ¯U ⊥) +(153) +and would like to show that for such a W there is a symmetric rank-k matrix X with the columns belonging +to the span of ¯U such that y = Avec(X), and the following holds +ℓ∗ +1(X + W) < ℓ∗ +1(X). +(154) +Existence of such an X would ensure that it both, satisfies all the constraints in (7) and is not the +solution of (7). Following the strategy of [39] one can reverse all the above steps from (153) to (149) with +strict inequalities and arrive at the first inequality in (149) which is exactly (154). There are two implications +that cause problems in such a reversal process, the one in (153) and the one in (149). If these implications +were equivalences everything would be fine. We address these two implications separately. +1) the implication in (152) – particular X to “overwhelm” W: Assume X = ¯UΛx ¯U T with Λx > +0 being a diagonal matrix with arbitrarily large elements on the main diagonal (here it is sufficient even to +choose diagonal of Λx so that its smallest element is larger than the maximum eigenvalue of ¯U T W ¯U). Now +one of course sees the main idea behind the “removing the absolute values” concept from [39,41]. Namely, +for such an X one has that ℓ∗ +1( ¯U T X + W) ¯U) = tr(ℓ∗ +1( ¯U T X + W) ¯U)) since for symmetric matrices the ℓ∗ +1(·) +(as the sum of the argument’s absolute eigenvalues) and tr (·) (as the sum of the argument’s eigenvalues) are +equal. That basically means that when going backwards the second inequality in (152) not only follows from +the first one but also implies it as well. In other words, for X = ¯UΛx ¯U T (with Λx > 0 and arbitrarily large) +tr ( ¯U T W ¯U) + ℓ∗ +1(( ¯U ⊥)T W ¯U ⊥) +≤ +0 +⇐⇒ +tr ( ¯U T (X + W) ¯U ) + ℓ∗ +1(( ¯U ⊥)T (W) ¯U ⊥) +≤ +ℓ∗ +1(X) +⇐⇒ +ℓ∗ +1( ¯U T (X + W) ¯U) + ℓ∗ +1(( ¯U ⊥)T (X + W) ¯U ⊥) +≤ +ℓ∗ +1(X), +(155) +which basically mans that there is an X that can “overwhelm” W (in the span of ¯U) and ensures that the +“removing the absolute values” is not only a sufficient but also a necessary concept for creating the +relaxation equivalence condition. +2) the implication in (149): One would now need to somehow show that the third inequality in (149) +not only follows from the second one but also implies it as well. This boils down to showing that inequality in +(150) can be replaced with an equality or, alternatively, that L0 and L are provisionally equivalent. Neither +of these statements is generically true. However, since we have a set of X at our disposal there might be an +X for which they actually hold. We continue to assume X = ¯UΛx ¯U T with Λx > 0 being a diagonal matrix +with arbitrarily large entries on the main diagonal. Then the last equality in (150) gives +ℓ∗ +1( ¯U T (X + W) ¯U) + ℓ∗ +1(( ¯U ⊥)T (X + W) ¯U ⊥) +≤ +ℓ∗ +1(X) +⇐⇒ +maxΛ∗=ΛT +∗ ∈L0∗ tr (Λ∗ +� ¯U +¯U ⊥�T (X + W) +� ¯U +¯U ⊥� +) +≤ +ℓ∗ +1(X). +(156) +Also, one has +maxΛ∗=ΛT +∗ ∈L0∗ tr (Λ∗ +� ¯U +¯U ⊥�T (X + W) +� ¯U +¯U ⊥� +) +≤ +ℓ∗ +1(X) +⇐⇒ +maxΛ∗,i=ΛT +∗,i,Λ∗,iΛT +∗,i≤I,i∈{1,2} tr (Λ∗,1 ¯U T X ¯U + Λ∗,2( ¯U ⊥)T W ¯U ⊥) +≤ +ℓ∗ +1(X) +⇐⇒ +maxΛ∗,i=ΛT +∗,i,Λ∗,iΛT +∗,i≤I,i∈{1,2} tr (Λ∗,1Λx + Λ∗,2( ¯U ⊥)T W ¯U ⊥) +≤ +tr (Λx). +(157) +Now, if at least one of the elements on the main diagonal of Λ∗,1, diag(Λ∗,1), is smaller than 1, then the +corresponding element on the diagonal of Λx can be made arbitrarily large compared to the other elements +30 + +of Λx and one would have +maxΛ∗,i=ΛT +∗,i,Λ∗,iΛT +∗,i≤I,i∈{1,2} tr (Λ∗,1Λx + Λ∗,2( ¯U ⊥)T W ¯U ⊥) +< +tr (Λx) +⇐⇒ +maxΛ∗=ΛT +∗ ∈L0∗ tr (Λ∗ +� ¯U +¯U ⊥�T (X + W) +� ¯U +¯U ⊥� +) +< +ℓ∗ +1(X) +⇐⇒ +maxΛ∗=ΛT +∗ ∈L∗ tr (Λ∗ +� ¯U +¯U ⊥�T (X + W) +� ¯U +¯U ⊥� +) +< +ℓ∗ +1(X), +(158) +where the last equivalence holds since the difference of the terms on the left-hand side in the last two +inequalities is bounded independently of X. Also, the last inequality in (158) together with the first equality +in (150) and the first inequality in (149) produces (154). Therefore the only scenario that is left as potentially +not producing (154) is when all the elements on the main diagonal are larger than or equal to 1. However, +the two lemmas preceding the theorem show that in such a scenario L0 = L and one consequently has an +equality instead of the inequality in (150) which then, together with (149), implies (154). This completes +the proof of the second (“the only if”) part of the theorem and therefore of the entire theorem. +31 + diff --git a/F9AyT4oBgHgl3EQf5Pp6/content/tmp_files/load_file.txt b/F9AyT4oBgHgl3EQf5Pp6/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..dc1fe56b6746a0e385930a56acc3a15aa059bd48 --- /dev/null +++ b/F9AyT4oBgHgl3EQf5Pp6/content/tmp_files/load_file.txt @@ -0,0 +1,1003 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf,len=1002 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='00801v1 [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='ML] 2 Jan 2023 Causal Inference (C-inf) — asymmetric scenario of typical phase transitions Agostino Capponi ∗ Mihailo Stojnic † Department of Industrial Engineering and Operations Research Columbia University, New York, NY 10027, USA Abstract In this paper, we revisit and further explore a mathematically rigorous connection between Causal in- ference (C-inf) and the Low-rank recovery (LRR) established in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Leveraging the Random duality Free probability theory (RDT-FPT) connection, we obtain the exact explicit typical C-inf asymmetric phase transitions (PT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' We uncover a doubling low-rankness phenomenon, which means that exactly two times larger low rankness is allowed in asymmetric scenarios compared to the symmetric worst case ones con- sidered in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Consequently, the final PT mathematical expressions are as elegant as those obtained in [10], and highlight direct relations between the targeted C-inf matrix low rankness and the time of treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Our results have strong implications for applications, where C-inf matrices are not necessarily symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Index Terms: Causal inference;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Random duality theory;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Algorithms;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Matrix completion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Spar- sity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' 1 Introduction Causal inference (C-inf) deals with the design of estimation strategies that allow researchers to draw causal conclusions based on data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' The overarching goal is to draw a conclusion regarding the effect of a causal variable, which is typically referred to as the “treatment” or the “intervention” on some outcome of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' For example, suppose we want to estimate the causal effect of a drug on deadly cancer progression (vs no exposure to the drug).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Then we want to compare metastasis in the patient’s body one month after the drug regime has begun versus metastasis in the absence of exposure to the drug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' The main challenge for causal inference is that we are not generally able to observe both of these states: at the point in time when we are measuring the outcomes, each individual either has had drug exposure or has not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' The problem of estimating the counterfactual, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=', what would have been the outcome in the absence of a treatement, is central in many disciplines, including economics, health, and social sciences (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' [2,11,13,14,32,33,50]), machine learning and theoretical computer science (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' [25–28]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Methodological developments to estimate causal effects have been based on experimental or observational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Experimental research offers the most plausibly unbiased estimates, but experiments are frequently infeasible because they are costly or subject to moral objections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Observational data instead are becoming increasing available due to technological advancements in the design of sensor and hardware devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Our focus is on causal inference in observational studies, and specifically on the design of efficient algorithmic techniques to estimate counterfactuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' The C-inf approaches can be broadly classified into three categories: 1) the unconfoundedness (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' [14, 32]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' 2) the synthetic control (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' [1, 2, 11]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' and 3) the matrix completion (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' [3, 4, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Matrix completion methods build upon the foundation works of [7,9,29]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Perhaps unexpectedly, all three methods heavily rely on mathematical, statistical, and ultimately algorithmic concepts with very deep roots ∗e-mail: ac3827@columbia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='edu †e-mail: flatoyer@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='com 1 in information theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Our work is positioned within the third line of work that mathematically resembles the matrix completion (MC) problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Along the same lines, our work extends significantly the analysis developed in the companion paper [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Therein, we obtained the exact explicit typical worst case C-inf phase transitions (PT), and further showed that these phase transitions are achievable by the symmetric targeted C-inf matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' In the present paper, we consider a generic asymmetric context, to deal with the situation that C-inf matrices are not necessarily always symmetric in real applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' This allows us improving upon the results from [10] in certain scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' We build further upon the RDT-FPT synergistic mechanisms considered in [10], and precisely characterize the corresponding asymmetric PTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' We also uncover a doubling low-rankness phenomenon, which means that exactly two times larger low rankness is allowed in asymmetric scenarios compared to the symmetric worst case ones of [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' 2 Causal inference mathematical setup In this section, we revisit the explicit causal inference (C-inf) ↔ matrix completion (MC) connection, established in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Therein, we have discussed the connection between low rank recovery (LRR), matrix completion (MC), and the causal inference (C-inf).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Mathematically speaking, one has that the MC is a special case of the LRR and the C-inf is a special case of the MC itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Consequently, the mathematical models that describe the LRR problems can be used to describe the MC and ultimately the C-inf ones as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Below we present the C-inf mathematical setup developed through such a connection in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' We start with a low rank matrix Xsol ∈ Rn×n with the singular value decomposition (SVD) X = UΣV T , (1) where σ(X) ≜ diag(Σ) and U T U = In×n and V T V = In×n, (2) with In×n being the n × n identity matrix and diag(·) being the operator that creates a column vector of the diagonal elements of its matrix argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' We then define ℓ∗ p(X) to be the so-called ℓp (quasi) norm of σ(X) (the vector of the singular values of X), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' ℓ∗ p(X) ≜ ℓp(σ(X)), p ∈ R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (3) The following limiting ℓp(·) connections are important as well ℓ∗ 0(Xsol) ≜ ℓ0(σ(Xsol)) = ∥σ(Xsol)∥0 = lim p−→0 ∥σ(Xsol)∥p = lim p−→0 ℓp(σ(Xsol)) = lim p−→0 ℓ∗ p(Xsol).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (4) Moreover, we also define the so-called block masking matrix M as (see Figure 1 as well) M matrix in block causal inference (C-inf): M ≜ M (l) ≜ 1n×11T n×1 − I(l)(I(l))T 1n×11T n×1I(l)(I(l))T and I(l) ≜ � 0l×(n−l) I(n−l)×(n−l) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (5) One then has the following two optimization problems that are at the heart of the C-inf ↔ MC connection ℓ∗ 0-minimization (C-inf – MMT) min X ℓ∗ 0(X) subject to Y = M ◦ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (6) ℓ∗ 1-minimization (C-inf – MMT) min X ℓ∗ 1(X) subject to Y = M ◦ X, (7) 2 M = 1 Matrix M – block causal inference (C-inf) 1 0 1 0 and 1 grouped in blocks l × l block of all 1s l × (n − l) block of all 1s (n − l) × (n − l) block of all 0s (n − l) × l block of all 1s Figure 1: Matrix M ≜ M (l) – block causal inference (C-inf) setup where ◦ stands for the component-wise multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Namely, keeping in mind that ℓ∗ 0(X) effectively counts the number of the nonzero singular values of X, the optimization problem in (6) is exactly the recovery of the C-inf targeted low rank matrix X from the linear observations Y obtained through a masking via M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Moreover, the problem in (6) (with a generic M) is a standard matrix completion setup which on the other hand is a special case of the LRR problems (expressed in the “masking matrix terminology” (MMT)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' On the other hand, the optimization problem in (7) is the tightest convex relaxation heuristic typically utilized in the matrix completion literature for solving NP-hard problem (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' For more on the origin of these two problems and their connection within the LRR and MC context we refer to the introductory LRR/MC papers [7,30,36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' More on their importance and different related algorithmic considerations can be found in many papers that followed (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' [8,16–22,31]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Here though, we would particularly like to point out reference [4] where the very same C-inf context was considered and the very same C-inf ↔ MC connection recognized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Considerations from [4] are in fact especially convenient to properly understand in what C-inf contexts the block structure of the matrix M might appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' To see that one can connect it to the so-called counterfactuals and the units/treatments terminology employed in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' First we note that M can be alternatively defined as Mi,j = � 1, (i, j)-th element of Xsol is observed 0, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (8) It is then rather clear that ones in M allow reading out the corresponding elements of Xsol while zeros block (mask) them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Then the context of [4] is roughly as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' One first assumes that the matrix X contains observations about a certain set of, say, n units (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' individuals, subpopulations, and geographic regions) over a period of say, n, time instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' After that the rows of X are allocated to the units and the columns to the time instances and one would like to estimate the effects that a certain treatment may have on the treated units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' A subset of the units (say those that correspond to the rows i > l) is then at time l exposed to an irreversible treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Examples of treatments include health therapies, socio-economic policies, and taxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' To ensure an appropriate assessment of the resulting treatment effects, in addition to having the values of X after the treatment, one would need to have the access to the so-called counterfactuals – the values of the treated units – had the treatment not been applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Relating back to the matrix completion terminology, one would basically need to estimate (a presumably low rank) X while not having access to its 3 portion covered by the block-mask M = M (l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' In other words, one would need to solve (6) with M = M (l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' The above describes the C-inf via counterfactuals and the underlying role of matrix M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Moreover, if one views things in the time domain, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' if the columns of M represent time axis, then the observations in ceratin rows will not be available after a fixed point in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' In the block scenario this point is fixed across the affected rows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' However, it does not necessarily need to be fixed (for more in this direction we refer to [2] (in particular, the California tobacco example), [49] (in particular, the latent factor modeling in the context of the simultaneous/staggered treatment adoption), and to [5, 6, 34] (in particular, the health care applications) as excellent references for understanding the need of various C-inf scenarios).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' As this and [10] are the introductory papers, where we present the overall methodology, we selected the block causal inference scenario as probably the most representative and well-known one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' In some of our companion papers we will show how the methodology that we are introducing here can be utilized to handle other C-inf scenarios as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' 3 ℓ∗ 0 − ℓ∗ 1 equivalence As mentioned earlier, solving the generic LRR (and consequently the C-inf as its a special case) might be difficult due to a highly non-convex objective function in (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Various heuristics can be employed depend- ing on the practical scenarios that one can face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' In the mathematically most challenging so-called linear regime, the above mentioned ℓ∗ 1-minimization relaxation heuristic (often called nuclear norm minimization) is typically viewed as the best known provably polynomial one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' We adopt the same view in what follows and take it as a current benchmark for the algorithmic handling of the C-inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' As mentioned above, a rather remarkable feature of this heuristic is that sometimes it can actually solve the underlying problems exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' When that happens we say that the following ℓ∗ 0 − ℓ∗ 1-equivalence phenomenon occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' ℓ∗ 0 − ℓ∗ 1-equivalence (C-inf): ℓ∗ 0 ⇐⇒ ℓ∗ 1 Let Xsol be the solution of (6) and let ˆX be a solution of (7) and set RMSE ≜ ∥vec( ˆX) − vec(Xsol)∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' If and only if ( ˆX = Xsol and RMSE = 0) then (ℓ∗ 0 − minimization ⇐⇒ ℓ∗ 1 − minimization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (9) The above basically means that when the ℓ∗ 0 − ℓ∗ 1-equivalence happens the optimization problems in (6) and (7) are equivalent and as such replaceable by each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' We denote such a phenomenon as ℓ∗ 0 ⇐⇒ ℓ∗ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' That would, of course, be an ideal scenario where it would be basically possible to replace the non-convex optimization problem with the convex one without losing anything in terms of the accuracy of the obtained solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Since the mere existence of such a phenomenon is rather remarkable we will in this paper be interested in uncovering the underlying intricacies that enable for it ro happen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Moreover, as it will turn out that its occurrence is not an anomaly but rather a consequence of a generic property, we will then raise the bar accordingly and attempt to provide not only the proof of its existence but also its a complete analytical characterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' This will include a full characterization as to how often and in what scenarios it might happen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' To do so we will combine the Random Duality Theory (RDT) tools from [37–44] and several advanced sophisticated probabilistic concepts that we will introduce along the way in the sections that follow below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' We start with some algebraic ℓ∗ 0 − ℓ∗ 1-equivalence preliminaries which are borrowed from the RDT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' The first one is a generic LRR ℓ∗ 0−ℓ∗ 1-equivalence result (the result is basically an adaptation of the corresponding CS equivalence condition from [39–41] (similar adaptation can also be found in [24])).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' ( [10] ℓ∗ 0 − ℓ∗ 1-equivalence condition (LRR) – general X) Consider a ¯U ∈ Rn×k such that ¯U T ¯U = Ik×k and a ¯V ∈ Rn×k such that ¯V T ¯V = Ik×k and a rank− k matrix Xsol = X ∈ Rn×n with all of its columns belonging to the span of ¯U and all of its rows belonging to the span of ¯V T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Also, let the orthogonal spans ¯U ⊥ ∈ Rn×(n−k) and ¯V ⊥ ∈ Rn×(n−k) be such that U ≜ � ¯U ¯U ⊥� and V ≜ � ¯V ¯V ⊥� and U T U ≜ � ¯U ¯U ⊥�T � ¯U ¯U ⊥� = In×n and V T V ≜ � ¯V ¯V ⊥�T � ¯V ¯V ⊥� = In×n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (10) 4 For a given matrix A ∈ Rm×n2 (m ≤ n2) assume that y = Avec(X) = Avec(Xsol) ∈ Rm and let ˆX be the solution of (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' If (∀W ∈ Rn×n|Avec(W) = 0m×1, W ̸= 0n×n) − tr ( ¯U T W ¯V ) < ℓ∗ 1(( ¯U ⊥)T W ¯V ⊥), (11) then ℓ∗ 0 ⇐⇒ ℓ∗ 1 and RMSE = ∥vec( ˆX) − vec(Xsol)∥2 = 0, (12) and the solutions of (6) and (7) coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Moreover, if (∃W ∈ Rn×n|Avec(W) = 0m×1, W ̸= 0n×n) − tr ( ¯U T W ¯V ) ≥ ℓ∗ 1(( ¯U ⊥)T W ¯V ⊥), (13) then there is an X from the above set of matrices with columns belonging to the span of ¯U and rows belonging to the span of ¯V such that the solutions of (6) and (7) are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' The proof is a trivial adaptation of the proof for symmetric matrices given in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Continuing further in the spirit of the RDT the following corollary is a matrix completion specific variant of the above theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' ( [10] ℓ∗ 0 − ℓ∗ 1-equivalence condition via masking matrix (MC/C-inf) – general X) Assume the setup of Theorem 1 with Xsol being the unique solution of (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Let the masking matrix M ∈ Rn×n have m ones and (n2−m) zeros and let A be generated via M, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' let A be the matrix obtained after removing all the zero rows from diag−1(vec(M))In2×n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' If and only if min W,W T W=1,M◦W=0n×n tr ( ¯U T W ¯V ) + ℓ∗ 1(( ¯U ⊥)T W ¯V ⊥) ≥ 0, (14) then ℓ∗ 0 ⇐⇒ ℓ∗ 1 and RMSE = ∥vec( ˆX) − vec(Xsol)∥2 = 0, (15) and the solutions of (6) and (7) coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Finally, the following spectral oriented corollary was proven in [10] as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' ( [10] ℓ∗ 0−ℓ∗ 1-equivalence condition via mask-modified bases spectra (C-inf) – general X) Assume the setup of Theorem 1 with k ≤ l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Let M ≜ M (l) ∈ Rn×n and I(l) ∈ Rn×(n−l) be as defined in (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Set ΛV ≜ ((I(l))T ¯V ⊥)−1(I(l))T ¯V ΛU ≜ ((I(l))T ¯U ⊥)−1(I(l))T ¯U Q = � (I(l))T ¯V ⊥( ¯V ⊥)T I(l)�−1 − I Q⊥ 1 = � (I(l))T ¯U ⊥( ¯U ⊥)T I(l)�−1 − I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (16) C-inf perfectly succeeds: ℓ∗ 0 ⇐⇒ ℓ∗ 1 and RMSE = ∥vec( ˆX) − vec(Xsol)∥2 = 0 If and only if λmax(ΛT V ΛV ΛT UΛU) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (17) Moreover, if λmax (Q) λmax � Q⊥ 1 � ≤ 1, (18) then again ℓ∗ 0 ⇐⇒ ℓ∗ 1 and RMSE = ∥vec( ˆX − vec(Xsol)∥2 = 0 and the C-inf perfectly succeeds as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' 5 Since we will be working in the mathematically most challenging large n linear regime, we find it useful to introduce the following large dimensional scalings β ≜ lim n→∞ k n and η ≜ lim n→∞ l n and α ≜ lim n→∞ m n2 = lim n→∞ n2 − (n − l)2 n2 = 1 − (1 − η)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (19) The key highlight result of [10] is the following theorem obtained through an analysis that relied on the above corollary and a combination of the Random duality theory (RDT) and Free probability theory (FPT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' It basically establishes the worst case phase-transition (PT) that ℓ∗ 1, tightest convex relaxation heuristic, exhibits when used for solving C-inf in a typical statistical scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (ℓ∗ 1 – phase transition – C-inf (typical worst case)) Consider a rank-k matrix Xsol = X ∈ Rn×n with the Haar distributed ( not necessarily independent) bases of its orthogonal row and column spans ¯U ⊥ ∈ Rn×(n−k) and ¯V ⊥ ∈ Rn×(n−k) (XT sol ¯U ⊥ = Xsol ¯V ⊥ = 0n×(n−k)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Let M ≜ M (l) ∈ Rn×n be as defined in (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Assume a large n linear regime with β ≜ limn→∞ k n and η ≜ limn→∞ l n and let βwc and η satisfy the following C-inf ℓ∗ 1 worst case phase transition (PT) characterization ξ(wc) η (β) ≜ β − 1 2 + � η − η2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (20) If and only if β ≤ βwc lim n→∞ P(ℓ∗ 0 ⇐⇒ ℓ∗ 1) = lim n→∞ P(RMSE = 0) = 1, (21) and the solutions of (6) and (7) coincide with overwhelming probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' The results obtained based on the above theorem are shown in Figure 2, where one can clearly see that the phase transition curve splits the entire (β, η) region into two subregions: 1) the first one (below (or to the right of) the curve) where the ℓ∗ 0 − ℓ∗ 1-equivalence phenomenon occurs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' and 2) the second one (above (or to the left of) the curve) where the ℓ∗ 0 − ℓ∗ 1-equivalence is lacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' This means that one can recover Xsol masked by M as in (6) via the ℓ∗ 1 heuristic from (7) with the residual mean square error (RMSE) equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' In other words, for the system parameters (β, η) that belong to the subregion below the curve one has a perfect recovery with Xsol and ˆX (the respective solutions of (6) and (7)) being equal to each other and consequently with RMSE = ∥vec( ˆX) − vec(Xsol)∥2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' On the other hand, in the subregion above the curve, the ℓ∗ 1 heuristic fails and one can even find an Xsol for which RMSE → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' 4 Analysis of the ℓ∗ 0 −ℓ∗ 1-equivalence – typical asymmetric scenario In this section we consider when the conditions given in Corollary 2 are met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' As in [10], we will be working in a “typical” statistical context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' On the other hand, differently from [10], instead of focusing on the worst case (symmetric) scenario we here consider a typical asymmetric scenario setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Practically this means two things: 1) as in [10], both ¯V and ¯U will be assumed as Haar distributed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' and 2) differently from Theorem 2 and [10], ¯V and ¯U will now be assumed as independent of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' In a way one can view the worst case scenario from [10] as an extreme where ¯V and ¯U are “not independent at all” (or, in other words, equal to each other).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Along similar lines, one can then view the scenario that we will consider here as another extreme where ¯V and ¯U are “not dependent at all” (or, in other words, completely independent).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' In situations where no particular structure of a low rank nonsymmetric Xsol is favored over any other this one would naturally be a most reasonable choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' In other words, it is not only an extreme case, but actually the one that typically might most faithfully describe the performance of the underlying C-inf heuristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='1 Free probability theory (FPT) – preliminaries Below we provide a short preview of the most basic FPT concepts needed for our analysis (we refer to our companion paper [10] for a more detail treatment).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' As is by now well known, the work od Dan Voiculescu 6 η 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='95 1 β 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='5 (β, η) region of success/failure — C-inf ℓ∗1 PT RMSE −→ ∞, ℓ∗1 fails RMSE = 0, ℓ∗1 succeeds ℓ∗1’s PT: ξ(wc) η (β) = β − 1 2 + �η − η2 = 0 Figure 2: Causal inference (C-inf) – typical worst case ℓ∗ 1 phase transition on group theories (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' [46–48]) established the foundations of the FPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' As the practical importance of FPT became immediately evident a substantial interest for further studying was generated and, in the years that followed, quite a few nice results appeared that made the whole theory more approachable and ultimately presentable in an easily understandable way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Along the same lines, we follow into the footsteps of [10], leave all the abstractions out and focus on the FPT’s key practically applicable components (for further details see also, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' [12,23,35,45–48]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='1 Basics of FPT – random matrix variables We assume large n linear regime and consider two symmetric matrices A = AT ∈ Rn×n and B = BT ∈ Rn×n with Haar distributed eigenspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' We also assume that their individual respective spectral laws are fA(·) and fB(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Three different transforms of these spectral densities will be needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' We start with the first one, the so-called Stieltjes (or G) transform G(z) ≜ � If f(x) z − xdx, z ∈ C \\ If, (22) where If is the domain of f(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' The following inverse relation is also well known f(x) = lim ǫ→0+ G(x − iǫ) − G(x + iǫ) 2iπ or f(x) = − lim ǫ→0+ imag(G(x + iǫ)) π .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (23) For the above to hold it makes things easier to implicitly assume that f(x) is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' We will, however, utilize it even in discrete (or semi-discrete) scenarios since the obvious asymptotic translation to continuity would make it fully rigorous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' A bit later though, when we see some concrete examples where things of this nature may appear, we will say a few more words and explain more thoroughly what exactly can be discrete and how one can deal with such a discreteness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' In the meantime we proceed with general principles not necessarily worrying about all the underlying technicalities that may appear in scenarios deviating from the typically seen ones and potentially requiring additional separate addressing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' To that end we continue by considering the R(·)- and S(·)-transforms that satisfy the following R(G(z)) + 1 G(z) = z, (24) 7 and S(z) = 1 R(zS(z)) and R(z) = 1 S(zR(z)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (25) Let fA(·) and fB(·) be the spectral distributions of A and B and let RA(z)/SA(z) and RB(z)/SB(z) be their associated R(·)-/S(·)-transforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' One then has the following Key Voiculescu’s FPT concepts [46, 47]: C = A + B =⇒ RC(z) = RA(z) + RB(z) C = AB =⇒ SC(z) = SA(z)SB(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (26) Now it is relatively easy to see that (22)-(26) are sufficient to determine the spectral distribution of the sum or the product of two independent matrices with given spectral densities and the Haar distributed bases of eigenspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' The above is of course a generic principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' It can be applied pretty much always as long as one has access to the statistics of the underlying matrices A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' In the following section we will raise the bar a bit higher and show that in the C-inf context one can use all of the above in such a manner that eventually all the quantities of interest are explicitly determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='2 Spectral preliminaries We start by recalling on Q from (16) and introducing Q1 Q1 ≜ ΛT V ΛV Q ≜ � (I(l))T ¯V ⊥( ¯V ⊥)T I(l)�−1 − I Sp(Q1) ⇐⇒\\0 Sp(Q), (27) where Sp(·) stands for the spectrum of the matrix argument and ⇐⇒\\0 means the equivalence of the parts of the spectra outside the zero eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' It is rather obvious that it will then be sufficient to handle the spectrum of D ≜ (I(l))T ¯V ⊥( ¯V ⊥)T I(l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (28) Consider Haar distributed ¯U ⊥ D ∈ Rn×(n−l) with ( ¯U ⊥ D)T ¯U ⊥ D = I(n−l)×(n−l) and let UD = � ¯UD ¯U ⊥ D � with U T DUD = In×n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (29) Also, we assume that ¯U ⊥ D (and UD) are independent of ¯V ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' After setting ¯D ≜ (I(l))T U T D ¯V ⊥( ¯V ⊥)T UDI(l), (30) we have that the spectra of D and ¯D are statistically identical, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Sp(D) ≜ Sp((I(l))T ¯V ⊥( ¯V ⊥)T I(l)) ⇐⇒P Sp((I(l))T U T D ¯V ⊥( ¯V ⊥)T UDI(l)) ≜ Sp( ¯D), (31) where ⇐⇒P stands for the statistical/probabilistic equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Two facts enable the above statistical iden- tity: 1) the spectrum of the projector ¯V ⊥( ¯V ⊥)T does not change under pre- and post-unitary multiplications on both sides;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' and 2) the Haar structure of ¯V ⊥ remains preserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Modulo zero eigenvalues, we then further have Sp((I(l))T U T D ¯V ⊥( ¯V ⊥)T UDI(l)) ⇐⇒P\\0 Sp( ¯V ⊥( ¯V ⊥)T UDI(l)(I(l))T U T D) ⇐⇒ Sp( ¯V ⊥( ¯V ⊥)T ¯U ⊥ D( ¯U ⊥ D)T ), (32) where, similarly as above, ⇐⇒P\\0 stands for the statistical/probabilistic equivalence in the part of the spectrum outside the zero eignevalues (introduced due to the non-square underlying matrices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Clearly, the 8 key object of our interest below will be ˜D ≜ ¯V ⊥( ¯V ⊥)T ¯U ⊥ D( ¯U ⊥ D)T , (33) where both ¯V ⊥ and ¯U ⊥ D are Haar distributed and independent of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' After setting V ≜ ¯V ⊥( ¯V ⊥)T U ≜ ¯U ⊥ D( ¯U ⊥ D)T , (34) we easily have from (33) ˜D ≜ VU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (35) The following lemma proven in [10] characterizes the G-transform of ˜D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='. Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' ( [10]) Let ¯V ⊥ ∈ Rn×(n−k) and ¯U ⊥ D ∈ Rn×(n−k) be Haar distributed unitary bases of (n − k)- dimensional subspaces of Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Let V and U be as in (34) and ˜D as in (35), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' V ≜ ¯V ⊥( ¯V ⊥)T U ≜ ¯U ⊥ D( ¯U ⊥ D)T ˜D ≜ VU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (36) In the large n linear regime, with β ≜ limn→∞ k n, the G-transform of the spectral density of ˜D, f ˜ D(·), is G± ˜ D(z) = z − (β + η) ± � (z − (β + η))2 + 4βη(z − 1) 2(z2 − z) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (37) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='2 Asymmetric scenario – FPT analysis of the ℓ∗ 0 − ℓ∗ 1-equivalence As in [10], we will again rely on the free probability theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' This time though things will be a bit more complicated as we will be determining, so to say, the “joint spectrum” of λT V λV λT UλU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' In other words, based on Corollary 2 and (17), we have ℓ∗ 0 − ℓ∗ 1 − −equivalence ⇐⇒ λmax(λT V λV λT UλU) ≤ 1, (38) and consequently determining the upper edge of the “joint spectrum” of λT V λV λT UλU would be then sufficient to establish ℓ∗ 0 − ℓ∗ 1-equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' We recall that in [10] we determined only the individual spectra λT V λV and λT UλU (which in the worst case was sufficient to ultimately obtain corresponding C-inf ℓ∗ 1 PT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' While the calculations and supporting technicalities might on occasion be a bit heavy the overall methodology will be fairly similar to what we presented in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' In fact, to make things easier to follow we will try to parallel the presentation from [10] as much as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' We start by recalling on Q1 and introducing Q⊥ 1 , and Q1 Q1 ≜ λT V λV Q⊥ 1 ≜ λT UλU Q1 ≜ Q1Q⊥ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (39) We also recall on the definitions of Q and Q⊥ and introduce Q in the following way Q ≜ � (I(l))T ¯V ⊥( ¯V ⊥)T I(l)�−1 − I Q⊥ ≜ � (I(l))T ¯U ⊥( ¯U ⊥)T I(l)�−1 − I Q ≜ QQ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (40) 9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='1 The spectrum of Q1 ≜ λT V λV λT UλU – theoretical considerations Since (Q1, Q) and (Q⊥ 1 , Q⊥) are statistically identical pairs, we will, for the time being, focus on only one of them, say (Q1, Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' To that end, we first recall the statistical relations within the pairs Q1 ≜ ΛT V ΛV Q ≜ � (I(l))T ¯V ⊥( ¯V ⊥)T I(l)�−1 − I = D−1 − I Sp(Q1) ⇐⇒\\0 Sp(Q), (41) where ⇐⇒\\0 stands for the spectral equivalence outside the zeros eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' We will also find it convenient to work with the spectrum of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Later on we will make the necessary adjustments so that the results fully fit the spectrum of Q1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' To start things off we first note GQ(z) = GD−1(z + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (42) To see that (42) indeed holds, we first observe that the spectral functions of Q and D−1, fQ(x) and fD−1(x), can be connected in the following way fQ(x) = fD−1(x + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (43) Then from (22) we have GQ(z) = � fQ(x) z − x dx = � fD−1(x + 1) z − x dx = � fD−1(x + 1) z + 1 − (x + 1)dx = � fD−1(x) z + 1 − xdx = GD−1(z + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (44) From (41) and (42) we also have RQ(z) = RD−1(z) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (45) Namely, (24) first gives RQ(GQ(z)) = z − 1 GQ(z), (46) and then RQ(z) = G−1 Q (z) − 1 z ⇐⇒ z = GQ � RQ(z) + 1 z � RD−1(z) = G−1 D−1(z) − 1 z ⇐⇒ z = GD−1 � RD−1(z) + 1 z � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (47) Combining (42) and (47) we obtain GQ � RQ(z) + 1 z � = GD−1 � RD−1(z) + 1 z � ⇐⇒ GQ � RQ(z) + 1 z � = GQ � RD−1(z) + 1 z − 1 � ⇐⇒ RQ(z) + 1 z = RD−1(z) + 1 z − 1 ⇐⇒ RQ(z) = RD−1(z) − 1, (48) which is exactly (46).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' From (25) we further have SQ(z) = 1 RQ(zSQ(z)) = 1 RD−1(zSQ(z)) − 1, (49) and RD−1(zSQ(z)) = 1 SQ(z) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (50) 10 Relying further on (25) we also have RD−1(zSQ(z)) = 1 SD−1(zSQ(z)RD−1(zSQ(z)) = 1 SD−1 � zSQ(z) � 1 SQ(z) + 1 �� = 1 SD−1 (z + zSQ(z)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (51) A combination of (50) and (51) gives a way to connect the S-transforms of D−1 and Q 1 SD−1 (z + zSQ(z)) = 1 SQ(z) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (52) From (40) and the key FPT principles (26) we find SQ(z) = SQ(z)SQ⊥(z) = (SQ(z))2, (53) where we used the fact that Q and Q⊥ are statistically identical and as such have the same S-transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' One can now rewrite (52) with z → zRQ(z) and utilize (25) to obtain 1 SD−1 (zRQ(z)+zRQ(z)SQ(zRQ(z))) = 1 SQ(zRQ(z)) + 1 ⇐⇒ 1 SD−1 � zRQ(z)+zRQ(z)√ SQ(zRQ(z)) � = 1 √ SQ(zRQ(z)) + 1 ⇐⇒ 1 SD−1 � zRQ(z)+z√ RQ(z) � = � RQ(z) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (54) Replacing z → GQ(z), (54) can be further rewritten 1 SD−1 � zRQ(z)+z√ RQ(z) � = � RQ(z) + 1 ⇐⇒ 1 SD−1 � GQ(z)RQ(GQ(z))+GQ(z)√ RQ(GQ(z)) � = � RQ(GQ(z)) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (55) From (24) we find RQ(GQ(z)) + 1 GQ(z) = z ⇐⇒ GQ(z)RQ(GQ(z)) = zGQ(z) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (56) Combining further (55) and (56) we also have 1 SD−1 � GQ(z)RQ(GQ(z))+GQ(z)√ RQ(GQ(z)) � = � RQ(GQ(z)) + 1 ⇐⇒ 1 SD−1 � zGQ(z)−1+√ GQ(z)√ zGQ(z)−1 � = � zGQ(z)−1 GQ(z) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (57) As in [12] one has for the connection between the S-transforms of the matrix and its inverse SD(z) = 1 SD−1(−1 − z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (58) Keeping (58) in mind, one can rewrite (57) in the following way 1 SD−1 � GQ(z)RQ(GQ(z))+GQ(z)√ RQ(GQ(z)) � = � RQ(GQ(z)) + 1 ⇐⇒ SD � −zGQ(z) − � GQ(z) � zGQ(z) − 1 � = � zGQ(z)−1 GQ(z) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (59) We will make a SD(z) − GD(z) connection below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' however, before doing so, we will need to make certain adjustments in the GD(z) transform itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' i) Adjusting GD(z) for the difference between ˜D and ¯D 11 We now briefly recall on the connection between ˜D, ¯D, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' First, from (32) and (19) we have ˜D = ¯V ⊥( ¯V ⊥)T ¯U ⊥ D( ¯U ⊥ D)T ¯D = ( ¯U ⊥ D)T ¯V ⊥( ¯V ⊥)T ¯U ⊥ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (60) The spectra of ˜D and ¯D are modulo scalings practically identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Since ¯D has all the eigenvalues that ˜D has with l = ηn zero eigenvalues removed one can connect their G-transforms in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' G ¯ D(z) = 1 1 − η � G ˜ D(z) − η z � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (61) To see that (61) is indeed true we first connect the spectral pdfs of ˜D and ¯D f ¯ D(x) = 1 1 − η (f ˜ D(x) − ηδ(x)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (62) Then from (22) we have G ¯ D(x) = � f ¯ D(x) z − x dx = 1 1 − η �� f ˜ D(x) z − x dx − η � δ(x) z − xdx � = 1 1 − η � G ˜ D(z) − η z � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (63) Connecting beginning and end in (63) we obtain (61).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' ii) Adjusting GD(z) for the difference between Q1 and Q We recall that Q1 has the same eigenvalues as Q minus n − l − k zero eigenvalues (when n − l − k ≤ 0 that means that Q1 has all the eigenvalues of Q plus |n−l−k| zero eigenvalues).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' To account for this difference we find it useful to introduce a matrix D1 obtained by removing/adding |n − (l + k)| ones into the spectrum of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' As these added ones are inversion invariant they remain in the spectrum after the inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' This means that after the inversion of D1 and subtraction of the identity matrix they become zeros and basically have an effect on Q as if |n − (l + k)| zeros were added or removed which is exactly what we need to account for the difference between Q1 and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' To put everything in the right mathematical context, let D1 be a matrix with the Haar distributed eigen-space basis and the spectral function defined int he following way fD1 = 1 1 − η − (1 − (β + η)) (f ˜ D − ηδ(x) − (1 − (β + η))δ(x − 1)) , (64) where we have now taken into the account the above mentioned adjusting between ˜D and ¯D ( ¯D and D have identical spectral functions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Utilizing again (22) we similarly to (63) have GD1(z) = 1 1 − η − (1 − β + η) � G ˜ D(z) − η z − 1 − (β + η) z − 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (65) Recalling once again on (24) we have RD1(GD1(z)) + 1 GD1(z) = z ⇐⇒ RD1(z) + 1 z = G−1 D1(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (66) After taking z → RD1(z) + 1 z we can rewrite (65) as GD1 � RD1(z) + 1 z � = 1 β � G ˜ D � RD1(z) + 1 z � − η RD1(z) + 1 z − 1 − (β + η) RD1(z) + 1 z − 1 � , (67) and after utilizing (66) z = 1 β � G ˜ D � RD1(z) + 1 z � − η RD1(z) + 1 z − 1 − (β + η) RD1(z) + 1 z − 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (68) 12 After another replacement, z → zSD1(z), (68) becomes zSD1(z) = 1 β � G ˜ D � RD1(zSD1(z)) + 1 zSD1(z) � − η RD1(zSD1(z)) + 1 zSD1 (z) − 1 − (β + η) RD1(zSD1(z)) + 1 zSD1 (z) − 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (69) Using (25) we from (69) further find zSD1(z) = 1 β � G ˜ D � 1 SD1(z) + 1 zSD1(z) � − η 1 SD1 (z) + 1 zSD1 (z) − 1 − (β + η) 1 SD1(z) + 1 zSD1(z) − 1 � , (70) and zSD1(z) = 1 β � G ˜ D � z + 1 zSD1(z) � − η z+1 zSD1 (z) − 1 − (β + η) z+1 zSD1(z) − 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (71) Taking D → D1 in (41) and correspondingly denoting Q → Q1 and Q → Q1, one can repeat all the steps between (41) and (59) to arrive at the following SD1 � −zGQ1(z) − � GQ1(z) � zGQ1(z) − 1 � = � zGQ1(z) − 1 GQ1(z) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (72) Setting z1(z) ≜ −zGQ1(z) − � GQ1(z) � zGQ1(z) − 1 y(z) ≜ z1(z) + 1 z1(z)SD1(z1(z)), (73) one has from (72) SD1(z1(z)) = � zGQ1(z) − 1 GQ1(z) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (74) After taking z → z1 and rewriting (71) one finally obtains z1(z) + 1 y(z) = 1 β � G ˜ D(y(z)) − η y(z) − 1 − (β + η) y(z) − 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (75) We summarize the above results in the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Let Q1 be as in (39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Then its G-transform, GQ1(z), satisfies z1(z) + 1 y(z) = 1 β � G ˜ D(y(z)) − η y(z) − 1 − (β + η) y(z) − 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (76) with z1(z) and y(z) as in (73), SD1(z1(z)) as in (74), and G ¯ D(y(z)) as in Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' A combination of Lemma 1 (where G ˜ D(·) is explicitly given) and (73)-(75) is then sufficient to determine GQ1(z) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Utilizing (23) then enables one to fully determine the spectral distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' This is a generic procedure that in principle works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Below we will move things a step further and provide a more detailed analysis of the edges of the spectrum as they play a critical role in the ℓ∗ 0 − ℓ∗ 1-equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' It will turn out that one can provide their a sufficiently explicit characterization so that the explicit closed form for the corresponding C-inf phase transitions can again be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Later on we will return to the above described procedure for determining the entire spectrum of Q1 and show what type of results such a procedure actually produces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' 13 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='2 Explicit characterization of Q1’s spectral edges As we have seen earlier, the upper edge of the spectrum of Q (or Q1), λmax(Q) = λmax(Q1) is directly related to the success of the ℓ∗ 1-minimization heuristic in causal inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' More precisely, as Corollary 2 states, one will have the ℓ∗ 0 − ℓ∗ 1-equivalence if and only if λmax(Q) = λmax(Q1) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Clearly, an explicit characterization of λmax(Q) = λmax(Q1) will be sufficient to explicitly characterize the ℓ∗ 0 − ℓ∗ 1-equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' That will then be enough to conclude when ℓ∗ 1 can be used reliable to handle the casual inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' To provide an explicit characterization of λmax(Q) = λmax(Q1) ≤ 1 we rely on the results that we presented in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' We start by observing that the spectral function of Q1, fQ1(x), can be obtained by utilizing (23) and the above discussed GQ1(z) transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Moreover, at the edge of the spectrum GQ1(z) should be real (the edge of the spectrum is actually the breaking point where the GQ1(z) becomes complex, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' starts having a nonzero imaginary part).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' That basically means that at the edge of the spectrum one should have (76) satisfied for a real GQ1(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Moreover, since our targeted edge of the spectrum is one that means that (76) needs to be satisfied for a real GQ1(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Rewriting (73)-(75) for z = 1 gives z1(1) = −GQ1(1) − � GQ1(1) � GQ1(1) − 1 y(1) ≜ z1(1) + 1 z1(1)SD1(z1(1)), (77) and SD1(z1(1)) = � GQ1(1) − 1 GQ1(1) + 1 = − z1(1) GQ1(1), (78) and z1(1) + 1 y(1) = 1 β � G ˜ D(y(1)) − η y(1) − 1 − (β + η) y(1) − 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (79) From (77) one further finds GQ1(1) = − (z1(1))2 1 + 2z1(1) y(1) = −(z1(1) + 1)GQ1(1) (z1(1))2 = z1(1) + 1 1 + 2z1(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (80) The second equality then also gives z1(1) = y(1) − 1 1 − 2y(1), (81) and z1(1) + 1 = y(1) 2y(1) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (82) Plugging (82) into (79) one has 1 2y(1) − 1 = 1 β � G ˜ D(y(1)) − η y(1) − 1 − (β + η) y(1) − 1 � , (83) or ζ1(y) ≜ − 1 2y − 1 + 1 β � G ˜ D(y) − η y − 1 − (β + η) y − 1 � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (84) Utilizing G ˜ D(z) (with the “−” sign as the lower edge in the bulk of the spectrum of ˜D corresponds to the 14 upper edge in the spectrum of Q) from Lemma 1 we further have ζ1(y) = − 1 2y − 1 + 2β − 1 2β(y − 1) + 1 2βy(y − 1) � −β + η − � (y − (β + η))2 + 4βη(y − 1) � = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (85) and ζ1(y) = −2βy(y − 1) + y(2β − 1)(2y − 1) + (2y − 1) � −β + η − � (y − (β + η))2 + 4βη(y − 1) � 2βy(y − 1)(2y − 1) = 2(β − 1)y2 + (1 − 2β + 2η)y + β − η − (2y − 1) � (y − (β + η))2 + 4βη(y − 1) 2βy(y − 1)(2y − 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (86) Setting ζ2(y) ≜ 2(β − 1)y2 + (1 − 2β + 2η)y + β − η − (2y − 1) � (y − (β + η))2 + 4βη(y − 1) ζ(y) ≜ (2(β − 1)y2 + (1 − 2β + 2η)y + β − η)2 − ((2y − 1) � (y − (β + η))2 + 4βη(y − 1))2, (87) we easily have ζ1(y) = 0 ⇐⇒ ζ2(y) = 0 ⇐⇒ ζ(y) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (88) We therefore below focus on ζ(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' After squaring and grouping the terms we have ζ(y) = 4β(c3y4 + c2y3 + c1y2 + c0y + c00), (89) with c3 = β − 2 c2 = 5 − 2β − 2η c1 = β − 4 + 3η c0 = 1 − η c00 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (90) From (89) we then also have ζ(y) = 4βy(c3y3 + c2y2 + c1y + c0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (91) Since we are interested in an edge or a breaking point of the spectrum ζ(y) should touch zero for certain y which means that it should have a stationary point at such y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' To find such a stationary point we take the derivative d � ζ(y) 4βy � dy = 3c3y2 + 2c2y + c1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (92) Solving over y gives y = −c2 + � c2 2 − 3c1c3 3c3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (93) Setting r ≜ c2 2 − 3c1c3 = 1 + β2 + 4η2 − 2β − 2η − βη, (94) 15 we have from (93) yopt = −c2 + √r 3c3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (95) First we set ζ3(y) ≜ c3y3 + c2y2 + c1y + c0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (96) Clearly, from (91) one has ζ(y) = 4βyζ3(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (97) Then we plug the value for yopt from (95) and after a bit of algebraic transformations obtain ζ3(yopt) = −2(√r)3 − c3 2 + 3rc2 + 27c2 3c0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (98) From (90) we first have c2 = −2c3 − 1 + 2c0 c1 = c3 − 3c0 + 1, (99) and then from (94) r = c2 3 + 1 + 4c2 0 + c3 − 4c0 + c0c3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (100) Combining (98)-(100) after a bit of additional algebraic transformations gives ζ3(yopt) = −2(√r)3 + 2c3 3 − 3c3 + 6c3c2 0 + 3c2 3 + 3c3c0 + 3c0c2 3 − 2 − 24c2 0 + 12c0 + 16c3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (101) Below we show that c2 3 = −1 − 2c3 + 4c0 − 4c2 0 ⇐⇒ ζ3(yopt) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (102) We first use (102) to systematically linearize ζ3(yopt) in c3 and obtain ζ3(yopt) = −2(√r)3 + (−3c3 + 5c3c0 − 2c3c2 0 − 1 − 8c2 0 + 5c0 + 4c3 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (103) Transforming further we also have ζ3(yopt) = −2(√r)3 + (−3c3 + 5c3c0 − 2c3c2 0 − 1 − 8c2 0 + 5c0 + 4c3 0) = −2(√r)3 + (c3(c0 − 1)(3 − 2c0) + (−4c0 + 1 + 4c2 0)(c0 − 1)) = 2(√c3 √ c0 − 1)3 + (c3(c0 − 1)(3 − 2c0) + (−4c0 + 1 + 4c2 0)(c0 − 1)) = � 2(√c3)3√ c0 − 1 + (c3(3 − 2c0) + (−4c0 + 1 + 4c2 0)) � (c0 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (104) where the third equality follows after noting that with condition (102) in place r in 100) becomes c2 3 = −1 − 2c3 + 4c0 − 4c2 0 =⇒ r = −c3 + c0c3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (105) We find it useful to rewrite (104) as ζ3(yopt) = ζ(1) 3 (yopt) + ζ(2) 3 (yopt), (106) where ζ(1) 3 (yopt) ≜ 2(√c3)3√ c0 − 1 ζ(2) 3 (yopt) ≜ c3(3 − 2c0) + (−4c0 + 1 + 4c2 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (107) 16 We then look at the squared values of these quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' First we start with ζ(1) 3 (yopt) (ζ(1) 3 (yopt))2 = 4c3 3(c0 − 1), (108) and utilize the condition (102) to systematically linearize in c3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' First we remove the cubic c3 term to arrive at the following (ζ(1) 3 (yopt))2 = 4c3 3(c0 − 1) = 4c3(−1 − 2c3 + 4c0 − 4c2 0)(c0 − 1) = −8c2 3c0 + 32c3c2 0 − 16c3c3 0 − 8 − 12c3 + 32c0 − 32c2 0 − 20c3c0, (109) and apply the same procedure again to arrive at a fully linearized form (ζ(1) 3 (yopt))2 = 4(c3((−2c2 0 + c0 + 1)(2c0 − 3)) − 2(1 − c0)(2c0 − 1)2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (110) Then we turn to ζ(2) 3 (yopt) (ζ(2) 3 (yopt))2 = (c3(3 − 2c0) + (−4c0 + 1 + 4c2 0))2, (111) and again utilize the condition (102) to linearize in c3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' This time the procedure is simpler as there is only a quadratic term in c3 and there is no need to apply the procedure from above in two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Instead only one step suffices and we have (ζ(2) 3 (yopt))2 = (c3(3 − 2c0) + (−4c0 + 1 + 4c2 0))2 = c2 3(3 − 2c0)2 + (−4c0 + 1 + 4c2 0)2 + 2(−4c0 + 1 + 4c2 0)c3(3 − 2c0) = (−1 − 2c3 + 4c0 − 4c2 0)(3 − 2c0)2 + (−4c0 + 1 + 4c2 0)2 + 2(−4c0 + 1 + 4c2 0)c3(3 − 2c0) = −2c3(9 − 12c0 + 4c2 0 − (−4c0 + 1 + 4c2 0)(3 − 2c0)) − (−4c0 + 1 + 4c2 0)(8 − 8c0) = 4(c3((−2c2 0 + c0 + 1)(2c0 − 3)) − 2(1 − c0)(2c0 − 1)2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (112) Comparing (110) and (112) we have (ζ(1) 3 (yopt))2 = (ζ(2) 3 (yopt))2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (113) Now we will show that one also has (ζ(1) 3 (yopt))2 = −(ζ(2) 3 (yopt))2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' We again look at the condition in (102) and replace the values for c0 and c3 from (91) to obtain c2 3 = −1 − 2c3 + 4c0 − 4c2 0 ⇐⇒ (β − 2)2 = −1 − 2(β − 2) + 4(1 − η) − 4(1 − η)2 ⇐⇒ (β − 2)2 + 2(β − 2) + 1 = 4(1 − η) − 4(1 − η)2 ⇐⇒ (β − 2 + 1)2 = 4η(1 − η) ⇐⇒ β = 1 − 2 � η(1 − η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (114) From (107) we then also have ζ(1) 3 (yopt) ≜ 2(√c3)3√c0 − 1 = 2( � β − 2)3√−η = 2(2 − β) � η(2 − β) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (115) Similarly, we have ζ(2) 3 (yopt) ≜ c3(3 − 2c0) + (−4c0 + 1 + 4c2 0) = (β − 2)(1 + 2η) + (1 − 2η)2 = (−1 − 2 � η(1 − η))(1 + 2η) + (1 − 2η)2 = −2 � η(1 − η)(1 + 2η) − 6η + 4η2 17 ≤ −2 � η(1 − η)(1 + 2η) − 6η + 4η = −2 � η(1 − η)(1 + 2η) − 2η ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (116) A combination of (106), (107), (113), (115), and (116) finally gives (ζ(1) 3 (yopt))2 (113) = (ζ(2) 3 (yopt))2 (115),(116) ⇐⇒ ζ(1) 3 (yopt) = −ζ(2) 3 (yopt) ⇐⇒ ζ(1) 3 (yopt) + ζ(2) 3 (yopt) = 0 (106) ⇐⇒ ζ3(yopt) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (117) Moreover, a combination of (102), (114), and (117) gives β = 1 − 2 � η(1 − η) ⇐⇒ c2 3 = −1 − 2c3 + 4c0 − 4c2 0 ⇐⇒ ζ3(yopt) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (118) After combining (97) and (118) one then also has β = 1 − 2 � η(1 − η) ⇐⇒ c2 3 = −1 − 2c3 + 4c0 − 4c2 0 ⇐⇒ ζ3(yopt) = 0 ⇐⇒ ζ(yopt) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (119) From (88) one then has that for yopt ζ1(yopt) = ζ2(yopt) = 0, (120) which means that y = yopt is indeed a choice for y that ensures that functional equation used to determine GQ1(z) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Moreover, since the derivative condition is met as well, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' since ζ(yopt) = 0, one has that not only is yopt a point where ζ(y) crosses zero, it is actually a point where it touches zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' That is exactly what is needed to determine an edge of the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Since we operated using the “−” sign in the definition of G ˜ D(z) that means (based on the considerations from [10]) that we have determined the lower edge in the corresponding spectrum of ˜D (or any of ¯D and D) which after the inversion means that we have determined the upper edge in the spectrum of Q1 or Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' One can even explicitly determine yopt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' From (90), (95), (99), and (105) we obtain yopt = −c2 + √r 3c3 = −(5 − 2β − 2η) + � η(2 − β) 3(β − 2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (121) In Figure 3 we show yopt as a function of η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' The whole mechanism of “touching zero” as β decreases is shown in Figure for η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' As can be seen from the figure, for β > 1−2 � η(1 − η) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='4 ζ1(y) remains below zero one therefore can not be a part of the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' On the other hand, for β ≤ 1−2 � η(1 − η) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='4 ζ1(y) does intersect zero line which implies that one is now in the spectrum (there is y = y(11) and consequently a real GQ1(1) such that ζ1(y) = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' The borderline or the breaking point happens exactly when the ζ1(y) curve touches the zero line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' As figure indicates that happens for y = yopt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='25, exactly as the theory predicts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' We summarize the above results in the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Assume the setup of Lemmas 1 and 2 with Q1 as in (39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Then we have for the upper edge of the Q1’s spectrum β = 1 − 2 � η(1 − η) ⇐⇒ λmax(Q1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (122) Moreover, β ≤ 1 − 2 � η(1 − η) ⇐⇒ λmax(Q1) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (123) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Follows from the above discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' 18 η 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='95 1 yopt 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='5 yopt as a function of η yopt = √1−η √η−√1−η η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='9 =⇒ yopt = √1−η √η+√1−η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='25 Figure 3: yopt as a function of η 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='3 The spectrum of Q1 ≜ λT V λV λT UλU – practical evaluations Now that we have fully characterized the upper edge of the Q1’s spectrum we can return to the consideration of the entire spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Relying on the above presented machinery we can establish the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Assume the setup of Lemmas 1 and 2 with Q1 as in (39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Let GQ1(z) be the solution of the following system of equations: y(z) = � zGQ1(z) − 1 � zGQ1(z) − 1 + z � GQ1(z) G ˜ D(y(z)) = y(z) − (β + η) ± � (y(z) − (β + η))2 + 4βη(y(z) − 1) 2((y(z))2 − y(z)) 1 β � G ˜ D(y(z)) − η y(z) − 1 − (β + η) y(z) − 1 � = −( � zGQ1(z) − 1 + � GQ1(z))( � zGQ1(z) − 1 + z � GQ1(z)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (124) Then the spectral function of Q1, fQ1(x), is obtained as fQ1(x) = − lim ǫ→0+ imag(GQ1(x + iǫ)) π .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (125) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Follows from Lemma 2 through a combination of the results of Lemma 1 (where G ˜ D(·) is explicitly given) and (73)-(75).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' The following two sequences of identities are then sufficient to prove the lemma y(z) = z1(z) + 1 z1(z)SD(z1(z)) = ((−zGQ1(z) − � GQ1(z) � zGQ1(z) − 1) + 1) � GQ1(z) (−zGQ1(z) − � GQ1(z) � zGQ1(z) − 1)( � zGQ1(z) − 1 + � GQ1(z)) = � zGQ1(z) − 1 � zGQ1(z) − 1 + z � GQ1(z) , (126) 19 y 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='25 ζ1(y) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='05 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='05 ζ1(y) as a function of y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='9 β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='41 β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='4 β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='39 β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='4, η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='9 =⇒ yopt = √1−η √η+√1−η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='25 Figure 4: ζ1y as a function of y and z1(z) + 1 y(z) = SD(z1(z)) z1(z) = (−zGQ1(z) − � GQ1(z) � zGQ1(z) − 1)( � zGQ1(z) − 1 + � GQ1(z)) � GQ1(z) = ( � zGQ1(z) − 1 + z � GQ1(z))( � zGQ1(z) − 1 + � GQ1(z)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (127) In Figure 5 we show the entire spectrum of fQ1(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' We chose β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='4 and η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='9 and ran the experiments with n = 4000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' As can be seen from the figure, the obtained numerical results are in a strong agreement with what the theory predicts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='4 ℓ∗ 0 − ℓ∗ 1-equivalence via the spectral limit – asymmetric scenario From Corollary 2, (17), and (38) one has in the asymmetric scenario ℓ∗ 0 − ℓ∗ 1 − equivalence ⇐⇒ λmax(λT V λV λT UλU) ≤ 1 ⇐⇒ λmax(Q1) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (128) From (123) and (128) we finally have ℓ∗ 0 − ℓ∗ 1 − equivalence ⇐⇒ β ≤ 1 − 2 � η − η2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (129) Analogously to Theorem 2 we can now establish a precise asymmetric scenario location of the phase transition in a typical statistical context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (ℓ∗ 1 – phase transition – C-inf (typical asymmetric scenario)) Assume the setup of Theorem 2 with rank-k matrix Xsol = X ∈ Rn×n that now has Haar distributed independent bases of its orthogonal row and column spans ¯U ⊥ ∈ Rn×(n−k) and ¯V ⊥ ∈ Rn×(n−k) (XT sol ¯U ⊥ = Xsol ¯V ⊥ = 0n×(n−k)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Let M ≜ M (l) ∈ Rn×n be as defined in (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Let βac and η satisfy the following 20 x 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='9 1 fQ1(x) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='8 2 Spectral distribution fQ1(x);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' η = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' β = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='9;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' n = 4000 simulated theory fQ1(x) Bulk Figure 5: fQ1(x) – spectral function of Q1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='4 and η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='9 C-inf ℓ∗ 1 asymmetric scenario phase transition (PT) characterization ξ(ac) η (β) ≜ β − 1 + 2 � η − η2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (130) If and only if β ≤ βac lim n→∞ P(ℓ∗ 0 ⇐⇒ ℓ∗ 1) = lim n→∞ P(RMSE = 0) = 1, (131) and the solutions of (6) and (7) coincide with overwhelming probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Follows from Lemma 3 and the above discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' The results related to the use of the ℓ∗ 1-minimization heuristic for solving the causal inference problems obtained based on the above theorem are shown in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' As in the worst case scenario, the phase transition (PT) curve splits the (β, η) region into two separate subregions where the ℓ∗ 0 − ℓ∗ 1-equivalence phenomenon either occurs or fails to occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Basically, below the curve one has a perfect recovery with the residual RMSE = ∥vec( ˆX) − vec(Xsol)∥2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Contrary to that, above the curve though, there is an Xsol for which RMSE → ∞ and ℓ∗ 1 fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' The following corollary adapts the above results so that they fit the standard (α, β) representation typically used in the compressed sensing (CS), low rank recovery (LRR), and matrix completion (MC) literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (ℓ∗ 1 – phase transition – C-inf (typical asymmetric scenario;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' standard (α, β) rep- resentation)) Assume the setup of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Let m be the total number of ones in matrix M and let α ≜ limn→∞ m n2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Let β and αw satisfy the C-inf ℓ∗ 1 asymmetric scenario PT (standard (α, β) representation) ξ(wc,s) β (α) ≜ β − 1 + 2 �√ 1 − α − 1 + α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (132) 21 η 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='95 1 β 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='9 1 (η, β) region of success/failure — C-inf ℓ∗1 PT;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' asymmetric scenario worst case average case RMSE −→ ∞, ℓ∗1 fails ℓ∗1’s PT: ξ(ac) η (β) = β − 1 + 2�η − η2 = 0 RMSE = 0, ℓ∗1 succeeds Doubling low rankness: ξ(ac) η (2β) = 2ξ(wc) η (β) Figure 6: Causal inference (C-inf) – typical asymmetric scenario ℓ∗ 1 phase transition If and only if α ≥ αw lim n→∞ P(ℓ∗ 0 ⇐⇒ ℓ∗ 1) = lim n→∞ P(RMSE = 0) = 1, (133) and the solutions of (6) and (7) coincide with overwhelming probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Follows as a direct consequence of Theorem 3 after noting that m = n2 − (n − l)2 and consequently α = 1 − (1 − η)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Figure 7 shows the results obtained based on the above corollary in the standard (α, β) region format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' As usual in the PT considerations, the entire (α, β) region is split in the part below the curve where RMSE = ∥vec( ˆX) − vec(Xsol)∥2 = 0 and the part above the curve where even RMSE → ∞ is achievable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' We should point out an interesting similarity between what we observed here in the above corollary and in Figure 7 on the one side and what is known to hold in generic LRR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Namely, as Corollary 3 states (and as is emphasized in Figure 7), for the same value of α one achieves exactly two times larger β in the asymmetric case than in the worst case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' As the worst case is basically symmetric, one has that the PTs of the symmetric and the nonsymmetric scenarios are distinguished by a factor of two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Similar observation was in place when it comes to the comparison between the LRR of the symmetric and the general (nonsymmetric) matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' However, one should keep in mind a fundamental difference as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' In LRR the underlying symmetry is a priori known and can be utilized in the algorithms design whereas here it is just the choice of the worst case problem instance and is not assumed to be known to the algorithm itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Of course, given the properties of the LRR, such a choice is not necessarily very surprising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='3 Numerical results To complement the above theoretical findings and see how successful in characterizing the utilization of the ℓ∗ 1-minimization in C-inf problems they indeed are, we conducted a set of numerical experiments and show the obtained results in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' As in [10], we again observe both the PT’s existence and a solid agreement between its theoretical prediction and the results obtained through the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' In the conducted numerical experiments we chose n = 80 and η in the range [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='6, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='95].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Clearly, such fairly small matrix sizes correspond to the settings quite opposite from the ones that we used in the theoretical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Still, even though the theory is predicated on the large n assumption, it is not impossible that its conclusions remain valid for smaller values of n as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' The results form Figure 8 confirm that this is 22 α 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='95 1 β/α 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='9 1 (α, β) region of success/failure — C-inf ℓ∗1 PT;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' asymmetric scenario worst case average case RMSE = 0, ℓ∗1 succeeds RMSE −→ ∞, ℓ∗1 fails ℓ∗1’s PT: ξ(ac,s) β (α) = β − 1 + 2 �√1 − α − 1 + α = 0 ξ(ac,s) 2β (α) = 2ξ(ac,s) β (α) Figure 7: Causal inference (C-inf) – typical asymmetric scenario ℓ∗ 1 phase transition ((α, β) region) indeed the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Moreover, one can then say that the large n regime, needed for the theoretical consideration, practically may start ro kick in already for rather small (of order of a few tens!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=') values of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' This ultimately means that the presented results, although theoretical in nature, have in themselves a strong practical component as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Finally, we should also add that for larger values of n an even better agreement between the theoretical and the simulated results is to be expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' A few additional points regarding the simulations setup might be useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' First, one should emphasize, that in order to be in an agreement with the theoretical considerations, we, in all numerical experiments, considered the so-called typical behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Following further into the footsteps of the theoretical consider- ations, the presented simulations results were obtained for the square matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' As was the case in [10], all theoretical considerations can be repeated assuming the non-square scenarios as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' We, however, (as in [10]) prioritized the clarity of the presentations over simple generalizations and opted for the square sce- narios which are substantially easier to present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Also, all the simulations needed for Figure 8 were done with the singular values of the unknown targeted matrices equal to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' While we refer to [10] for a bit more complete discussion regarding such a choice, we here briefly mention that choices of this type are known to serve as the worst case examples in establishing the reversal ℓ0 − ℓ1-equivalence conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' As in [10], we also ran the simulations where the singular values were randomly chosen with results either identically matching or improving on the ones shown in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' 5 Conclusion In this paper, we have built on the mathematical Causal inference (C-inf) ↔ low-rank recovery (LRR) connection established in [10] to deal with asymmetric PTs phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' The results of [10] proved that the nuclear norm (ℓ∗ 1) minimization heuristic, when used for solving the low rank recovery C-inf problems, exhibits the so-called phase transition (PT) phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Moreover, in a typical statistical scenario, [10] characterized the exact location of the worst case PT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' This effectively meant that there are problem in- stances where the ℓ∗ 1 predicated behavior might be improved upon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Here we showed that this is indeed true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Considering an asymmetric scenario (in contrast with the symmetric worst case one from [10]) we deter- mined the underlying exact phase transitions locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Moreover, we uncovered a doubling low rankness phenomenon, which means that, throughout the entire region of allowed system parameters, matrices of exactly two times larger rank can be recovered when compared to the worst case scenario from [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Such a phenomenon also ensures that the simplicity of the worst case PTs from [10] is preserved in the asymmetric 23 η 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='9 1 β 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='9 1 (η, β) region of success/failure — ℓ∗1’s PT;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' simulated/theory ℓ∗ 1’s PT – simulated ℓ∗ 1’s PT – theory 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='9 1 Success Failure Figure 8: C-inf ℓ∗ 1’s asymmetric scenario phase transition (PT) scenarios as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Consequently, one is again able to elegantly pin down the relation between the low rankness of the target C-inf matrix and the time when the treatment is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Throughout the process of creating the theoretical phase transitions characterizations we also established several mathematical results that are of independent interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' All of our theoretical findings we supplemented with the results obtained from the corresponding numerical experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Moreover, in all cases we observed a rather overwhelming agreement between what the theory predicts and what the simulations provide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' To achieve the desired phase transition results we relied on a combination of the ideas from the Random duality theory (RDT) and the Free probability theory (FPT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' As a result, we obtained a very powerful and generic mathematical apparatus that will serve as a theoretical platform in further explorations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' As this and the companion paper [10] are of 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Now Publishers, Hanover, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' [46] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Voiculescu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Addition of certain non-commuting random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' J.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Operator Theory, 18:2223– 2235, 1987.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' [48] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Voiculescu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Limit laws for random matrices and free products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Invent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=', 104(1):201–220, 1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' [49] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Xiong and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Pelger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Large dimensional latent factor modeling with missing observations and appli- cations to causal inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Electronic copy available at: https://ssrn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='com/abstract=3465357.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' [50] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Xu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Generalized synthetic control method: Causal inference with interactive fixed effects models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Political Analysis, 25:57–76, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' A Proof of Theorem 1 As mentioned earlier, the proof of Theorem 1 is conceptually identical to the corresponding proof when matrix X is symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' A detailed proof for the symmetric matrices is given below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Before being able to present the proof we need a couple of technical lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Let C = CT ∈ Rn×n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Also let all eigenvalues of C belong to the interval [−1, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Finally, let the first k entries on the main diagonal, Ci,i, 1 ≤ i ≤ k, be larger than or equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Then the upper k × k left block of C, C1:k,1:k, is an identity matrix, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' C1:k,1:k = Ik×k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (134) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Let λmax(C) be the maximum eigenvalue of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Then λmax(C) ≜ max ∥c∥2=1 cT Cc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (135) Since by assumptions 1 ≤ Ci,i, 1 ≤ i ≤ k and λmax(C) ≤ 1 we also have for any 1 ≤ i ≤ k 1 ≤ Ci,i ≤ max ∥c∥2=1 cT Cc ≜ λmax(C) ≤ 1, (136) which implies C(i, i) = 1, 1 ≤ i ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' The proof that all other elements of C1:k,1:k are equal to zero proceeds inductively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' 1) Induction move from l = 1 to l = 2: First we look at the upper block of size 2 × 2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' at C1:2,1:2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' We then have 1 ≥ max ∥c∥2=1 cT Cc ≥ max ∥c1:2∥2=1 cT 1:2C1:2,1:2c1:2 ≥ max ∥c1:2∥2=1 (∥c1:2∥2 + 2|c1c2C1,2|) ≥ max ∥c1:2∥2=1 (1 + 2|c1c2C1,2|) ≥ 1, (137) which implies C1,2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' 2) Induction move from l to l + 1: Now we look at the upper block of size (l + 1) × (l + 1), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' at C1:l+1,1:l+1 while assuming that C1:l,1:l = Il×l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' We then have 1 ≥ max ∥c∥2=1 cT Cc ≥ max ∥c1:l+1∥2=1 cT 1:l+1C1:l+1,1:l+1c1:l+1 ≥ max ∥c1:l+1∥2=1 � ∥c1:l+1∥2 + 2|cT 1:lC1:l,l+1cl+1| � ≥ max ∥c1:l+1∥2=1 � 1 + 2|cT 1:lC1:l,l+1cl+1| � ≥ 1, (138) which implies C1:l,l+1 = 0l×1 and completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' 27 Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Assume the setup of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Then the upper k × k left block of C, C1:k,1:k, is an identity matrix and the upper k × (n − k) right block of C, C1:k,n−k+1:n is a zero matrix, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' C1:k,1:k = Ik×k C1:k,n−k+1:n = 0k×(n−k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (139) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' The first part follows by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' We now focus on the second part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Consider the following partition of matrix C C = � C1:k,1:k C1:k,n−k+1:n Cn−k+1:n,1:k Cn−k+1:n,n−k+1:n � = � Ik×k C1:k,n−k+1:n Cn−k+1:n,1:k Cn−k+1:n,n−k+1:n � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (140) Then assuming that the largest nonzero singular value of C1:k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='n−k+1:n is equal to b > 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' we have 1 ≥ max ∥c∥2=1 cT Cc ≥ max ∥c1:k∥2=a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='cn−k+1:n � cT 1:kC1:k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='1:kc1:k + 2|cT 1:kC1:k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='n−k+1:ncn−k+1:n| + cT n−k+1:nCn−k+1:n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='n−k+1:ncn−k+1:n � ≥ max ∥c1:k∥2=a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='cn−k+1:n � a2 + 2|cT 1:kC1:k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='n−k+1:ncn−k+1:n| + cT n−k+1:nCn−k+1:n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='n−k+1:ncn−k+1:n � ≥ max ∥c1:k∥2=a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='cn−k+1:n � a2 + 2|cT 1:kC1:k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='n−k+1:ncn−k+1:n| − cT n−k+1:ncn−k+1:n � ≥ max a∈[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='1] � a2 + 2ba � 1 − a2 − (1 − a2) � = max a∈[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='1] � 2a2 − 1 + 2ba � 1 − a2 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (141) where the fourth inequality follows since the minimum eigenvalue of Cn−k+1:n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='n−k+1:n is larger than or equal to the minimum eigenvalue of C which is by the lemma’s assumption larger than or equal to -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Now, we further have c ≜ 2a � 1 − a2 and 2a2 − 1 + 2ba � 1 − a2 = � 1 − c2 + bc, (142) and d( √ 1 − c2 + bc) dc = −c √ 1 − c2 + b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (143) From (143) we then easily obtain c = b √ 1 + b2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (144) A combination of (141), (142), and (144) gives 1 ≥ max ∥c∥2=1 cT Cc ≥ max a∈[0,1] � 2a2 − 1 + 2ba � 1 − a2 � = √ 1 + b2, (145) which implies b = 0 and automatically C1:k,n−k+1:n = 0k×1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Now we can consider the above mentioned theorem that adapts the general ℓ1 equivalence condition result from [39–41] to the corresponding one for the ℓ1 norm of the singular/eigenvalues (similar adaptation can also be found in [24]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (ℓ∗ 0 − ℓ∗ 1-equivalence condition (LRR) – symmetric X) Consider a ¯U ∈ Rn×k such that ¯U T ¯U = Ik×k and a rank − k a priori known to be symmetric matrix Xsol = X ∈ Rn×n with all of its columns belonging to the span of ¯U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' For concreteness, and without loss of generality, assume that X has only 28 positive nonzero eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' For a given matrix A ∈ Rm×n2 (m ≤ n2) assume that y = Avec(X) ∈ Rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' If (∀W ∈ Rn×n|Avec(W) = 0m×1, W = W T ̸= 0n×n) − tr ( ¯U T W ¯U) < ℓ∗ 1(( ¯U ⊥)T W ¯U ⊥), (146) then the solutions of (6) and (7) coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Moreover, if (∃W ∈ Rn×n|Avec(W) = 0m×1, W = W T ̸= 0n×n) − tr ( ¯U T W ¯U) ≥ ℓ∗ 1(( ¯U ⊥)T W ¯U ⊥), (147) then there is an X from the above set of the symmetric matrices with columns belonging to the span of ¯U such that the solutions of (6) and (7) are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' The proof follows literally step-by-step the proof of the corresponding theorem in [39–41] and adapts it to matrices or their singular/eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' For experts in the field this adaptation is highly likely to be viewed as trivial and certainly doesn’t need to be as detailed as we will make it to be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Nonetheless, to ensure a perfect clarity of all arguments we provide a step-by-step instructional derivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' For concreteness and without loss of generality we also assume that the eigen-decomposition of X is X = UΛU T = � ¯U ¯U ⊥� � ¯ΛX 0k×(n−k) 0(n−k)×k ¯Λ⊥ X � � ¯U ¯U ⊥�T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (148) (i) =⇒ (the if part): Following step-by-step the proof of Theorem 2 in [41], we start by assuming that ˆX is the solution of (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Then we want to show that if (146) holds then ˆX = X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' As usual, we instead of that, assume opposite, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' we assume that (146) holds but ˆX ̸= X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Then since y = Avec( ˆ X) and y = Avec(X) must hold simultaneously there must exist W such that ˆX = X + W with W ̸= 0, Avec(W) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Moreover, since ˆX is the solution of (7) one must also have ℓ∗ 1(X + W) = ℓ∗ 1( ˆX) ≤ ℓ∗ 1(X) ⇐⇒ ℓ∗ 1( � ¯U ¯U ⊥�T (X + W) � ¯U ¯U ⊥� ) ≤ ℓ∗ 1(X) =⇒ ℓ∗ 1( ¯U T (X + W) ¯U) + ℓ∗ 1(( ¯U ⊥)T (X + W) ¯U ⊥) ≤ ℓ∗ 1(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (149) The last implication follows after one trivially notes ℓ∗ 1( � ¯U ¯U ⊥�T (X + W) � ¯U ¯U ⊥� ) = max Λ∗=ΛT ∗ ∈L∗ tr (Λ∗ � ¯U ¯U ⊥�T (X + W) � ¯U ¯U ⊥� ) ≥ max Λ∗=ΛT ∗ ∈L0∗ tr (Λ∗ � ¯U ¯U ⊥�T (X + W) � ¯U ¯U ⊥� ) = ℓ∗ 1( ¯U T (X + W) ¯U) + ℓ∗ 1(( ¯U ⊥)T (X + W) ¯U ⊥), (150) where L0 ∗ ≜ � Λ∗ ∈ Rn×n|Λ∗ = ΛT ∗ , Λ∗ΛT ∗ ≤ I, Λ∗ = � Λ∗,1 0k×(n−k) 0(n−k)×k Λ∗,2 �� ⊆ � Λ∗ ∈ Rn×n|Λ∗ = ΛT ∗ , Λ∗ΛT ∗ ≤ I � ≜ L∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (151) The key observation – “Removing the absolute values”: Now, the key observation made in [41] comes into play.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Namely, one notes that the absolute values can be removed in the nonzero part and that the ℓ∗ 1(·) can be “replaced” by tr (·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Such a simple observation is the most fundamental reason for all the success of the RDT when used for the exact performance characterization of the structured objects’ recovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' From (149) we then have ℓ∗ 1( ¯U T (X + W) ¯U) + ℓ∗ 1(( ¯U ⊥)T (X + W) ¯U ⊥) ≤ ℓ∗ 1(X) =⇒ tr ( ¯U T (X + W) ¯U) + ℓ∗ 1(( ¯U ⊥)T (W) ¯U ⊥) ≤ ℓ∗ 1(X) ⇐⇒ tr ( ¯U T W ¯U) + ℓ∗ 1(( ¯U ⊥)T W ¯U ⊥) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (152) 29 We have arrived at a contradiction as the last inequality in (152) is exactly the opposite of (146).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' This implies that our initial assumption ˆX ̸= X cannot hold and we therefore must have ˆX = X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' This is precisely the claim of the first part of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (ii) ⇐= (the only if part): We now assume that (147) holds, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (∃W ∈ Rn×n|Avec(W) = 0m×1, W ̸= 0n×n) − tr (( ¯U)T W ¯U) ≥ ℓ∗ 1(( ¯U ⊥)T W ¯U ⊥) (153) and would like to show that for such a W there is a symmetric rank-k matrix X with the columns belonging to the span of ¯U such that y = Avec(X), and the following holds ℓ∗ 1(X + W) < ℓ∗ 1(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (154) Existence of such an X would ensure that it both, satisfies all the constraints in (7) and is not the solution of (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Following the strategy of [39] one can reverse all the above steps from (153) to (149) with strict inequalities and arrive at the first inequality in (149) which is exactly (154).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' There are two implications that cause problems in such a reversal process, the one in (153) and the one in (149).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' If these implications were equivalences everything would be fine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' We address these two implications separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' 1) the implication in (152) – particular X to “overwhelm” W: Assume X = ¯UΛx ¯U T with Λx > 0 being a diagonal matrix with arbitrarily large elements on the main diagonal (here it is sufficient even to choose diagonal of Λx so that its smallest element is larger than the maximum eigenvalue of ¯U T W ¯U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Now one of course sees the main idea behind the “removing the absolute values” concept from [39,41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Namely, for such an X one has that ℓ∗ 1( ¯U T X + W) ¯U) = tr(ℓ∗ 1( ¯U T X + W) ¯U)) since for symmetric matrices the ℓ∗ 1(·) (as the sum of the argument’s absolute eigenvalues) and tr (·) (as the sum of the argument’s eigenvalues) are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' That basically means that when going backwards the second inequality in (152) not only follows from the first one but also implies it as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' In other words,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' for X = ¯UΛx ¯U T (with Λx > 0 and arbitrarily large) tr ( ¯U T W ¯U) + ℓ∗ 1(( ¯U ⊥)T W ¯U ⊥) ≤ 0 ⇐⇒ tr ( ¯U T (X + W) ¯U ) + ℓ∗ 1(( ¯U ⊥)T (W) ¯U ⊥) ≤ ℓ∗ 1(X) ⇐⇒ ℓ∗ 1( ¯U T (X + W) ¯U) + ℓ∗ 1(( ¯U ⊥)T (X + W) ¯U ⊥) ≤ ℓ∗ 1(X),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (155) which basically mans that there is an X that can “overwhelm” W (in the span of ¯U) and ensures that the “removing the absolute values” is not only a sufficient but also a necessary concept for creating the relaxation equivalence condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' 2) the implication in (149): One would now need to somehow show that the third inequality in (149) not only follows from the second one but also implies it as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' This boils down to showing that inequality in (150) can be replaced with an equality or, alternatively, that L0 and L are provisionally equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Neither of these statements is generically true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' However, since we have a set of X at our disposal there might be an X for which they actually hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' We continue to assume X = ¯UΛx ¯U T with Λx > 0 being a diagonal matrix with arbitrarily large entries on the main diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Then the last equality in (150) gives ℓ∗ 1( ¯U T (X + W) ¯U) + ℓ∗ 1(( ¯U ⊥)T (X + W) ¯U ⊥) ≤ ℓ∗ 1(X) ⇐⇒ maxΛ∗=ΛT ∗ ∈L0∗ tr (Λ∗ � ¯U ¯U ⊥�T (X + W) � ¯U ¯U ⊥� ) ≤ ℓ∗ 1(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (156) Also, one has maxΛ∗=ΛT ∗ ∈L0∗ tr (Λ∗ � ¯U ¯U ⊥�T (X + W) � ¯U ¯U ⊥� ) ≤ ℓ∗ 1(X) ⇐⇒ maxΛ∗,i=ΛT ∗,i,Λ∗,iΛT ∗,i≤I,i∈{1,2} tr (Λ∗,1 ¯U T X ¯U + Λ∗,2( ¯U ⊥)T W ¯U ⊥) ≤ ℓ∗ 1(X) ⇐⇒ maxΛ∗,i=ΛT ∗,i,Λ∗,iΛT ∗,i≤I,i∈{1,2} tr (Λ∗,1Λx + Λ∗,2( ¯U ⊥)T W ¯U ⊥) ≤ tr (Λx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (157) Now,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' if at least one of the elements on the main diagonal of Λ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' diag(Λ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' is smaller than 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' then the corresponding element on the diagonal of Λx can be made arbitrarily large compared to the other elements 30 of Λx and one would have maxΛ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='i=ΛT ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='Λ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='iΛT ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='i≤I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='i∈{1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='2} tr (Λ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='1Λx + Λ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content='2( ¯U ⊥)T W ¯U ⊥) < tr (Λx) ⇐⇒ maxΛ∗=ΛT ∗ ∈L0∗ tr (Λ∗ � ¯U ¯U ⊥�T (X + W) � ¯U ¯U ⊥� ) < ℓ∗ 1(X) ⇐⇒ maxΛ∗=ΛT ∗ ∈L∗ tr (Λ∗ � ¯U ¯U ⊥�T (X + W) � ¯U ¯U ⊥� ) < ℓ∗ 1(X),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' (158) where the last equivalence holds since the difference of the terms on the left-hand side in the last two inequalities is bounded independently of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Also, the last inequality in (158) together with the first equality in (150) and the first inequality in (149) produces (154).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' Therefore the only scenario that is left as potentially not producing (154) is when all the elements on the main diagonal are larger than or equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' However, the two lemmas preceding the theorem show that in such a scenario L0 = L and one consequently has an equality instead of the inequality in (150) which then, together with (149), implies (154).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' This completes the proof of the second (“the only if”) part of the theorem and therefore of the entire theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} +page_content=' 31' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQf5Pp6/content/2301.00801v1.pdf'} diff --git a/FtFKT4oBgHgl3EQfbC5a/content/2301.11810v1.pdf b/FtFKT4oBgHgl3EQfbC5a/content/2301.11810v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..0dda2513179710e9b532334314c9d13bd43ca807 --- /dev/null +++ b/FtFKT4oBgHgl3EQfbC5a/content/2301.11810v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:6a219af5adf52db7ff9ec43d3613dcfbb5150bd68025aca911d3244de90bcb45 +size 523609 diff --git 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100644 index 0000000000000000000000000000000000000000..c29fd8b6564d06f52b0e231a6fec147742782b32 --- /dev/null +++ b/G9E1T4oBgHgl3EQfrQVc/content/tmp_files/2301.03352v1.pdf.txt @@ -0,0 +1,1131 @@ +MEMRISTIVE MEMORY ENHANCEMENT BY DEVICE +MINIATURIZATION FOR NEUROMORPHIC COMPUTING +A PREPRINT +A. S. Goossens 1,2,*, M. Ahmadi 1,2, D. Gupta 1,2, I. Bhaduri 1,2, B. J. Kooi 1,2, and T. Banerjee 1,2.* +1Groningen Cognitive Systems and Materials Center, University of Groningen, Groningen, The Netherlands +2Zernike Institute for Advanced Materials, University of Groningen, Groningen, The Netherlands +*{a.s.goossens,t.banerjee}@rug.nl +January, 2023 +ABSTRACT +The areal footprint of memristors is a key consideration in material-based neuromorophic comput- +ing and large-scale architecture integration. Electronic transport in the most widely investigated +memristive devices is mediated by filaments, posing a challenge to their scalability in architecture +implementation. Here we present a compelling alternative memristive device and demonstrate that +areal downscaling leads to enhancement in memristive memory window, while maintaining analogue +behavior, contrary to expectations. Our device designs directly integrated on semiconducting Nb- +SrTiO3 allows leveraging electric field effects at edges, increasing the dynamic range in smaller +devices. Our findings are substantiated by studying the microscopic nature of switching using scan- +ning transmission electron microscopy, in different resistive states, revealing an interfacial layer +whose physical extent is influenced by applied electric fields. The ability of Nb-SrTiO3 memristors +to satisfy hardware and software requirements with downscaling, while significantly enhancing +memristive functionalities, makes them strong contenders for non-von Neumann computing, beyond +CMOS. +Keywords Interface memristor, Areal scaling, Beyond CMOS, Neuromorphic computing, Scanning transmission +electron microscopy (STEM) +1 +Introduction +The growing demand for applications such as artificial intelligence and the Internet of Things has given rise to critical +challenges in the storage and processing of big data using existing computational architectures [1]. The currently +employed von Neumann architecture, using complementary metal-oxide-semiconductor (CMOS) hardware, suffers +from limited transmission speed [2, 3, 4] due to a memory throughput bottleneck as well as energy inefficiency and +limited scalability [4, 5, 6]. Moving away from CMOS technology, towards logic-in-memory chips would alleviate +some of the above issues but requires us to massively rethink every aspect of computing [7]. The first step towards this +is identifying novel materials and devices with suitable physical properties. Resistive switching devices, or memristors, +are one such class of devices where the resistance can be switched between several states. Reported in different ionic +materials, they are distinguished by the switching mechanism as either occurring through the material bulk between +two electrodes or interface-type where switching takes place in a localized region underneath the area of the electrodes +[8]. Their ability to co-locate memory and computation, and exhibit characteristics absent in digital computing makes +them important for novel computing approaches. Given the robust way in which the human brain is able to process +large amounts of data with remarkably low power, it is unsurprising that it serves as a source of inspiration to the +development of computing beyond using CMOS. As the brain utilizes a vast network, downscaling memristive devices +is a crucial area of research to develop large scale neuromorphic systems. +arXiv:2301.03352v1 [cs.ET] 9 Jan 2023 + +MEMRISTIVE MEMORY ENHANCEMENT BY DEVICE MINIATURIZATION +For this material-driven research, the areal footprint in unconventional computing architectures that seek to integrate +in-memory computing devices such as memristors is a prime consideration. Considerable research has been devoted to +this in the realm of non-volatile conventional filamentary devices. The challenges in their implementation in such novel +architectures, besides the requirement for unfavourable electroforming processes, lie in their switching endurance [9], +and their efficacy to exhibit discernible analogue resistance states. Memristive devices that exhibit more than two stable +states also greatly enhance integration density because each device can store multiple data bits in an analogue manner. +In valence change memristors, where switching originates from filaments, such behavior is observed in large areal +dimensions but is lost when devices are downscaled and conduction is mediated by a single nanoscale filament causing +an abrupt transition between the two resistance states [10]. Further, the effects of Joule heating on filaments are +an important consideration as devices shrink; Joule heating can cause a wide distribution of switching voltages and +endurance deterioration. These limitations in device stability, endurance and associated enhanced power of operation +are major roadblocks in the successful implementation of filamentary devices in large scale architectures. +Memristive devices have the potential to be integrated in large scale architectures, for which they should exhibit large +memory windows, high endurance and low variability [11]. Herein the areal switching mechanism is a strong contender. +A model system in which this mechanism is dominant is Schottky contacts on Nb-doped SrTiO3 (Nb:STO), formed at +the interface with a high work function metal. It is widely accepted that in these material systems it is not the bulk of +the device, but an area close to the interface that is responsible for the switching, a more detailed discussion on the +proposed mechanisms is presented in Supporting Information section S3. +Distinguishing Nb:STO from conventional semiconductors such as Si, widely used in conventional architectures, is its +dielectric permittivity which is comparatively large (300) and is strongly dependent on electric field. This property +extends the parameter space for designing functionality: electric fields can be used to tune the barrier height and width +relevant for memristive device design. We have previously shown that such Schottky contacts form robust memristors, +exhibiting non-linear transport and continuous conductance modulation [12], and that their behavior can be described +by a power-law which can be successfully implemented as a learning algorithm [13]. However, for the applicability of +Nb:STO-based memristors as hardware elements for non-von Neumann computing architecture beyond CMOS, the +focus should be on establishing their memristive performance with device miniaturization, which has not been shown +on such semiconducting platforms. In this work, we demonstrate that memristive devices of Co Schottky contacts +on Nb:STO exhibit an increase in the analogue memristive memory window in devices down to 1 μm, contrary to +expectations. Ionic defects are at the heart of memristive behavior, hence one of the following two scenarios is expected. +For a homogeneous areal mechanism, the current density will scale with device area so that the device resistance in +both the high resistance state (HRS) and the low resistance state (LRS) scales with the electrode size, but the ratio +between them is area independent. Alternatively, the resistance window can be severely reduced or even vanish with +downscaling due to insufficient ionic defects. However, we observe an enhancement in the memory window as the +device area is reduced, with minimal device-to-device variation, an unforeseen finding. +To understand the microscopic nature of the switching, we conducted scanning transmission electron microscopy +(STEM) on virgin samples and on samples subjected to either a positive (SET) or negative (RESET) voltage . Using +integrated differential phase contrast (iDPC) we image oxygen atomic columns next to the heavy metal atomic columns. +Figure 1: State stability and multilevel memristive operation. a Schematic of the fabricated devices on Nb-doped +SrTiO3, electrical connections. Black lines are used to represent the varying overall electric fields acting over each area. +The field strengths at the interface are also indicated by a color gradient, showing the fields are weakest in the central +area (blue) and strongest around the perimeter (red). b Current read at +0.3 V for device sizes of 100 μm (black), 10 μm +(blue) and 1 μm (red). c Current read at 0.3 V after switching between a SET voltage of +1 V (black, red and blue) or ++2 V (green, purple and orange) and a RESET voltage of -2 V (black and green), -2.5 V (red and purple) or -3 V (blue +and orange). Each combination was repeated over 100 cycles. +2 + ++2 V ++2 V +2.5 +Nb-doped SrTiO. +3 VMEMRISTIVE MEMORY ENHANCEMENT BY DEVICE MINIATURIZATION +Figure 2: Characterization of memristive devices in the virgin state. Electrical characteristics of virgin devices. The +compliance current was fixed at 100 mA for all measurements. Results are shown for four devices of each area in a-f. +Virgin samples show the existence of a layer near the interface with neither the perovskite structure of the substrate nor +that of the Co electrode. Applying a bias across the interface results in oxygen vacancy movement, which is a key factor +controlling the resistance states. These new revelations are consolidated with a mathematical model describing the +kinetics of trapping and de-trapping in dielectric materials and relates experimental results to the effective trapping +density. Surprisingly, this is found to be larger for smaller junctions, suggesting that an increase in the density of traps +is responsible for the increased resistance ratio and attributed to inhomogeneous distribution of the electric field due to +device edges. +These memristive devices, integrated directly on a semiconducting platform, demonstrate multistate analogue switching +with remarkably high memory windows with downscaling, as well as high endurance and low device and cycle variation +down to the smallest devices. Their ability to meet both hardware and software requirements for unconventional +computing, make Nb:STO memristors strong material contenders for physical computing beyond CMOS. +2 +Results +2.1 +Electrical Characterization +Figure 1a shows a schematic of the device structure used for the electrical measurements. An array of circular Co +electrodes of varying sizes are fabricated on a semiconducting Nb:STO single crystalline substrate. The bottom of the +substrate serves as a back contact for the devices. The top electrodes were patterned by a two-step electron lithography +process using aluminium oxide as an insulation layer to define the contact areas and to prevent electronic cross talk. +After fabrication, we performed small range voltage sweeps to characterize the virgin states of each device on a chip. +The results for devices with radial dimension from 100 μm to 800 nm are shown in Fig. 2, where each sweep followed a +voltage sequence from 0 to +1 V to -1 V and back to 0 V. We show four devices of each area, which are plotted in Fig. +2a-f. +The current magnitudes for different devices of the same area show no significant differences down to 1 μm, indicating +device-to-device variations are minimal. Establishing this is important as this signifies the sole influence of device area +in determining the resistance ratio and rules out contributions from device-to-device variation. The 800 nm devices +show a greater degree of variation; this is likely due to small differences in their areas and edges arising from the +fabrication process and not inherent to the material or due to device fallibility. No significant differences in the current +3 + +MEMRISTIVE MEMORY ENHANCEMENT BY DEVICE MINIATURIZATION +Figure 3: Resistance ratio, cycling endurance and state stability. a-f show 1000 consecutive current-voltage sweeps +from +2 V to -3 V to +2 V at a rate of 1.52 Vs−1 for devices of 100 μm down to 800 nm. Starting from a SET voltage +of +2 V, each device is in an LRS, represented by the upper branch reaching the RESET voltage of -3 V and sweeping +back, the devices are switched to an HRS (represented by the lower branch. +densities at low bias values are found in the virgin state, confirming that the entire device area contributes to the charge +transport (Supporting Information Fig. S1). For all the devices, the current gradually increases and exhibits a small +hysteretic effect from the virgin state, indicating that no forming step is required. +Figure 3a-f shows 1000 consecutive current-voltage (I-V) sweeps of these devices. Starting from a SET voltage of +2 V, +each device is in an LRS, represented by the upper branches. After reaching the RESET voltage of -3 V and sweeping +back, the devices are switched to an HRS (represented by the lower branches). In all device areas both the SET and +RESET operation remain continuous, indicating the resistive switching retains its analog nature when downscaling. The +cycling endurance was measured for over 105 switching cycles without device failure, illustrating an endurance of >105. +The current in the HRSs scales approximately with area at low bias values, while the low resistance current, is less +closely correlated to the area. As a result, the resistance window increases with decreasing device area in both forward +and reverse bias. Figure 1b and Supporting Information Fig. S2 show the current and current density at a low read +voltage of 0.3 V, respectively. Minimal cycle-to-cycle variations at low reading voltages are found with reproducible +switching between clearly distinguishable states without degradation in device performance. This also establishes +the low power operation of these devices after downscaling, which is important for memristor operation. As shown +in Supporting Information Fig. S3, the device-to-device variation remains low down to 1 μm. The variation in the +resistance ratio in the 800 nm devices is larger (Fig. S4), and will be discussed later. +The SET and RESET transitions are gradual and highly tunable. To demonstrate this, a 1 μm device was subjected to +voltage sweeps varying between different positive (SET) and negative (RESET) voltages. Figure 1c shows that a wide +range of stable states is available at a low read voltage of +0.3 V. The wide dynamic range combined with the large +number of distinct addressable states ensures device reliability and increased memory storage capabilities. Each state +maintains a narrow distribution of current values over the 100 cycles shown, reiterating the stability of the switching +process. +2.2 +Scanning Transmission Electron Microscopy +A microscopy study of the Schottky interface was carried out using STEM. Figure 4 shows atomic resolution cross- +section STEM-integrated Differential Phase Contrast (iDPC) images of the Co/Nb:STO interface for samples in the +4 + +MEMRISTIVE MEMORY ENHANCEMENT BY DEVICE MINIATURIZATION +Figure 4: Visualization of oxygen vacancy migration using STEM. iDPC-STEM images of Co/Nb:STO samples in +a the virgin (unbiased) state, b the LRS and (c the HRS, highlighting the structure close and far from the interface. The +perovskite unit cell of STO, showing Sr in green, O in dark red and Ti in light red, viewed along the <110> in d the +pristine state and (e with oxygen vacancies. The deficiency of O causes Ti atoms to move away from the vacancies as +shown by the arrows. f shows a schematic representation of how the interfacial layer is affected by biasing. +unbiased virgin condition (Fig. 4a), the LRS state (Fig. 4b) and the HRS state (Fig. 4c). To image lighter oxygen atoms, +integrated into a matrix with heavier Sr and Ti atoms, we utilized STEM-integrated Differential Phase Contrast (iDPC) +instead of the more commonly employed STEM-High-angle annular dark-field (HAADF) imaging technique.[14, 15]. +The STEM images in Fig. 4a show that, apart from a thin interfacial region, the bulk STO consists of a cubic perovskite +lattice and no defects are observable. All images taken within the bulk did not show any dislocation and possessed +the expected perovskite structure as shown in Fig. 4d. However, the structure close to the interface deviates from this +perovskite structure and is deficient in oxygen. The migration of oxygen ions near the interface towards Co causes +positively charged Ti ions to be displaced so that they no longer sit equidistantly from the Sr ions along <001>. Figure +4e illustrates how the loss of O ions gives rise to Ti displacements along the <001> direction away from the interface +as well as along <1-10> (see Supporting Information Fig. S7) and is similar to what was reported in ref. [16] in +La0.67Sr0.33MnO3/Hf0.5Zr0.5O2. We believe the creation of this thin layer to be related to the formation of a Schottky +barrier. The analysis for a non-memristive interface with Ti contacts can be found in Supporting Information Fig. S6. +Figure 4b shows analogous results to Fig. 4a, but now for the sample switched to the LRS, representing the upper +branch in Fig. 3, after the application of a positive bias voltage of 2 V. Comparing the two figures shows that in the +LRS state the extent of the interfacial layer has decreased. This suggests that under the influence of a positive voltage, +the labile bonds between O and interfacial Co atoms are broken and oxygen moves back into the STO substrate. A +negative bias voltage of -3 V (corresponding to the lower branch in Fig. 3), on the other hand, causes oxygen to move +from STO to cobalt causing the formation of CoO and more oxygen vacancies in the STO, highlighted by a larger +region over which Ti ions are displaced (see Fig. 4c). This indicates that the formation of the CoO switches the sample +to the HRS state. It has been shown [17, 18] that the oxygen vacancy distribution inside the system will determine +how the oxygen vacancies are affected by the applied voltage. The formation of an oxygen deficient interfacial layer +confirms that in these samples the oxygen vacancies are concentrated near the interface. In this case, it is expected +that the application of a positive voltage will cause oxygen vacancies to be repelled from the interface while a negative +voltage will cause oxygen vacancies to be attracted to the interface, consistent with our findings. After removing the +5 + +(a +middle of STO +middle of STO +Virgin state +Low resistance state +High resistance state +(d) +(f) +(e) +<001> +Virgin state +LR state +HR state +Co +Co +Co +oxygen deficient areal +oxygen deficient area +STO +STO +STO +→ <1-10> +Zone axis: <110>MEMRISTIVE MEMORY ENHANCEMENT BY DEVICE MINIATURIZATION +voltage, the interfacial layer did not reform over time, suggesting the presence of an oxygen-migration blocking layer. +These results are summarized in Fig. 4f. +Our results directly confirm the existence of a homogeneous oxygen deficient layer at the interface. The homogeneous +nature of the defect state layer ensures ionic defects are retained with downscaling. We furthermore show that the +physical extent of the layer is reduced or extended when a positive or negative voltage is applied respectively. Although +the uniform nature of the ionic contribution to switching is now verified, this does not explain the origin of the +unexpected enhancement of the resistance window with downscaling. This we discuss next by considering the trapping +of electronic charges at oxygen vacancy sites. +2.3 +Model +In order to understand how the electrical properties of the devices are influenced by these oxygen vacancies, we +consider the interaction between electrons and defect states. This interaction is most strongly evidenced by the retention +characteristics, which have a slow decaying component. This behavior is caused by the detrapping of charges. It has +been shown that this occurs over long timescales and the different states will remain clearly distinguishable for long +time periods of hours and that the retention time is tunable by the applied stimuli [12]. We utilized short voltage pulses +to measure the retention characteristics of each device in both an HRS and LRS. This was done by applying alternating +SET and RESET pulses of +2 V and -3 V respectively, and reading the small-signal current at either +0.3 V or -0.5 V +after each writing event. The state retention characteristics of the different devices are shown in Fig. 5 for the LRS (red) +and HRS (black). Over time, the current in both states tends to an intermediate value. For the LRS, the rate of change +follows a power law that is commonly observed for charge trapping under bias in high-κ dielectrics, referred to as the +Curie-von Schweidler law. +This law describes a non-Debye type relaxation in dielectrics. Empirical evidence of this behavior is seen in a wide +variety of materials, but the precise physical origin remains unclear. Here we consider the effect of injected electrons +becoming trapped in defects states within the dielectric. The space charge generated by these trapped electrons lowers +the electric field, in turn reducing the flow of current through the dielectric. In this case the trapping rate can be +expressed as: +dn +dt = n0σ Jvth +qvd +e− nh +V +(1) +where n0 is the maximum number of traps available, J/q is the net flux density, vth and vd are the thermal and drift +velocities respectively, and σ is capture cross-section. Solving this equation yields the following expression for n: +n = V +h ln +� Q +Q∗ + 1 +� +(2) +where Q = +� +Jdt is the total injected charge and +Q∗ = +V vdq +n0hvthσ +(3) +Expressing the current as J = Jst−α and extending this analysis results in: +α +1 − α ln(Js) ≈ mEap + n +(4) +where m is a constant. +We can also directly relate the trapping rate to the current. QT represents the charge that is trapped when charge Q is +injected into the dielectric. The ratio dQT +dQ is a function of current. The current can be written as: +J = Js +t +t0 +−1/(α+1) +(5) +where α ≥0 and Js depends on the transport mechanism. For conduction following an exponential relation: +Js ∝ e +(1−1/(α+1))V +V0 +(6) +Here, V0 is a constant. The full derivation is shown in Supporting Information Section S1 and Fig. S8, and is also +extended to show that it holds for other transport mechanisms. +Equations 4 and 6 serve as a direct mathematical proof that the exponent α in the power law is related to the effective +trap density or capacity of the dielectric to trap electrons. This derivation is applicable to a wider range of systems, +irrespective of the choice of dielectric material. In Table 1 we show the LRS exponents, α for each device. Larger +values are observed for smaller devices indicating that the trap density is higher in the smallest device compared to the +larger device. +6 + +MEMRISTIVE MEMORY ENHANCEMENT BY DEVICE MINIATURIZATION +Figure 5: Trapping dynamics and Schottky interface energy landscapes. Retention characteristics of differently +sized devices read at +0.3 V a-c) and -0.5 V (d-f) after a SET voltage of +2 V (red) or -3 V (black). g shows the energy +landscape of a Schottky interface in equilibrium when the dielectric constant does not depend on electric field (solid +line) and when the dielectric constant is field-dependent (dashed line). EF and EC are the Fermi level and conduction +band respectively. The energy landscapes at the center and edge of a device are compared in h in equilibrium and i in +reverse bias. Red circles represent oxygen vacancy states and the green arrow indicates electron tunneling. +area +��������� +|Exponent| +Radius (μm) +Read at +0.3 V +Read at -0.5 V +1 +0.85±0.03 +2.17±0.02 +10 +0.47±0.02 +0.987±0.002 +100 +0.041±0.004 +0.626±0.007 +��������� +trapping density +Table 1: Magnitude of exponents, α, extracted by fitting a power-law to the low resistance states in the graphs in Fig. 5. +3 +Discussion +While this model provides a clear correlation between trapping density and device area, it does not give information +about the traps; we implicitly take all traps to be of the same kind, while in reality, the nature of traps can vary greatly. +The trapping rate can depend on the spatial location of the traps and new traps can be generated via defect migration. +For a more precise picture of the mechanism, we need to consider a distribution of traps with respect to their location +within the dielectric. Evidenced by the STEM study, oxygen vacancies are the most important class of trapping defects +to consider. They are abundantly present in SrTiO3 due to their low formation (0.51 eV[19]) and migration (0.62 +eV[20]) enthalpies and their locations within the energy landscape are well documented[21]. +7 + +MEMRISTIVE MEMORY ENHANCEMENT BY DEVICE MINIATURIZATION +From the discussion above, it is clear that the energy landscape of these Schottky junctions is far more complex than is +captured by the most commonly used models that are based solely on parameters of the individual materials forming +the contact [22, 23]. Transport through these junctions is usually described by the thermionic emission equation, which +includes an ideality factor accounting for the deviating transport from this ideal diode equation. This model furthermore +does not consider that the interfacial area is not spatially homogeneous and that in devices of finite areas the boundary +of the device will be relevant. In particular, it is known that near the edges crowding of the field lines leads to an +enhancement in the field strength which can decrease the barrier width [24, 25]. This is supported by the results of +the finite element simulations in Supporting Information Fig. S11 and S12, showing a significant enhancement in the +electric field around the edge and when downscaling. From the simulations it is evident that there is still a clear field +gradient in the 1 μm devices, indicating that a further increase in ratio with downscaling can be expected, and the areal +field shows no apparent saturation till around 10 nm (Fig. S13). +The observed enhancement is especially important in Nb:STO-based memristive devices as the dielectric constant of +the substrate strongly depends on electric fields [26, 27]. This will further alter the potential landscape of the Schottky +interface in such memristive devices. In particular, the dielectric permittivity of Nb:STO rapidly decreases in the +presence of large electric fields which results in a decrease in the effective Schottky barrier width as illustrated in Fig. +5g. Consequently, a large reduction in the barrier width is expected to occur near the device edges (Fig. 5h). It has also +been shown that an electric field can modify the defect states and significantly affect trapping parameters[28]. +Given that the charge transport is governed by the potential landscape, this will hugely impact the measured current, +pictured in Fig. 5i. Tunneling through the barrier will be enhanced near the device edges leading to a larger current near +the device perimeter. This will be especially important in the LRS where the interface is depleted of trapped charges +and the Schottky barrier is narrower, leading to more tunneling [29, 12]. +Transport across the interface is comprised of thermionic emission and tunneling. The thermionic current density is +expected to be independent of area and is the dominant mechanism in the HRS at low bias voltages, giving rise to the +decreasing current in the HRS around zero with downscaling observed in Fig. 3. At higher voltage values, however, +tunneling will also contribute to the current; the tunneling current density will increase with decreasing area. In Fig. 1b, +the current is read at +0.3 V where we expect both thermionic emission and tunneling to contribute to transport, giving +rise to similar currents measured for the 10 and 1 μm devices in the HRS. The tunneling contribution increases in the +LRS, especially in smaller devices due to the larger electric fields, resulting in the observed increase in current density +with reducing area. +By applying a potential over the Schottky barrier, the Fermi level is shifted such that tunneling electrons sample different +oxygen vacancy energy levels. As the reverse bias voltage is increased, electrons are gradually exposed to larger ranges +of states in which they can become trapped. In addition, in reverse bias, the electric field at the interface becomes +larger leading to a reduction in the dielectric constant and a corresponding decrease of the Schottky barrier widths. This +decrease in width will be more pronounced in regions closer to the edge due to the local field enhancement. As a result +of the narrower barrier, electron-electron scattering will be reduced and the trap states will act as the main barrier for +transport. The stronger edge field may additionally facilitate the migration of oxygen vacancies resulting in a higher +number of vacancies accumulating around the perimeter. Consequently, the trapping efficiency will be greater near the +edge than in the center. This is a unique effect enabled by the electric field control of the dielectric permittivity, does +not occur in conventional semiconductors and is relevant for Nb:STO memristive device design. +We can express the area and perimeter of a device with radius r as A = πr2 and p = 2πr respectively. The ratio of the +perimeter to area: +p +A = 2πr +πr2 = 2 +r +(7) +indicates that the edge effects become more dominant as the device area is reduced. As a result, current flow at the +perimeter will constitute a larger percentage to the overall transport behavior in smaller devices. This explains the +enhanced current densities observed when downscaling after applying large bias voltages as well as the larger effective +trapping densities for smaller devices. Specifically, this field enhancement around the device edges gives rise to an +increase in the dynamic range in smaller devices, and explains the unexpected resistance window scaling. +4 +Conclusions +As a first demonstration of exploiting edge effect related additional electric fields, our work successfully demonstrates +the ability to increase the resistance window by device miniaturization of interface memristors from 100 μm down +to 1 μm, contrary to expectations, with exceptional robustness to device-to-device and cycle variability. Scanning +transmission electron microscopy images taken in the virgin, high and low resistance states prove the existence of a +homogeneous interfacial layer, deficient in oxygen, whose physical extent is influenced by applying an electric field. +8 + +MEMRISTIVE MEMORY ENHANCEMENT BY DEVICE MINIATURIZATION +This, however, does not explain the enhancement in the resistance window with device downscaling. A model describing +the interaction of electrons with oxygen vacancy trap states shows an increase in the effective trapping density with +downscaling. The advantage of direct integration of devices on a semiconducting platform of Nb-doped SrTiO3 allows +for the locally enhanced fields to controllably tune the interfacial energy landscape at the interface, leading to a greater +contribution of edge effects in smaller devices as confirmed by finite element simulations. With rapid advances made in +the palette of materials and devices available for neuromorphic hardware, the thrust now should be in their efficient +integration on semiconducting platforms for on-chip applications with substantial reduction in areal footprint. In this, +our work provides an encouraging direction. +5 +Experimental Section +5.1 +Electrical Device Fabrication +We investigated a series of Co/Nb-doped SrTiO3 devices, where the device area was varied across the series over a +range spanning five orders of magnitude ranging from 10−12 to 10−8 m2, with radii between 800 nm and 100 μm. +The devices were fabricated using Nb-doped SrTiO3 (001) substrates with a doping concentration of 0.1 wt% from +Crystec. SrTiO3 consists of alternating SrO and TiO2 planes along the [001] direction. The as-received substrates +have a slight miscut from the exact crystallographic direction and as a result, a mixture of both terminations exists at +the surface. It has been shown that the local properties of Schottky barriers grown on the different terminations may +differ, hence to minimize the variation of different areas on the substrate a single termination is desired. To ensure that +the terminating layer is TiO2, a chemical treatment was carried out with buffered hydrofluoric acid (BHF). A further +annealing treatment at 960 ◦C in an O2 flow of 300 ccmin−1 to facilitate the reorientation of surface atoms to form +an atomically flat and straight terraced surface. Atomic force microscopy images were taken at different parts of the +substrate and confirmed the existence of uniform terraces. The substrate was then coated with a negative resist (AZ +nLOF 2020) and using electron beam lithography circles of different areas were patterned. A thick insulation layer of +AlOx was deposited using electron beam evaporation and lift-off was carried out to define a set of direct contacts to +the substrate. By means of a second lithography step with a positive resist (950 K PMMA), square contact pads were +defined, each covering a hole and part of the surrounding AlOx: the dimensions of these pads were identical for each +device to minimize spurious effects arising from significantly different contact resistances. Co (20 nm) and a capping +layer of Au (100 nm) were then deposited using electron beam evaporation in high vacuum (∼10−6 Torr). +5.2 +Electrical Characterization +Electrical measurements were conducted using probes connected to two remote-sense and switch units (RSU) of a +Keysight B1500A Semiconductor Device Parameter Analyzer. During the voltage sweeping measurements, conducted +using a sweeping measurement unit (SMU), the bottom of the substrate is held at 0 V while a voltage is applied to +the top electrode. Due to the diodic nature of the devices in conjunction with large degrees of resistive switching, +the measured currents during a single sweeping measurement span up to 9 orders of magnitude. For this reason, the +measurements were performed using auto range for the measured current. The effects of this can be observed in the +endurance cycling measurements which were performed at high sweeping rates in the form of plateaus in the current +whenever a limit of the SMU range is reached. +5.3 +Scanning Transmission Electron Microscopy +The samples discussed in this work use SrTiO3 (001) substrates with an Nb-doping in place of Ti of 0.1 wt% from +Crystec. The surface was prepared using a chemical treatment with buffered hydrofluoric acid (BHF). Next, the +substrates were annealed at 960◦C in an O2 flow of 300 ccmin−1. For STEM samples films were deposited by electron +beam evaporation of 20 nm of Co capped with 20 nm of Au and 20 nm of Pt. From this, three types of STEM lamellae +were prepared: virgin (unbiased) samples, low resistance state (LRS) samples and high resistance state (HRS) samples. +Using a probe station, samples are subjected to bias values of +2 V and -3 V to prepare samples in the LRS and HRS +respectively. STEM lamellae were extracted from samples along the <110> direction using a Helios G4 CX dual beam +system with a Ga focused ion beam. The lamellae were thinned to make them transparent to electrons using the focused +ion beam. Imaging was carried out using a Thermo Fisher Scientific Themis Z S/TEM system operating at 300 kV. +STEM-High-angle annular dark-field (HAADF) images are most widely used, because they are readily interpretable +with atomic columns being bright spots in a dark surrounding, where the brightness of the spots scale with the average +atomic number Z (∼Z1.7). This technique is well suited to image heavy elements, but lighter elements, such as oxygen, +are harder to detect, and cannot be detected properly when integrated into a matrix with much heavier elements (like Sr). +Therefore, to gain more insight into the important role played here by the oxygen ions, we utilized here STEM-iDPC +9 + +MEMRISTIVE MEMORY ENHANCEMENT BY DEVICE MINIATURIZATION +instead of STEM-HAADF imaging. This technique uses a four-quadrant annular bright field detector and can be used +to acquire the projected local electrostatic potential of the sample (when thin) and has clear advantages over traditional +annular bright field (ABF) imaging [14, 15]. +5.4 +Simulations +Finite element modeling of the electric field profile at the interface was carried out using COMSOL Multiphysics. +5.5 +Statistical Analysis +For the |current|-voltage graphs, the absolute value of the measured current is taken; to determine the current density, +the measured current was divided by the area of the Co contact. The values in Table 1 were derived by iteratively fitting +the data in Fig. 5a-f using a power-law equation of the form I = I0(t − t0)−α by means of the Levenberg-Marquardt +algorithm; the reported errors are the standard errors calculated by this method. The fits are shown in supporting Fig. +S9. The inverse scaling of the exponent and device area was verified for different devices and different reading and SET +voltages. Plotting and analysis of electrical measurements was done using OriginPro 8.5. Measurements were repeated +on four devices of each area to check reproducibility and validity of results. +For STEM images, multiple regions for each one of the three bias conditions were taken to verify the results. The idpc +images were filtered by applying a high-pass Gaussian filter using Velox. +Acknowledgements +A.G. is supported by the CogniGron Center, University of Groningen. Device fabrication was realized using NanoLab +NL facilities. We acknowledge technical support from J. G. Holstein, H. H. de Vries, A. Joshua, T. Schouten, and H. +Adema. We thank R. J. E Hueting, P. Nukala, S. de Graaf and T. Kenyon for useful discussions. A.G., D.G, I.B, and +T.B. benefited from helpful discussions with the members of the Spintronics of Functional Materials group. +Conflict of Interest +The authors declare no conflict of interest +Author Contributions +A.G. and T.B. conceived the idea and designed the devices. A.G. and D.G. fabricated devices for electrical measurements +and performed electrical measurements, along with I.B.. D.G. derived the mathematical model discussed in the +manuscript. M.A. and A.G. fabricated lamalae for STEM and M.A. took STEM images. Finite element simulations +were done by A.G.. All authors analyzed the data, discussed the results and agreed on their implications. All authors +contributed to the preparation of the manuscript. +References +[1] S. Hamdioui, M. Taouil, H. A. Du Nguyen, A. Haron, L. Xie, and K. Bertels. Memristor: the enabler of computation- +in-memory architecture for big-data. In 2015 International Conference on Memristive Systems (MEMRISYS), pages +1–3. IEEE, 2015. +[2] M. V. 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Journal of Applied Physics, 37(7):2797– +2804, 1966. +[29] E. Mikheev, B. D. Hoskins, D. B. Strukov, and S. Stemmer. Resistive switching and its suppression in Pt/Nb:SrTiO3 +junctions. Nature Communications, 5(1):1–9, 2014. +11 + +MEMRISTIVE MEMORY ENHANCEMENT BY DEVICE MINIATURIZATION +Supplementary Data +Figure S1: Current densities of virgin devices. Results are shown for devices of radial dimensions of 100 μm (black), +10 μm (blue) and 1 μm (red). +Figure S2: Cycle-to-cycle variation. The current density read at +0.3 V for device sizes of 100 μm (black), 10 μm +(blue) and 1 μm (red). +S1 + +MEMRISTIVE MEMORY ENHANCEMENT BY DEVICE MINIATURIZATION +Figure S3: Device-to-device variation. Current-voltage sweeps from +2 V to -3 V to +2 V at a rate of 1.52 Vs−1 +measured between a SET voltage of +2 V and a -3 V RESET voltage. Measurements are shown for different devices to +demonstrate the low device-to-device variability. +Figure S4: Measurements of 800 nm devices: device-to-device variation when controlled with a larger voltage range. +The red graph is the device presented in the main text. These devices show a greater degree of variation, due to small +differences in their areas and edges arising from the fabrication process. There resistance ratios, however remain high. +S2 + +32 μm +10 μm +3.2 μm +1 μm10'3 +10-5 +ICurrentl (A) +10' +109 +10 +11 +TT +-3 +-2 +-1 +0 +1 +2 +Voltage (V)MEMRISTIVE MEMORY ENHANCEMENT BY DEVICE MINIATURIZATION +Figure S5: Side wall profile of electrical measurement device: (a) and (b) STEM-HAADF images. The inset in (b) +marks the interfacial region close to the edge. STEM-energy-dispersive X-ray spectroscopy (STEM-EDX) elemental +mapping image of (c) Au, (d) Sr, (e) O, (f) Ti, (g) Al and (h) Co. +S3 + +(a) +(b) +400nm +200nm +Au +Sr +(c) +(d) +0 +Ti +(e) +(f) +(g) +Al +(h) +CoMEMRISTIVE MEMORY ENHANCEMENT BY DEVICE MINIATURIZATION +Figure S6: Nb:STO/Ti interface: (a) STEM-EDX elemental map of Sr Ti and O. (b) elemental intensity as a function +of position along the line scan in (a). STEM-iDPC images of (c) the interface and (d) away from the interface. +S4 + +(a) +Sr +Ti +line scan +2 nm +(b) +Net Intensity / Counts +Intensity / kCounts +PositionMEMRISTIVE MEMORY ENHANCEMENT BY DEVICE MINIATURIZATION +Figure S7: Ti-column displacement: iDPC-STEM image inside Nb:STO substrate close to the interface. Some of the +Ti ions occupying ideal perovskite positions are marked in yellow while displaced ions are marked in red with arrows +highlighting the direction of displacement. +S5 + +QQ0 +000 +000 +000 +0Q0 +unmMEMRISTIVE MEMORY ENHANCEMENT BY DEVICE MINIATURIZATION +S1. Derivation of the relation between trapping density and exponent. +Figure S8: Schematic of parameters in section S1: Eap and x represent the applied electric field and centroid of +trapped charge, defined with respect to the interface, respectively. The number density of trapped charges, n, is depicted +by the black curve as a function of position in the dielectric. +If we assume that the rate of trapping has no dependence on the location of traps, the electric field, E, can be expressed +as: +E = Eap − qnx +η +(S1) +where Eap is the applied electric field, q the electric charge, n is the number density of trapped charges, x is the centroid +of the trapped charge with respect to the interface and η is the dielectric permittivity. In [S1], charge trapping was +analyzed on the basis of three mechanisms, namely first-order trapping, first-order trapping with Coulombic interactions, +and trapping which increases during injection due to the generation of states. The expressions for current they derive +are qualitatively similar for each mechanism. Hence, for simplicity, we consider the rate of trapping density to be a +decay in first order with the addition of electron-electron interactions. Coulombic repulsion may inactivate trapping +sites surrounding a trapped electron. This is included in the rate equation by multiplying a probability factor. If the +volume of dielectric rendered inactive by a trap is h, then the trapping is reduced by a factor of (1 − h +V ), where V is the +volume of the dielectric. For n trapped charges, the factor is (1 − h +V )n. The trapping rate can be expressed as: +dn +dt = (n0 − n)σ Jvth +qvd +� +1 − h +V +�n +(S2) +where n0 is the maximum number of traps available, J/q is the net flux density, vth and vd are the thermal and drift +velocities respectively, and σ is capture cross-section. Assuming the total volume of the dielectric to be much larger +than the volume deactivated by trapping events so that, 1≫ h/V and n0 ≫ n, this expression can be simplified to: +dn +dt = n0σ Jvth +qvd +e− nh +V +(S3) +S6 + +E +apMEMRISTIVE MEMORY ENHANCEMENT BY DEVICE MINIATURIZATION +Solving this equation yields the following expression for n: +n = V +h ln +� Q +Q∗ + 1 +� +(S4) +where Q = +� +Jdt is the total injected charge and +Q∗ = +V vdq +n0hvthσ +(S5) +We express the current in terms of the electric field as: +ln +� J +J0 +� += E +E0 += 1 +E0 +� +Eap − V +h +qx +η ln +� Q +Q∗ + 1 +�� +(S6) +The current follows a decaying power law with time, J = Jst−α, and the injected charge as a function of time is given +by: +Q(t) = +� +Jdt = Jst1−α +1 − α +(S7) +Substituting S7 into S6 when Q/Q∗ ≫ yields and noting β = V qx +hE0ϵ: +ln +� J +J0 +� += 1 +E0 +� +Eap − β ln +� +Jst1−α +Q∗(1 − α) +�� += 1 +E0 +� +Eap − β ln +� +Jst +Q∗(1 − α) +� +− β(1 − α) ln t +� +(S8) +Comparing S8 with J = Jst−α implies +α = β(1 − α) += +β +1 + β +(S9) +and +Js ≈ mEap + n − β ln(Js) +(S10) +Where m encompasses several material parameters. Writing β in terms of α, and since measured currents are less than +10−4 A, Js can be neglected in comparison to ln(Js), leading to: +α +1 − α ln(Js) ≈ mEap + n +(S11) +β is positive, we know from Eq. S9 that α lies between 0 and 1, and is a monotonically increasing function of β. +Considering that β = V +h +qx +E0 , an increase in either the effective density, V/h or in x gives rise to an increase in α, with +the former being physically more likely. +Instead of deriving an explicit expression for the number density of trapped charge, we can also directly relate the +trapping rate to the current, as was done in for example [S2]. We use QT to denote the charge that is trapped when +charge Q is injected into the dielectric. The ratio dQT +dQ is assumed to be a function of current, i.e. +dQT +dQ = f +� J +J0 +� +(S12) +Substituting Eq. S12 into Eq. S1 gives: +dE +dt = Jx +lη +dQT +dQ +(S13) +where l is the length of the dielectric and J = dQ +dt . To relate this to the power law, we assume a solution of the form +dQT +dQ = +� J +J0 +�α +(S14) +with α ≥ 0. A general expression for the current assumes the form: +J(E) = J0(E)eg(E0) +(S15) +S7 + +MEMRISTIVE MEMORY ENHANCEMENT BY DEVICE MINIATURIZATION +where the specific functions are determined by the relevant conduction mechanisms. Specifically here, using Eq. S14 +we can express the current as +J = Js +t +t0 +−1/(α+1) +(S16) +For conduction given by an exponential relation as in Eq. S6: +Js ∝ e +(1−1/(α+1)V +V0 +(S17) +while for conduction determined by Frenkle-Poole equation, we arrive at: +Js ∝ V e +(1−1/(α+1)V +1 +2 +V0 +(S18) +and for Fowler-Nordheim conduction we get: +Js ∝ V 2e +(1−1/(α+1))V −1 +V0 +(S19) +Here, V0 is a constant. +The dominant transport mechanism in a system can be determined by plotting the current versus voltage on a double +logarithmic scale. Equations S9, S11, S17, S18 and S19 indicate a clear theoretical proof that the exponent in the power +law is directly proportional to the effective trap density or capacity of the dielectric to trap electrons, independent of +which transport mechanism is dominant. +S8 + +MEMRISTIVE MEMORY ENHANCEMENT BY DEVICE MINIATURIZATION +Figure S9: Fits of the retention data to extract the exponents α: The model used is |I| = I0(t − t0)−α), where |I| +and t are the absolute current and time respectively, and I0 and t0 are fitting parameters. The adjusted R2 values of the +fits are (a) 0.99237, (b) 0.99475, (c) 0.99689, (d) 0.99717, (e) 0.99995, and (f) 0.99996. (g) shows the dependence of +the exponents on area. +S2. Modeling the edge effects +To visualize the field profiles in our devices we used finite element analysis (COMSOL). The modeling geometry is +shown in Fig S10. In each simulation, the Nb:STO substrate was modelled as a cube with a dielectric constant of 300 +and a thickness of 0.5 mm (along z), corresponding to the thickness used in the experimental study. A circular Co +electrode of radius 1 μm, 10 μm or 100 μm was placed on the top surface of the substrate (z=0.5 mm). A ground node +was placed on the bottom of the substrate (z=0), while a voltage was applied to the top Co electrode. For the simulations +in Fig. S13. the size of the substrate was reduced to improve the resolution of the mesh. This was required to retain the +circular nature of the electrodes for the 10 nm devices; this was determined not to influence the electric field strength. +S9 + +100 μum read at +0.3 V +10 μum read at +0.3 V +1 μm read at +0.3 V +α=0.041±0.004 +α=0.47±0.02 +α=0.85±0.03 +104 +10~5 +105 +I (A) +[Currentl +ICurrentl +ICurrentl +10° +10 +(a) +(b) +107 +(c) +10° +10° +Time (a.u.) +Time (a.u.) +Time (a.u.) +100 μm read at -0.5 V +10 μm read at -0.5 V +1 μm read at -0.5 V +α=0.626±0.007 +α=0.987±0.002 +10~5 +α=2.17±0.02 +10°5 +10° +10*6 +Currentl (A) +W10* +ICurrentl (A) +10- +ICurrentl +10° +10 +10° +109 +(d) +10~8 +(e) +() +10° +10- +Time (a.u.) +Time (a.u.) +Time (a.u.) +Read at +0.3 +2.0 +Read at -0.5 +1.5 +1.0 +0.5 +0.0 +(g) +10-12 +10-11 +10-10 +10-9 +108 +Area (m²)MEMRISTIVE MEMORY ENHANCEMENT BY DEVICE MINIATURIZATION +Figure S10: Model sample geometry: the Nb:STO substrate is represented by a cube with a thickness of 0.5 mm +(along z). Circular Co electrode of radii (a) 1 μm, (b) 10 μm and (c) 100 μm is placed on the top surface of the substrate +(z=0.5 mm). A ground node is placed on the bottom of the substrate (z=0), while a voltage is applied to the top Co +electrode. +Figure S11: Electric field at -3 V: along the surface normal (z direction) for (a)+(d) 1 μm, (b)+(e) 10 μm and (c)+(f) +100 μm devices, The plots on the top row ((a)-(c)) have the same scale bar, with a maximum field value of 3.5 ×106 +Vm−1. For the bottom row, the scale bar has a maximum value of (d) 2 ×107 Vm−1, (e) 2 ×106 Vm−1 and (f) 3 ×105 +Vm−1. +S10 + +(a) +×10*4 m +(b) +×10'4 m +(c) +×10'4 m +2 +2 +2 +0 +×104 m +2 +×10*4 m +×104 m +×104 m(a) +X10° +(b) +310° +(c) +X10% +3 +3 +3 +2.5 +2.5 +2.5 +2 +２ +2 +1.5 +1.5 +1.5 +1 +0.5 +0.5 +0.5 +(d) +X107 +(e) +(f) +×106 +X105 +2 +2 +1.8 +1.8 +2.5 +1.6 +1.6 +1.4 +1.4 +1.2 +1.2 +1 +1 +1.5 +0.8 +0.8 +0.6 +0.6 +0.4 +0.4 +0.5 +0.2 +0.2MEMRISTIVE MEMORY ENHANCEMENT BY DEVICE MINIATURIZATION +Figure S12: Electric field at +2 V: along the surface normal (z direction) for (a)+(d) 1 μm, (b)+(e) 10 μm and (c)+(f) +100 μm devices, The plots on the top row ((a)-(c)) have the same scale bar, with a maximum field strength of -3 ×106 +Vm−1. For the bottom row, the scale bar has a maximum value of (d) -1 ×107 Vm−1, (e) -1.5 ×106 Vm−1 and (f) -2 +×105 Vm−1. +Figure S13: Electric field at -3 V: along the surface normal (z direction) for (a) 100 nm, (b) 50 nm and (c) 10 nm +devices. No saturation of the field is observed in (a) and (b) and the field appears to saturate in the 10 nm devices. +S11 + +(a) +(b) +(c) +X106 +X106 +0 +0 +-0.5 +-0.5 +-0.5 +-1 +-1 +-1 +-1.5 +-1.5 +-1.5 +-2 +-2 +-2.5 +-2.5 +-2.5 +-3 +-3 +-3 +(d) +(e) +(f) +×107 +X106 +X105 +10 +0 +10 +-0.1 +-0.2 +-0.2 +-0.2 +-0.4 +-0.4 +-0.3 +-0.6 +-0.4 +-0.6 +-0.8 +0.5 +1 +-0.8 +-0.6 +-1.2 +-1 +0.7 +-1.4 +-0.8 +-1.2 +-1.6 +-0.9 +-1.8 +-1.4 +.1 +-2(a) +(b) +(c) +X107 +×107 +X108 +9 +1.3 +8.8 +1.29 +5.5 +8.6 +1.28 +8.4 +1.27 +8.2 +1.26 +5 +8 +1.25 +7.8 +1.24. +4.5 +7.6 +1.23 +7.4 +1.22 +4 +7.2 +1.21 +3.5 +7 +1.2MEMRISTIVE MEMORY ENHANCEMENT BY DEVICE MINIATURIZATION +S3. Literature survey of interfacial switching +It is often suggested that a layer close to the interface layer is responsible for the switching [29, S4, S5]. Some groups +have shown that both high and low resistance states show an area-independent current density, eluding to a switching +mechanism that occurs homogeneously over the entire device area [S6]. Often this is explained in terms of a change +in the Schottky barrier height and width induced by charge trapping at the interface [S7, 29, S4, S8, 12, S10] and +movement of oxygen vacancies [S8, S10]. Other explanations are proposed where the barrier profile is unchanged +and interfacial changes happen at local regions. Explanation of this type includes It has also been proposed that the +application of a positive bias results in the generation of oxygen vacancies, forming tunnelling paths and giving rise to a +LRS where tunnelling, rather than thermionic emission dominate charge transport. The application of a negative bias +results in the accumulated of large amounts of oxygen in the vacancies which prevents tunnelling and gives a HRS +[S11, S12]. +Rodenbücher et al. used local-conductivity AFM on highly doped Nb:STO to show the presence of nanoscale +conducting and switchable clusters. Suggesting that in this case switching is a local phenomenon related to the presence +of conducting clusters with higher Nb content than their surroundings [S13]. +Finally Chen et al. used scanning tunnelling microscopy and spectroscopy to study the resistive switching in Nb-doped +SrTiO3 without an electrode, demonstrating that oxygen migration is the results in a variation of electronic structure +during the switching. With a negative voltage, oxygen anions at the interface near the STM tip were oxidised into oxygen +molecules and left the lattice. Simultaneously, oxygen vacancies diffuse into the sample, which act like donor-like +levels causing distortions in LDOS near conduction band and enhance the carrier concentration with electron hopping, +thus increasing the sample’s conducting. With a positive voltage, oxygen anions return into the sample and the influence +of the donor-like level became weak and the conductivity decreased [S14]. +Despite a large number of contradictory results and explanations, factors of importance that have been identified include +the semiconductor doping concentration, electrode material and the quality of the interface. +S12 + +MEMRISTIVE MEMORY ENHANCEMENT BY DEVICE MINIATURIZATION +References +[S1] D. R. Wolters and J. J. van der Schoot. Kinetics of charge trapping in dielectrics. Journal of Applied Physics, +58(2):831–837, 1985. +[S2] R. H. Walden. A method for the determination of high-field conduction laws in insulating films in the presence +of charge trapping. 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Kawasaki, A. Sawa, Y. Kawazoe, H. Akoh, and Y. Tokura. Electrical properties and colossal +electroresistance of heteroepitaxial SrRuO3/SrTi1−xNbxO3 (0.0002≤x≤0.02) Schottky junctions. Physical +Review B, 75(16):165101, 2007. +[S12] D.-J. Seong, D. Lee, M. Pyun, J. Yoon, and H. Hwang. Understanding of the switching mechanism of a +Pt/Ni-doped SrTiO3 junction via current–voltage and capacitance–voltage measurements. Japanese Journal of +Applied Physics, 47(12R):8749, 2008. +[S13] C. Rodenbücher, W. Speier, G. Bihlmayer, U. Breuer, R. Waser, and K. Szot. Cluster-like resistive switching of +SrTiO3:Nb surface layers. New Journal of Physics, 15(10):103017, 2013. +[S14] Y. L. Chen, J. Wang, C. M. Xiong, R. F. Dou, J. Y. Yang, and J. C. Nie. +Scanning tunneling mi- +croscopy/spectroscopy studies of resistive switching in Nb-doped SrTiO3. +Journal of Applied Physics, +112(2):023703, 2012. +S13 + diff --git a/G9E1T4oBgHgl3EQfrQVc/content/tmp_files/load_file.txt b/G9E1T4oBgHgl3EQfrQVc/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7c09edb3c180ce45715f352387f69ac9cba1b809 --- /dev/null +++ b/G9E1T4oBgHgl3EQfrQVc/content/tmp_files/load_file.txt @@ -0,0 +1,1034 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf,len=1033 +page_content='MEMRISTIVE MEMORY ENHANCEMENT BY DEVICE MINIATURIZATION FOR NEUROMORPHIC COMPUTING A PREPRINT A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Goossens 1,2,*, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Ahmadi 1,2, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Gupta 1,2, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Bhaduri 1,2, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Kooi 1,2, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Banerjee 1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' * 1Groningen Cognitive Systems and Materials Center, University of Groningen, Groningen, The Netherlands 2Zernike Institute for Advanced Materials, University of Groningen, Groningen, The Netherlands {a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='goossens,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='banerjee}@rug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='nl January, 2023 ABSTRACT The areal footprint of memristors is a key consideration in material-based neuromorophic comput- ing and large-scale architecture integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Electronic transport in the most widely investigated memristive devices is mediated by filaments, posing a challenge to their scalability in architecture implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Here we present a compelling alternative memristive device and demonstrate that areal downscaling leads to enhancement in memristive memory window, while maintaining analogue behavior, contrary to expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Our device designs directly integrated on semiconducting Nb- SrTiO3 allows leveraging electric field effects at edges, increasing the dynamic range in smaller devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Our findings are substantiated by studying the microscopic nature of switching using scan- ning transmission electron microscopy, in different resistive states, revealing an interfacial layer whose physical extent is influenced by applied electric fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' The ability of Nb-SrTiO3 memristors to satisfy hardware and software requirements with downscaling, while significantly enhancing memristive functionalities, makes them strong contenders for non-von Neumann computing, beyond CMOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Keywords Interface memristor, Areal scaling, Beyond CMOS, Neuromorphic computing, Scanning transmission electron microscopy (STEM) 1 Introduction The growing demand for applications such as artificial intelligence and the Internet of Things has given rise to critical challenges in the storage and processing of big data using existing computational architectures [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' The currently employed von Neumann architecture, using complementary metal-oxide-semiconductor (CMOS) hardware, suffers from limited transmission speed [2, 3, 4] due to a memory throughput bottleneck as well as energy inefficiency and limited scalability [4, 5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Moving away from CMOS technology, towards logic-in-memory chips would alleviate some of the above issues but requires us to massively rethink every aspect of computing [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' The first step towards this is identifying novel materials and devices with suitable physical properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Resistive switching devices, or memristors, are one such class of devices where the resistance can be switched between several states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Reported in different ionic materials, they are distinguished by the switching mechanism as either occurring through the material bulk between two electrodes or interface-type where switching takes place in a localized region underneath the area of the electrodes [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Their ability to co-locate memory and computation, and exhibit characteristics absent in digital computing makes them important for novel computing approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Given the robust way in which the human brain is able to process large amounts of data with remarkably low power, it is unsurprising that it serves as a source of inspiration to the development of computing beyond using CMOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' As the brain utilizes a vast network, downscaling memristive devices is a crucial area of research to develop large scale neuromorphic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='03352v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='ET] 9 Jan 2023 MEMRISTIVE MEMORY ENHANCEMENT BY DEVICE MINIATURIZATION For this material-driven research, the areal footprint in unconventional computing architectures that seek to integrate in-memory computing devices such as memristors is a prime consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Considerable research has been devoted to this in the realm of non-volatile conventional filamentary devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' The challenges in their implementation in such novel architectures, besides the requirement for unfavourable electroforming processes, lie in their switching endurance [9], and their efficacy to exhibit discernible analogue resistance states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Memristive devices that exhibit more than two stable states also greatly enhance integration density because each device can store multiple data bits in an analogue manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' In valence change memristors, where switching originates from filaments, such behavior is observed in large areal dimensions but is lost when devices are downscaled and conduction is mediated by a single nanoscale filament causing an abrupt transition between the two resistance states [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Further, the effects of Joule heating on filaments are an important consideration as devices shrink;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Joule heating can cause a wide distribution of switching voltages and endurance deterioration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' These limitations in device stability, endurance and associated enhanced power of operation are major roadblocks in the successful implementation of filamentary devices in large scale architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Memristive devices have the potential to be integrated in large scale architectures, for which they should exhibit large memory windows, high endurance and low variability [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Herein the areal switching mechanism is a strong contender.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' A model system in which this mechanism is dominant is Schottky contacts on Nb-doped SrTiO3 (Nb:STO), formed at the interface with a high work function metal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' It is widely accepted that in these material systems it is not the bulk of the device, but an area close to the interface that is responsible for the switching, a more detailed discussion on the proposed mechanisms is presented in Supporting Information section S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Distinguishing Nb:STO from conventional semiconductors such as Si, widely used in conventional architectures, is its dielectric permittivity which is comparatively large (300) and is strongly dependent on electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' This property extends the parameter space for designing functionality: electric fields can be used to tune the barrier height and width relevant for memristive device design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' We have previously shown that such Schottky contacts form robust memristors, exhibiting non-linear transport and continuous conductance modulation [12], and that their behavior can be described by a power-law which can be successfully implemented as a learning algorithm [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' However, for the applicability of Nb:STO-based memristors as hardware elements for non-von Neumann computing architecture beyond CMOS, the focus should be on establishing their memristive performance with device miniaturization, which has not been shown on such semiconducting platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' In this work, we demonstrate that memristive devices of Co Schottky contacts on Nb:STO exhibit an increase in the analogue memristive memory window in devices down to 1 μm, contrary to expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Ionic defects are at the heart of memristive behavior, hence one of the following two scenarios is expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' For a homogeneous areal mechanism, the current density will scale with device area so that the device resistance in both the high resistance state (HRS) and the low resistance state (LRS) scales with the electrode size, but the ratio between them is area independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Alternatively, the resistance window can be severely reduced or even vanish with downscaling due to insufficient ionic defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' However, we observe an enhancement in the memory window as the device area is reduced, with minimal device-to-device variation, an unforeseen finding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' To understand the microscopic nature of the switching, we conducted scanning transmission electron microscopy (STEM) on virgin samples and on samples subjected to either a positive (SET) or negative (RESET) voltage .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Using integrated differential phase contrast (iDPC) we image oxygen atomic columns next to the heavy metal atomic columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Figure 1: State stability and multilevel memristive operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' a Schematic of the fabricated devices on Nb-doped SrTiO3, electrical connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Black lines are used to represent the varying overall electric fields acting over each area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' The field strengths at the interface are also indicated by a color gradient, showing the fields are weakest in the central area (blue) and strongest around the perimeter (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' b Current read at +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='3 V for device sizes of 100 μm (black), 10 μm (blue) and 1 μm (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' c Current read at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='3 V after switching between a SET voltage of +1 V (black, red and blue) or +2 V (green, purple and orange) and a RESET voltage of -2 V (black and green), -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='5 V (red and purple) or -3 V (blue and orange).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Each combination was repeated over 100 cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' 2 +2 V +2 V 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='5 Nb-doped SrTiO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' 3 VMEMRISTIVE MEMORY ENHANCEMENT BY DEVICE MINIATURIZATION Figure 2: Characterization of memristive devices in the virgin state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Electrical characteristics of virgin devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' The compliance current was fixed at 100 mA for all measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Results are shown for four devices of each area in a-f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Virgin samples show the existence of a layer near the interface with neither the perovskite structure of the substrate nor that of the Co electrode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Applying a bias across the interface results in oxygen vacancy movement, which is a key factor controlling the resistance states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' These new revelations are consolidated with a mathematical model describing the kinetics of trapping and de-trapping in dielectric materials and relates experimental results to the effective trapping density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Surprisingly, this is found to be larger for smaller junctions, suggesting that an increase in the density of traps is responsible for the increased resistance ratio and attributed to inhomogeneous distribution of the electric field due to device edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' These memristive devices, integrated directly on a semiconducting platform, demonstrate multistate analogue switching with remarkably high memory windows with downscaling, as well as high endurance and low device and cycle variation down to the smallest devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Their ability to meet both hardware and software requirements for unconventional computing, make Nb:STO memristors strong material contenders for physical computing beyond CMOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' 2 Results 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='1 Electrical Characterization Figure 1a shows a schematic of the device structure used for the electrical measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' An array of circular Co electrodes of varying sizes are fabricated on a semiconducting Nb:STO single crystalline substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' The bottom of the substrate serves as a back contact for the devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' The top electrodes were patterned by a two-step electron lithography process using aluminium oxide as an insulation layer to define the contact areas and to prevent electronic cross talk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' After fabrication, we performed small range voltage sweeps to characterize the virgin states of each device on a chip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' The results for devices with radial dimension from 100 μm to 800 nm are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' 2, where each sweep followed a voltage sequence from 0 to +1 V to -1 V and back to 0 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' We show four devices of each area, which are plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' 2a-f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' The current magnitudes for different devices of the same area show no significant differences down to 1 μm, indicating device-to-device variations are minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Establishing this is important as this signifies the sole influence of device area in determining the resistance ratio and rules out contributions from device-to-device variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' The 800 nm devices show a greater degree of variation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' this is likely due to small differences in their areas and edges arising from the fabrication process and not inherent to the material or due to device fallibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' No significant differences in the current 3 MEMRISTIVE MEMORY ENHANCEMENT BY DEVICE MINIATURIZATION Figure 3: Resistance ratio, cycling endurance and state stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' a-f show 1000 consecutive current-voltage sweeps from +2 V to -3 V to +2 V at a rate of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='52 Vs−1 for devices of 100 μm down to 800 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Starting from a SET voltage of +2 V, each device is in an LRS, represented by the upper branch reaching the RESET voltage of -3 V and sweeping back, the devices are switched to an HRS (represented by the lower branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' densities at low bias values are found in the virgin state, confirming that the entire device area contributes to the charge transport (Supporting Information Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' For all the devices, the current gradually increases and exhibits a small hysteretic effect from the virgin state, indicating that no forming step is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Figure 3a-f shows 1000 consecutive current-voltage (I-V) sweeps of these devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Starting from a SET voltage of +2 V, each device is in an LRS, represented by the upper branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' After reaching the RESET voltage of -3 V and sweeping back, the devices are switched to an HRS (represented by the lower branches).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' In all device areas both the SET and RESET operation remain continuous, indicating the resistive switching retains its analog nature when downscaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' The cycling endurance was measured for over 105 switching cycles without device failure, illustrating an endurance of >105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' The current in the HRSs scales approximately with area at low bias values, while the low resistance current, is less closely correlated to the area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' As a result, the resistance window increases with decreasing device area in both forward and reverse bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Figure 1b and Supporting Information Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' S2 show the current and current density at a low read voltage of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='3 V, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Minimal cycle-to-cycle variations at low reading voltages are found with reproducible switching between clearly distinguishable states without degradation in device performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' This also establishes the low power operation of these devices after downscaling, which is important for memristor operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' As shown in Supporting Information Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' S3, the device-to-device variation remains low down to 1 μm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' The variation in the resistance ratio in the 800 nm devices is larger (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' S4), and will be discussed later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' The SET and RESET transitions are gradual and highly tunable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' To demonstrate this, a 1 μm device was subjected to voltage sweeps varying between different positive (SET) and negative (RESET) voltages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Figure 1c shows that a wide range of stable states is available at a low read voltage of +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='3 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' The wide dynamic range combined with the large number of distinct addressable states ensures device reliability and increased memory storage capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Each state maintains a narrow distribution of current values over the 100 cycles shown, reiterating the stability of the switching process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='2 Scanning Transmission Electron Microscopy A microscopy study of the Schottky interface was carried out using STEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Figure 4 shows atomic resolution cross- section STEM-integrated Differential Phase Contrast (iDPC) images of the Co/Nb:STO interface for samples in the 4 MEMRISTIVE MEMORY ENHANCEMENT BY DEVICE MINIATURIZATION Figure 4: Visualization of oxygen vacancy migration using STEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' iDPC-STEM images of Co/Nb:STO samples in a the virgin (unbiased) state, b the LRS and (c the HRS, highlighting the structure close and far from the interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' The perovskite unit cell of STO, showing Sr in green, O in dark red and Ti in light red, viewed along the <110> in d the pristine state and (e with oxygen vacancies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' The deficiency of O causes Ti atoms to move away from the vacancies as shown by the arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' f shows a schematic representation of how the interfacial layer is affected by biasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' unbiased virgin condition (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' 4a), the LRS state (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' 4b) and the HRS state (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' 4c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' To image lighter oxygen atoms, integrated into a matrix with heavier Sr and Ti atoms, we utilized STEM-integrated Differential Phase Contrast (iDPC) instead of the more commonly employed STEM-High-angle annular dark-field (HAADF) imaging technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' [14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' The STEM images in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' 4a show that, apart from a thin interfacial region, the bulk STO consists of a cubic perovskite lattice and no defects are observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' All images taken within the bulk did not show any dislocation and possessed the expected perovskite structure as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' 4d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' However, the structure close to the interface deviates from this perovskite structure and is deficient in oxygen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' The migration of oxygen ions near the interface towards Co causes positively charged Ti ions to be displaced so that they no longer sit equidistantly from the Sr ions along <001>.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Figure 4e illustrates how the loss of O ions gives rise to Ti displacements along the <001> direction away from the interface as well as along <1-10> (see Supporting Information Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' S7) and is similar to what was reported in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' [16] in La0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='67Sr0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='33MnO3/Hf0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='5Zr0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='5O2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' We believe the creation of this thin layer to be related to the formation of a Schottky barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' The analysis for a non-memristive interface with Ti contacts can be found in Supporting Information Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Figure 4b shows analogous results to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' 4a, but now for the sample switched to the LRS, representing the upper branch in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' 3, after the application of a positive bias voltage of 2 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Comparing the two figures shows that in the LRS state the extent of the interfacial layer has decreased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' This suggests that under the influence of a positive voltage, the labile bonds between O and interfacial Co atoms are broken and oxygen moves back into the STO substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' A negative bias voltage of -3 V (corresponding to the lower branch in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' 3), on the other hand, causes oxygen to move from STO to cobalt causing the formation of CoO and more oxygen vacancies in the STO, highlighted by a larger region over which Ti ions are displaced (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' 4c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' This indicates that the formation of the CoO switches the sample to the HRS state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' It has been shown [17, 18] that the oxygen vacancy distribution inside the system will determine how the oxygen vacancies are affected by the applied voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' The formation of an oxygen deficient interfacial layer confirms that in these samples the oxygen vacancies are concentrated near the interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' In this case, it is expected that the application of a positive voltage will cause oxygen vacancies to be repelled from the interface while a negative voltage will cause oxygen vacancies to be attracted to the interface, consistent with our findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' After removing the 5 (a middle of STO middle of STO Virgin state Low resistance state High resistance state (d) (f) (e) <001> Virgin state LR state HR state Co Co Co oxygen deficient areal oxygen deficient area STO STO STO → <1-10> Zone axis: <110>MEMRISTIVE MEMORY ENHANCEMENT BY DEVICE MINIATURIZATION voltage, the interfacial layer did not reform over time, suggesting the presence of an oxygen-migration blocking layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' These results are summarized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' 4f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Our results directly confirm the existence of a homogeneous oxygen deficient layer at the interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' The homogeneous nature of the defect state layer ensures ionic defects are retained with downscaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' We furthermore show that the physical extent of the layer is reduced or extended when a positive or negative voltage is applied respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Although the uniform nature of the ionic contribution to switching is now verified, this does not explain the origin of the unexpected enhancement of the resistance window with downscaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' This we discuss next by considering the trapping of electronic charges at oxygen vacancy sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='3 Model In order to understand how the electrical properties of the devices are influenced by these oxygen vacancies, we consider the interaction between electrons and defect states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' This interaction is most strongly evidenced by the retention characteristics, which have a slow decaying component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' This behavior is caused by the detrapping of charges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' It has been shown that this occurs over long timescales and the different states will remain clearly distinguishable for long time periods of hours and that the retention time is tunable by the applied stimuli [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' We utilized short voltage pulses to measure the retention characteristics of each device in both an HRS and LRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' This was done by applying alternating SET and RESET pulses of +2 V and -3 V respectively, and reading the small-signal current at either +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='3 V or -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='5 V after each writing event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' The state retention characteristics of the different devices are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' 5 for the LRS (red) and HRS (black).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Over time, the current in both states tends to an intermediate value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' For the LRS, the rate of change follows a power law that is commonly observed for charge trapping under bias in high-κ dielectrics, referred to as the Curie-von Schweidler law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' This law describes a non-Debye type relaxation in dielectrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Empirical evidence of this behavior is seen in a wide variety of materials, but the precise physical origin remains unclear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Here we consider the effect of injected electrons becoming trapped in defects states within the dielectric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' The space charge generated by these trapped electrons lowers the electric field, in turn reducing the flow of current through the dielectric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' In this case the trapping rate can be expressed as: dn dt = n0σ Jvth qvd e− nh V (1) where n0 is the maximum number of traps available, J/q is the net flux density, vth and vd are the thermal and drift velocities respectively, and σ is capture cross-section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Solving this equation yields the following expression for n: n = V h ln � Q Q∗ + 1 � (2) where Q = � Jdt is the total injected charge and Q∗ = V vdq n0hvthσ (3) Expressing the current as J = Jst−α and extending this analysis results in: α 1 − α ln(Js) ≈ mEap + n (4) where m is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' We can also directly relate the trapping rate to the current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' QT represents the charge that is trapped when charge Q is injected into the dielectric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' The ratio dQT dQ is a function of current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' The current can be written as: J = Js t t0 −1/(α+1) (5) where α ≥0 and Js depends on the transport mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' For conduction following an exponential relation: Js ∝ e (1−1/(α+1))V V0 (6) Here, V0 is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' The full derivation is shown in Supporting Information Section S1 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' S8, and is also extended to show that it holds for other transport mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Equations 4 and 6 serve as a direct mathematical proof that the exponent α in the power law is related to the effective trap density or capacity of the dielectric to trap electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' This derivation is applicable to a wider range of systems, irrespective of the choice of dielectric material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' In Table 1 we show the LRS exponents, α for each device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Larger values are observed for smaller devices indicating that the trap density is higher in the smallest device compared to the larger device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' 6 MEMRISTIVE MEMORY ENHANCEMENT BY DEVICE MINIATURIZATION Figure 5: Trapping dynamics and Schottky interface energy landscapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Retention characteristics of differently sized devices read at +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='3 V a-c) and -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='5 V (d-f) after a SET voltage of +2 V (red) or -3 V (black).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' g shows the energy landscape of a Schottky interface in equilibrium when the dielectric constant does not depend on electric field (solid line) and when the dielectric constant is field-dependent (dashed line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' EF and EC are the Fermi level and conduction band respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' The energy landscapes at the center and edge of a device are compared in h in equilibrium and i in reverse bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Red circles represent oxygen vacancy states and the green arrow indicates electron tunneling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' area ��������� |Exponent| Radius (μm) Read at +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='3 V Read at -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='5 V 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='85±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='03 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='17±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='02 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='47±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='987±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='002 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='041±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='626±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='007 ��������� trapping density Table 1: Magnitude of exponents, α, extracted by fitting a power-law to the low resistance states in the graphs in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' 3 Discussion While this model provides a clear correlation between trapping density and device area, it does not give information about the traps;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' we implicitly take all traps to be of the same kind, while in reality, the nature of traps can vary greatly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' The trapping rate can depend on the spatial location of the traps and new traps can be generated via defect migration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' For a more precise picture of the mechanism, we need to consider a distribution of traps with respect to their location within the dielectric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Evidenced by the STEM study, oxygen vacancies are the most important class of trapping defects to consider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' They are abundantly present in SrTiO3 due to their low formation (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='51 eV[19]) and migration (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='62 eV[20]) enthalpies and their locations within the energy landscape are well documented[21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' 7 MEMRISTIVE MEMORY ENHANCEMENT BY DEVICE MINIATURIZATION From the discussion above, it is clear that the energy landscape of these Schottky junctions is far more complex than is captured by the most commonly used models that are based solely on parameters of the individual materials forming the contact [22, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Transport through these junctions is usually described by the thermionic emission equation, which includes an ideality factor accounting for the deviating transport from this ideal diode equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' This model furthermore does not consider that the interfacial area is not spatially homogeneous and that in devices of finite areas the boundary of the device will be relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' In particular, it is known that near the edges crowding of the field lines leads to an enhancement in the field strength which can decrease the barrier width [24, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' This is supported by the results of the finite element simulations in Supporting Information Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' S11 and S12, showing a significant enhancement in the electric field around the edge and when downscaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' From the simulations it is evident that there is still a clear field gradient in the 1 μm devices, indicating that a further increase in ratio with downscaling can be expected, and the areal field shows no apparent saturation till around 10 nm (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' S13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' The observed enhancement is especially important in Nb:STO-based memristive devices as the dielectric constant of the substrate strongly depends on electric fields [26, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' This will further alter the potential landscape of the Schottky interface in such memristive devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' In particular, the dielectric permittivity of Nb:STO rapidly decreases in the presence of large electric fields which results in a decrease in the effective Schottky barrier width as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' 5g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Consequently, a large reduction in the barrier width is expected to occur near the device edges (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' 5h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' It has also been shown that an electric field can modify the defect states and significantly affect trapping parameters[28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Given that the charge transport is governed by the potential landscape, this will hugely impact the measured current, pictured in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' 5i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Tunneling through the barrier will be enhanced near the device edges leading to a larger current near the device perimeter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' This will be especially important in the LRS where the interface is depleted of trapped charges and the Schottky barrier is narrower, leading to more tunneling [29, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Transport across the interface is comprised of thermionic emission and tunneling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' The thermionic current density is expected to be independent of area and is the dominant mechanism in the HRS at low bias voltages, giving rise to the decreasing current in the HRS around zero with downscaling observed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' At higher voltage values, however, tunneling will also contribute to the current;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' the tunneling current density will increase with decreasing area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' 1b, the current is read at +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='3 V where we expect both thermionic emission and tunneling to contribute to transport, giving rise to similar currents measured for the 10 and 1 μm devices in the HRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' The tunneling contribution increases in the LRS, especially in smaller devices due to the larger electric fields, resulting in the observed increase in current density with reducing area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' By applying a potential over the Schottky barrier, the Fermi level is shifted such that tunneling electrons sample different oxygen vacancy energy levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' As the reverse bias voltage is increased, electrons are gradually exposed to larger ranges of states in which they can become trapped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' In addition, in reverse bias, the electric field at the interface becomes larger leading to a reduction in the dielectric constant and a corresponding decrease of the Schottky barrier widths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' This decrease in width will be more pronounced in regions closer to the edge due to the local field enhancement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' As a result of the narrower barrier, electron-electron scattering will be reduced and the trap states will act as the main barrier for transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' The stronger edge field may additionally facilitate the migration of oxygen vacancies resulting in a higher number of vacancies accumulating around the perimeter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Consequently, the trapping efficiency will be greater near the edge than in the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' This is a unique effect enabled by the electric field control of the dielectric permittivity, does not occur in conventional semiconductors and is relevant for Nb:STO memristive device design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' We can express the area and perimeter of a device with radius r as A = πr2 and p = 2πr respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' The ratio of the perimeter to area: p A = 2πr πr2 = 2 r (7) indicates that the edge effects become more dominant as the device area is reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' As a result, current flow at the perimeter will constitute a larger percentage to the overall transport behavior in smaller devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' This explains the enhanced current densities observed when downscaling after applying large bias voltages as well as the larger effective trapping densities for smaller devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Specifically, this field enhancement around the device edges gives rise to an increase in the dynamic range in smaller devices, and explains the unexpected resistance window scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' 4 Conclusions As a first demonstration of exploiting edge effect related additional electric fields, our work successfully demonstrates the ability to increase the resistance window by device miniaturization of interface memristors from 100 μm down to 1 μm, contrary to expectations, with exceptional robustness to device-to-device and cycle variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Scanning transmission electron microscopy images taken in the virgin, high and low resistance states prove the existence of a homogeneous interfacial layer, deficient in oxygen, whose physical extent is influenced by applying an electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' 8 MEMRISTIVE MEMORY ENHANCEMENT BY DEVICE MINIATURIZATION This, however, does not explain the enhancement in the resistance window with device downscaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' A model describing the interaction of electrons with oxygen vacancy trap states shows an increase in the effective trapping density with downscaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' The advantage of direct integration of devices on a semiconducting platform of Nb-doped SrTiO3 allows for the locally enhanced fields to controllably tune the interfacial energy landscape at the interface, leading to a greater contribution of edge effects in smaller devices as confirmed by finite element simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' With rapid advances made in the palette of materials and devices available for neuromorphic hardware, the thrust now should be in their efficient integration on semiconducting platforms for on-chip applications with substantial reduction in areal footprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' In this, our work provides an encouraging direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' 5 Experimental Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='1 Electrical Device Fabrication We investigated a series of Co/Nb-doped SrTiO3 devices, where the device area was varied across the series over a range spanning five orders of magnitude ranging from 10−12 to 10−8 m2, with radii between 800 nm and 100 μm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' The devices were fabricated using Nb-doped SrTiO3 (001) substrates with a doping concentration of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='1 wt% from Crystec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' SrTiO3 consists of alternating SrO and TiO2 planes along the [001] direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' The as-received substrates have a slight miscut from the exact crystallographic direction and as a result, a mixture of both terminations exists at the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' It has been shown that the local properties of Schottky barriers grown on the different terminations may differ, hence to minimize the variation of different areas on the substrate a single termination is desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' To ensure that the terminating layer is TiO2, a chemical treatment was carried out with buffered hydrofluoric acid (BHF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' A further annealing treatment at 960 ◦C in an O2 flow of 300 ccmin−1 to facilitate the reorientation of surface atoms to form an atomically flat and straight terraced surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Atomic force microscopy images were taken at different parts of the substrate and confirmed the existence of uniform terraces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' The substrate was then coated with a negative resist (AZ nLOF 2020) and using electron beam lithography circles of different areas were patterned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' A thick insulation layer of AlOx was deposited using electron beam evaporation and lift-off was carried out to define a set of direct contacts to the substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' By means of a second lithography step with a positive resist (950 K PMMA), square contact pads were defined, each covering a hole and part of the surrounding AlOx: the dimensions of these pads were identical for each device to minimize spurious effects arising from significantly different contact resistances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Co (20 nm) and a capping layer of Au (100 nm) were then deposited using electron beam evaporation in high vacuum (∼10−6 Torr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='2 Electrical Characterization Electrical measurements were conducted using probes connected to two remote-sense and switch units (RSU) of a Keysight B1500A Semiconductor Device Parameter Analyzer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' During the voltage sweeping measurements, conducted using a sweeping measurement unit (SMU), the bottom of the substrate is held at 0 V while a voltage is applied to the top electrode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Due to the diodic nature of the devices in conjunction with large degrees of resistive switching, the measured currents during a single sweeping measurement span up to 9 orders of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' For this reason, the measurements were performed using auto range for the measured current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' The effects of this can be observed in the endurance cycling measurements which were performed at high sweeping rates in the form of plateaus in the current whenever a limit of the SMU range is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='3 Scanning Transmission Electron Microscopy The samples discussed in this work use SrTiO3 (001) substrates with an Nb-doping in place of Ti of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='1 wt% from Crystec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' The surface was prepared using a chemical treatment with buffered hydrofluoric acid (BHF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Next, the substrates were annealed at 960◦C in an O2 flow of 300 ccmin−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' For STEM samples films were deposited by electron beam evaporation of 20 nm of Co capped with 20 nm of Au and 20 nm of Pt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' From this, three types of STEM lamellae were prepared: virgin (unbiased) samples, low resistance state (LRS) samples and high resistance state (HRS) samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Using a probe station, samples are subjected to bias values of +2 V and -3 V to prepare samples in the LRS and HRS respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' STEM lamellae were extracted from samples along the <110> direction using a Helios G4 CX dual beam system with a Ga focused ion beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' The lamellae were thinned to make them transparent to electrons using the focused ion beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Imaging was carried out using a Thermo Fisher Scientific Themis Z S/TEM system operating at 300 kV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' STEM-High-angle annular dark-field (HAADF) images are most widely used, because they are readily interpretable with atomic columns being bright spots in a dark surrounding, where the brightness of the spots scale with the average atomic number Z (∼Z1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' This technique is well suited to image heavy elements, but lighter elements, such as oxygen, are harder to detect, and cannot be detected properly when integrated into a matrix with much heavier elements (like Sr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Therefore, to gain more insight into the important role played here by the oxygen ions, we utilized here STEM-iDPC 9 MEMRISTIVE MEMORY ENHANCEMENT BY DEVICE MINIATURIZATION instead of STEM-HAADF imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' This technique uses a four-quadrant annular bright field detector and can be used to acquire the projected local electrostatic potential of the sample (when thin) and has clear advantages over traditional annular bright field (ABF) imaging [14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='4 Simulations Finite element modeling of the electric field profile at the interface was carried out using COMSOL Multiphysics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='5 Statistical Analysis For the |current|-voltage graphs, the absolute value of the measured current is taken;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' to determine the current density, the measured current was divided by the area of the Co contact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' The values in Table 1 were derived by iteratively fitting the data in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' 5a-f using a power-law equation of the form I = I0(t − t0)−α by means of the Levenberg-Marquardt algorithm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' the reported errors are the standard errors calculated by this method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' The fits are shown in supporting Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' S9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' The inverse scaling of the exponent and device area was verified for different devices and different reading and SET voltages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Plotting and analysis of electrical measurements was done using OriginPro 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Measurements were repeated on four devices of each area to check reproducibility and validity of results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' For STEM images, multiple regions for each one of the three bias conditions were taken to verify the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' The idpc images were filtered by applying a high-pass Gaussian filter using Velox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Acknowledgements A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' is supported by the CogniGron Center, University of Groningen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Device fabrication was realized using NanoLab NL facilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' We acknowledge technical support from J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Holstein, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' de Vries, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Joshua, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Schouten, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Adema.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' We thank R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' E Hueting, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Nukala, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' de Graaf and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Kenyon for useful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=', D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='G, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='B, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' benefited from helpful discussions with the members of the Spintronics of Functional Materials group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Conflict of Interest The authors declare no conflict of interest Author Contributions A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' conceived the idea and designed the devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' fabricated devices for electrical measurements and performed electrical measurements, along with I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='. D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' derived the mathematical model discussed in the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' fabricated lamalae for STEM and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' took STEM images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Finite element simulations were done by A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='. 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Wolf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Field dependent permittivity in metal-semiconducting SrTiO3 Schottky diodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Applied Physics Letters, 66(6):697–699, 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' [28] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Dussel and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Bube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Electric field effects in trapping processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Journal of Applied Physics, 37(7):2797– 2804, 1966.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' [29] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Mikheev, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Hoskins, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Strukov, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Stemmer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Resistive switching and its suppression in Pt/Nb:SrTiO3 junctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Nature Communications, 5(1):1–9, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' 11 MEMRISTIVE MEMORY ENHANCEMENT BY DEVICE MINIATURIZATION Supplementary Data Figure S1: Current densities of virgin devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Results are shown for devices of radial dimensions of 100 μm (black), 10 μm (blue) and 1 μm (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Figure S2: Cycle-to-cycle variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' The current density read at +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='3 V for device sizes of 100 μm (black), 10 μm (blue) and 1 μm (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' S1 MEMRISTIVE MEMORY ENHANCEMENT BY DEVICE MINIATURIZATION Figure S3: Device-to-device variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Current-voltage sweeps from +2 V to -3 V to +2 V at a rate of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='52 Vs−1 measured between a SET voltage of +2 V and a -3 V RESET voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Measurements are shown for different devices to demonstrate the low device-to-device variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Figure S4: Measurements of 800 nm devices: device-to-device variation when controlled with a larger voltage range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' The red graph is the device presented in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' These devices show a greater degree of variation, due to small differences in their areas and edges arising from the fabrication process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' There resistance ratios, however remain high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' S2 32 μm 10 μm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content="2 μm 1 μm10'3 10-5 ICurrentl (A) 10' 109 10 11 TT 3 2 1 0 1 2 Voltage (V)MEMRISTIVE MEMORY ENHANCEMENT BY DEVICE MINIATURIZATION Figure S5: Side wall profile of electrical measurement device: (a) and (b) STEM-HAADF images." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' The inset in (b) marks the interfacial region close to the edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' STEM-energy-dispersive X-ray spectroscopy (STEM-EDX) elemental mapping image of (c) Au, (d) Sr, (e) O, (f) Ti, (g) Al and (h) Co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' S3 (a) (b) 400nm 200nm Au Sr (c) (d) 0 Ti (e) (f) (g) Al (h) CoMEMRISTIVE MEMORY ENHANCEMENT BY DEVICE MINIATURIZATION Figure S6: Nb:STO/Ti interface: (a) STEM-EDX elemental map of Sr Ti and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' (b) elemental intensity as a function of position along the line scan in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' STEM-iDPC images of (c) the interface and (d) away from the interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' S4 (a) Sr Ti line scan 2 nm (b) Net Intensity / Counts Intensity / kCounts PositionMEMRISTIVE MEMORY ENHANCEMENT BY DEVICE MINIATURIZATION Figure S7: Ti-column displacement: iDPC-STEM image inside Nb:STO substrate close to the interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Some of the Ti ions occupying ideal perovskite positions are marked in yellow while displaced ions are marked in red with arrows highlighting the direction of displacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' S5 QQ0 000 000 000 0Q0 unmMEMRISTIVE MEMORY ENHANCEMENT BY DEVICE MINIATURIZATION S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Derivation of the relation between trapping density and exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Figure S8: Schematic of parameters in section S1: Eap and x represent the applied electric field and centroid of trapped charge, defined with respect to the interface, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' The number density of trapped charges, n, is depicted by the black curve as a function of position in the dielectric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' If we assume that the rate of trapping has no dependence on the location of traps, the electric field, E, can be expressed as: E = Eap − qnx η (S1) where Eap is the applied electric field, q the electric charge, n is the number density of trapped charges, x is the centroid of the trapped charge with respect to the interface and η is the dielectric permittivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' In [S1], charge trapping was analyzed on the basis of three mechanisms, namely first-order trapping, first-order trapping with Coulombic interactions, and trapping which increases during injection due to the generation of states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' The expressions for current they derive are qualitatively similar for each mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Hence, for simplicity, we consider the rate of trapping density to be a decay in first order with the addition of electron-electron interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Coulombic repulsion may inactivate trapping sites surrounding a trapped electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' This is included in the rate equation by multiplying a probability factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' If the volume of dielectric rendered inactive by a trap is h, then the trapping is reduced by a factor of (1 − h V ), where V is the volume of the dielectric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' For n trapped charges, the factor is (1 − h V )n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' The trapping rate can be expressed as: dn dt = (n0 − n)σ Jvth qvd � 1 − h V �n (S2) where n0 is the maximum number of traps available, J/q is the net flux density, vth and vd are the thermal and drift velocities respectively, and σ is capture cross-section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Assuming the total volume of the dielectric to be much larger than the volume deactivated by trapping events so that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' 1≫ h/V and n0 ≫ n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' this expression can be simplified to: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='dn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='dt = n0σ Jvth ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='qvd ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='e− nh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='(S3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='S6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='apMEMRISTIVE MEMORY ENHANCEMENT BY DEVICE MINIATURIZATION ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='Solving this equation yields the following expression for n: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='n = V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='h ln ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='� Q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='Q∗ + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='(S4) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='where Q = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='Jdt is the total injected charge and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='Q∗ = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='V vdq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='n0hvthσ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='(S5) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='We express the current in terms of the electric field as: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='ln ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='� J ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='J0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='= E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='E0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='= 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='E0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='Eap − V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='h ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='qx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='η ln ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='� Q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='Q∗ + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='(S6) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='The current follows a decaying power law with time,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' J = Jst−α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' and the injected charge as a function of time is given by: Q(t) = � Jdt = Jst1−α 1 − α (S7) Substituting S7 into S6 when Q/Q∗ ≫ yields and noting β = V qx hE0ϵ: ln � J J0 � = 1 E0 � Eap − β ln � Jst1−α Q∗(1 − α) �� = 1 E0 � Eap − β ln � Jst Q∗(1 − α) � − β(1 − α) ln t � (S8) Comparing S8 with J = Jst−α implies α = β(1 − α) = β 1 + β (S9) and Js ≈ mEap + n − β ln(Js) (S10) Where m encompasses several material parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Writing β in terms of α, and since measured currents are less than 10−4 A, Js can be neglected in comparison to ln(Js), leading to: α 1 − α ln(Js) ≈ mEap + n (S11) β is positive, we know from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' S9 that α lies between 0 and 1, and is a monotonically increasing function of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Considering that β = V h qx E0 , an increase in either the effective density, V/h or in x gives rise to an increase in α, with the former being physically more likely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Instead of deriving an explicit expression for the number density of trapped charge, we can also directly relate the trapping rate to the current, as was done in for example [S2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' We use QT to denote the charge that is trapped when charge Q is injected into the dielectric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' The ratio dQT dQ is assumed to be a function of current, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' dQT dQ = f � J J0 � (S12) Substituting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' S12 into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' S1 gives: dE dt = Jx lη dQT dQ (S13) where l is the length of the dielectric and J = dQ dt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' To relate this to the power law, we assume a solution of the form dQT dQ = � J J0 �α (S14) with α ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' A general expression for the current assumes the form: J(E) = J0(E)eg(E0) (S15) S7 MEMRISTIVE MEMORY ENHANCEMENT BY DEVICE MINIATURIZATION where the specific functions are determined by the relevant conduction mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Specifically here, using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' S14 we can express the current as J = Js t t0 −1/(α+1) (S16) For conduction given by an exponential relation as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' S6: Js ∝ e (1−1/(α+1)V V0 (S17) while for conduction determined by Frenkle-Poole equation, we arrive at: Js ∝ V e (1−1/(α+1)V 1 2 V0 (S18) and for Fowler-Nordheim conduction we get: Js ∝ V 2e (1−1/(α+1))V −1 V0 (S19) Here, V0 is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' The dominant transport mechanism in a system can be determined by plotting the current versus voltage on a double logarithmic scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Equations S9, S11, S17, S18 and S19 indicate a clear theoretical proof that the exponent in the power law is directly proportional to the effective trap density or capacity of the dielectric to trap electrons, independent of which transport mechanism is dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' S8 MEMRISTIVE MEMORY ENHANCEMENT BY DEVICE MINIATURIZATION Figure S9: Fits of the retention data to extract the exponents α: The model used is |I| = I0(t − t0)−α), where |I| and t are the absolute current and time respectively, and I0 and t0 are fitting parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' The adjusted R2 values of the fits are (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='99237, (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='99475, (c) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='99689, (d) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='99717, (e) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='99995, and (f) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='99996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' (g) shows the dependence of the exponents on area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Modeling the edge effects To visualize the field profiles in our devices we used finite element analysis (COMSOL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' The modeling geometry is shown in Fig S10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' In each simulation, the Nb:STO substrate was modelled as a cube with a dielectric constant of 300 and a thickness of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='5 mm (along z), corresponding to the thickness used in the experimental study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' A circular Co electrode of radius 1 μm, 10 μm or 100 μm was placed on the top surface of the substrate (z=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='5 mm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' A ground node was placed on the bottom of the substrate (z=0), while a voltage was applied to the top Co electrode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' For the simulations in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' S13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' the size of the substrate was reduced to improve the resolution of the mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' This was required to retain the circular nature of the electrodes for the 10 nm devices;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' this was determined not to influence the electric field strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' S9 100 μum read at +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='3 V 10 μum read at +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='3 V 1 μm read at +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='3 V α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='041±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='004 α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='47±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='02 α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='85±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='03 104 10~5 105 I (A) [Currentl ICurrentl ICurrentl 10° 10 (a) (b) 107 (c) 10° 10° Time (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=') Time (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=') Time (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=') 100 μm read at -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='5 V 10 μm read at -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='5 V 1 μm read at -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='5 V α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='626±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='007 α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='987±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='002 10~5 α=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='17±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='02 10°5 10° 10*6 Currentl (A) W10* ICurrentl (A) 10- ICurrentl 10° 10 10° 109 (d) 10~8 (e) () 10° 10- Time (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=') Time (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=') Time (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=') Read at +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='0 Read at -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='0 (g) 10-12 10-11 10-10 10-9 108 Area (m²)MEMRISTIVE MEMORY ENHANCEMENT BY DEVICE MINIATURIZATION Figure S10: Model sample geometry: the Nb:STO substrate is represented by a cube with a thickness of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='5 mm (along z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Circular Co electrode of radii (a) 1 μm, (b) 10 μm and (c) 100 μm is placed on the top surface of the substrate (z=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='5 mm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' A ground node is placed on the bottom of the substrate (z=0), while a voltage is applied to the top Co electrode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Figure S11: Electric field at -3 V: along the surface normal (z direction) for (a)+(d) 1 μm, (b)+(e) 10 μm and (c)+(f) 100 μm devices, The plots on the top row ((a)-(c)) have the same scale bar, with a maximum field value of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='5 ×106 Vm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' For the bottom row, the scale bar has a maximum value of (d) 2 ×107 Vm−1, (e) 2 ×106 Vm−1 and (f) 3 ×105 Vm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=" S10 (a) ×10*4 m (b) ×10'4 m (c) ×10'4 m 2 2 2 0 ×104 m 2 ×10*4 m ×104 m ×104 m(a) X10° (b) 310° (c) X10% 3 3 3 2." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='5 2 ２ 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='5 (d) X107 (e) (f) ×106 X105 2 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='2 1 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='2MEMRISTIVE MEMORY ENHANCEMENT BY DEVICE MINIATURIZATION Figure S12: Electric field at +2 V: along the surface normal (z direction) for (a)+(d) 1 μm, (b)+(e) 10 μm and (c)+(f) 100 μm devices, The plots on the top row ((a)-(c)) have the same scale bar, with a maximum field strength of -3 ×106 Vm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' For the bottom row, the scale bar has a maximum value of (d) -1 ×107 Vm−1, (e) -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='5 ×106 Vm−1 and (f) -2 ×105 Vm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Figure S13: Electric field at -3 V: along the surface normal (z direction) for (a) 100 nm, (b) 50 nm and (c) 10 nm devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' No saturation of the field is observed in (a) and (b) and the field appears to saturate in the 10 nm devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' S11 (a) (b) (c) X106 X106 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='5 1 1 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='5 2 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='5 3 3 3 (d) (e) (f) ×107 X106 X105 10 0 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='2 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='1 2(a) (b) (c) X107 ×107 X108 9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='3 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='29 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='5 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='28 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='27 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='26 5 8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='25 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='23 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='22 4 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='21 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='5 7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content='2MEMRISTIVE MEMORY ENHANCEMENT BY DEVICE MINIATURIZATION S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Literature survey of interfacial switching It is often suggested that a layer close to the interface layer is responsible for the switching [29, S4, S5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Some groups have shown that both high and low resistance states show an area-independent current density, eluding to a switching mechanism that occurs homogeneously over the entire device area [S6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Often this is explained in terms of a change in the Schottky barrier height and width induced by charge trapping at the interface [S7, 29, S4, S8, 12, S10] and movement of oxygen vacancies [S8, S10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Other explanations are proposed where the barrier profile is unchanged and interfacial changes happen at local regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Explanation of this type includes It has also been proposed that the application of a positive bias results in the generation of oxygen vacancies, forming tunnelling paths and giving rise to a LRS where tunnelling, rather than thermionic emission dominate charge transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' The application of a negative bias results in the accumulated of large amounts of oxygen in the vacancies which prevents tunnelling and gives a HRS [S11, S12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Rodenbücher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' used local-conductivity AFM on highly doped Nb:STO to show the presence of nanoscale conducting and switchable clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Suggesting that in this case switching is a local phenomenon related to the presence of conducting clusters with higher Nb content than their surroundings [S13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Finally Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' used scanning tunnelling microscopy and spectroscopy to study the resistive switching in Nb-doped SrTiO3 without an electrode, demonstrating that oxygen migration is the results in a variation of electronic structure during the switching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' With a negative voltage, oxygen anions at the interface near the STM tip were oxidised into oxygen molecules and left the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Simultaneously, oxygen vacancies diffuse into the sample, which act like donor-like levels causing distortions in LDOS near conduction band and enhance the carrier concentration with electron hopping, thus increasing the sample’s conducting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' With a positive voltage, oxygen anions return into the sample and the influence of the donor-like level became weak and the conductivity decreased [S14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Despite a large number of contradictory results and explanations, factors of importance that have been identified include the semiconductor doping concentration, electrode material and the quality of the interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' S12 MEMRISTIVE MEMORY ENHANCEMENT BY DEVICE MINIATURIZATION References [S1] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' Wolters and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfrQVc/content/2301.03352v1.pdf'} +page_content=' J.' 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a/IdAyT4oBgHgl3EQf5vqw/content/tmp_files/2301.00811v1.pdf.txt b/IdAyT4oBgHgl3EQf5vqw/content/tmp_files/2301.00811v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..0225ffda7bbc7a93ac3939e2cb369bad83f0d160 --- /dev/null +++ b/IdAyT4oBgHgl3EQf5vqw/content/tmp_files/2301.00811v1.pdf.txt @@ -0,0 +1,2623 @@ +The Fermi-Dirac staircase occupation of Floquet bands and current rectification inside +the optical gap of metals: a rigorous perspective +Oles Matsyshyn,1, 2 Justin C. W. Song,2 Inti Sodemann Villadiego,3, 1, ∗ and Li-kun Shi1, 3, † +1Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Straße 38, 01187 Dresden, Germany +2Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, +Nanyang Technological University, Singapore 637371, Republic of Singapore +3Institut für Theoretische Physik, Universität Leipzig, Brüderstraße 16, 04103, Leipzig, Germany +(Dated: January 4, 2023) +We consider a model of a Bloch band subjected to an oscillating electric field and coupled to a +featureless fermionic heat bath, which can be solved exactly. We demonstrate rigorously that in the +limit of vanishing coupling to this bath (so that it acts as an ideal thermodynamic bath) the occu- +pation of the Floquet band is not a simple Fermi-Dirac distribution function of the Floquet energy, +but instead it becomes a “staircase” version of this distribution. We show that this distribution +generically leads to a finite rectified electric current within the optical gap of a metal even in the +limit of vanishing carrier relaxation rates, providing a rigorous demonstration that such rectification +is generically possible and clarifying previous statements in the optoelectronics literature. We show +that this current remains non-zero even up to the leading perturbative second order in the amplitude +of electric fields, and that it approaches the standard perturbative expression of the Jerk current +obtained from a simpler Boltzmann description within a relaxation time approximation when the +frequencies are small compared to the bandwidth. +INTRODUCTION +Quantum many body systems that are periodically +driven in time have garnered attention over the last +decades as rich platforms to realize novel collective phe- +nomena and non-equilibrium states beyond those realized +in equilibrium [1–25]. A phenomenon that can arise in +such periodically driven systems and which is forbidden +in equilibrium, is the existence of an average net recti- +fied particle flow or ratchet effect. In particular for the +case of electrons in crystals driven by oscillating electric +fields, these effects have enjoyed recent renewed attention +due to their interesting interplay with the dispersions and +Berry phases of band structures and their potential for +novel opto-electronic devices [26–32]. Despite their nat- +ural connection, only a handful of studies have studied +current rectification effects in Bloch bands through the +lens of Floquet theory [33–35]. This is in part due to the +difficulty that even for non-interacting systems, there is +no simple general formula dictating the occupation of +Floquet bands analogous to the Fermi-Dirac distribution +that dictates the occupation of bands in equilibrium [36– +41]. +One of the central goals of our study is to provide +simple analytical formulae for the occupation of a single +Floquet band coupled to a “featureless fermionic bath” +(which is a commonly used model of bath that for exam- +ple was employed in Refs. [33, 35, 42]). This featureless +bath is a physical system that has a finite coupling to +the fermionic system of interest and is characterized by +a single relaxation scale Γ. The dynamics of the system +∗ sodemann@itp.uni-leipzig.de +† shi@itp.uni-leipzig.de +0.5 +1 +0 +-2 +0 +-1 +1 +2 +3 +4 +(a) +(b) +FIG. 1. +(a) Time independent limΓ→0 pn as a function of +¯ϵn/ω calculated using Eq. (46) showing a ladder-like occupa- +tion. Parameters used: β/ω = 50, µ/ω = 1. (b) Schematic of +the ladder-like occupation for a Bloch band. +coupled to this bath can be described in an exact man- +ner thanks to the fact that we will take the combined +system plus bath as a non-interacting fermionic system. +As we will see, this featureless fermionic bath behaves +as an ideal thermodynamic bath in the limit in which +its coupling to system is vanishingly small (Γ → 0), and +in particular in this limit it relaxes the system towards +the equilibrium Fermi-Dirac occupation of the bands in +arXiv:2301.00811v1 [cond-mat.mes-hall] 2 Jan 2023 + +2 +the absence of an external periodic drive. +As we will +show, however, when the system is periodically driven in +time, this bath leads to a self-consistent occupation the +Floquet bands that is sharply different from that of the +equilibrium Fermi-Dirac distribution, but which we can +determine analytically with no approximations. This oc- +cupation is instead a staircase version of the Fermi-Dirac +distribution with several jumps that occur at copies of +the chemical potential shifted by all the harmonics of +the driving frequency [see Fig. 1(a)]. In the limit of an +ideal bath (Γ → 0), we will show that this distribution +coincides with the distribution that has been previously +obtained within the Boltzmann approach to Floquet sys- +tems (see in particular Eq. (12) of Ref. [38]). +Another central purpose of our study is to exploit the +Floquet formalism to further elaborate on our recent find- +ing [43] that it is indeed possible for time dependent os- +cillating electric fields with a frequency that lies within +the optical gap of a metal, to induce a net rectified DC +electric current. We will see that this is true even when +the electric field has a single monochromatic frequency +ω that is much larger than the relaxation rates and this +current remains finite in the limit when these rates van- +ish (Γ → 0) (and therefore does not rely on the fre- +quency difference effect [44] or in the Raman scattering +effect [45, 46]). We will demonstrate that this is possi- +ble by choosing a simple model containing a single Bloch +band with no Berry curvature, which in a simpler relax- +ation time Boltzmann description would give rise to the +so-called “Jerk” effect described in Refs. [43, 47]. +Our +aim is to use this simple model because it will allow us +to carry out calculations of its response coupled to the +fermionic bath in a clear and exact analytical manner. +We are motivated to do this rigorously in order to +clarify a series of misconceptions that originated from +the work of Belinicher, Ivchenko and Pikus [48] and that +have propagated into some of the subsequent literature +[43, 45, 46, 49, 50]. In Appendix E we comment in more +detail about some of these previous works and point out +more specifically some of their imprecisions. +One of the central messages of our study is that it is in- +deed possible to have a net rectified current in response +to a monochromatic oscillating electric field whose fre- +quency lies within the optical gap of a system, in the limit +of vanishing carrier relaxation rates. We will demonstrate +this within a self-consistent picture of the occupation of +Floquet Bloch band in the steady state of the system. +More specifically we will show that in the limit of an +ideal bath (Γ → 0) the average rectified current in the +non-equilibrium steady state of the system is given by: +¯j = +� +k +pk∇k¯ϵk, +(1) +where ¯ϵk is the Floquet energy of the band, pk is the occu- +pation of the Floquet band, ¯j is the current averaged over +one period, and the integral is over the crystal momen- +tum in the Brillouin zone with the usual normalization +of 1/(2π)d. +The crucial difference between the above expression +and that for the average current in an equilibrium sys- +tem, is that the occupation function pk is not simply the +Fermi-Dirac distribution associated with ¯ϵk, but instead +it is precisely the stair-case occupation function depicted +in Fig. 1. Crucially, we will show that generically this +stair-case occupation is a function that depends on all the +information of the time dependence of the Hamiltonian, +and cannot be expressed as a function of only ¯ϵk. We +will then show that as a consequence of this, the rectified +current is in fact generically non-zero in the optical gap +of a metal that breaks inversion and time-reversal sym- +metries. This result remains true even to second order +in the amplitude of the oscillating electric field, which is +the leading order at which rectification currents appear, +and therefore implies a non-vanishing rectification con- +ductivity within the optical gap of metals, in agreement +with our previous results [43]. For other previous dis- +cussions of the possibility of in-gap rectification see also +Refs. [46, 51–55]. +Our paper is organized as follows. +In Section I, we +setup the approach to open quantum systems, obtain ex- +act occupation functions for diagonal and time-periodic +Hamiltonians coupled to a featureless fermionic bath. In +Section II, based on the exact occupation functions, we +calculate exact linear and rectification conductivities and +show that there is a net rectified current in response to a +monochromatic oscillating electric field whose frequency +lies within the optical gap of a metal, in the limit of van- +ishing carrier relaxation rates. +I. +THE OPEN-SYSTEM SCHRÖDINGER +EQUATION APPROACH TO OPEN QUANTUM +SYSTEMS +In descriptions of quantum open systems it is typically +natural to view the combined Hilbert space of the “sys- +tem” and the “bath” as a tensor product of their Hilbert +spaces in isolation. There are situations, however, where +it is possible to alternatively cast this separation of sys- +tem and bath as a direct sum of their individual Hilbert +spaces. As we will show, such separation into sums of +Hilbert spaces is extremely powerful and convenient, be- +cause it allows to integrate out the dynamics of the “bath” +in an exact manner and to obtain a simple non-Hermitian +generalization of Schrödinger’s equation for the system +which captures its coupling to the bath without any ap- +proximations. +One example of the class of models which admits such +direct sum separation into system and bath are those of +non-interacting particles. To see this let us imagine that +the system and the bath as a whole can be described by a +non-interacting model. For concreteness we can imagine +this to be a tight biding model of particles hopping on +a lattice. Because the problem is non-interacting, then +the dynamics can be analyzed by computing the trajecto- +ries of single individual particles and then adding them + +3 +up. However, for a single particle the Hilbert space of +the “system” and the “bath” can be naturally viewed as +a direct sum. For example, in the case of a tight-binding +model, some sites can be viewed as belonging to the sys- +tem and the remainder sites as belonging to the bath. +Let us then consider that the Hilbert space of the sys- +tem and the bath can be decomposed into a direct sum, +namely their Hamiltonian and states have block form as +follows: +H(t) = +� +HS(t) +HSB(t) +HBS(t) +HB(t) +� +, +|ψ(t)⟩ = +� +|ψS(t)⟩ +|ψB(t)⟩ +� +, +(2) +where HBS(t) = H† +SB(t). +From Eq. (2), the coupled +Schrödinger equations for system and bath states then +read: +i∂t |ψS(t)⟩ = HS(t) |ψS(t)⟩ + HSB(t) |ψB(t)⟩ , +(3) +i∂t |ψB(t)⟩ = HBS(t) |ψS(t)⟩ + HB(t) |ψB(t)⟩ , +(4) +where we set ℏ = 1 throughout the paper. By integrating +Eq. (4) over time and inserting it into Eq. (3) allows +to formally eliminate the bath state dynamics |ψB(t)⟩ +and to obtain the open-system Schrödinger equation for +|ψS(t)⟩: +i∂t |ψS(t)⟩ = HS(t) |ψS(t)⟩ + HSB(t)UB(t, t0) |ψB(t0)⟩ +− iHSB(t) +� t +t0 +dt′ UB(t, t′)HBS(t′) |ψS(t′)⟩ , +(5) +where UB(t, t′) is the bath (intrinsic) evolution operator +satisfying i∂tUB(t, t0) = HB(t)UB(t, t0). This procedure +is often carried within the Schwinger-Keldysh formalism +by integrating out part of the action describing the de- +grees of freedom of the bath (see e.g. Refs. [33, 35, 42]). +But this is easier and more physically transparent in our +first quantization notation and the final results would be +identical. +A. +Featureless fermionic bath +We will now specialize the above equation to a model of +a “featureless fermionic bath”, which we define as having +the following characteristics: +(i) In a featureless fermionic bath every state of the +system is coupled to a collection of identical sites with +the same energy spectrum and the same coupling λ [see +Fig. 2(a)]. If the system states (basis) are denoted by +|χn⟩ and the bath states (basis) by |ϕn,j⟩, the bath and +the system-bath coupling are +HB = +� +n,j +εj |ϕn,j⟩ ⟨ϕn,j| , +(6) +HSB = λ +� +n,j +|χn⟩ ⟨ϕn,j| , +(7) +where εj is the energy for the bath state |ϕn,j⟩. This +model of the bath is identical to that employed in +Refs. [35, 43, 56–62]. +(ii) The featureless fermionic bath is prepared in an ini- +tial condition at t0 with a Fermi-Dirac distribution that +only has weight on the bath sites, described by +ρS(t0) = 0, +ρB(t0) = � +n,jf0(εj) |ϕn,j⟩ ⟨ϕn,j| , +(8) +f0(εj) = +1 +exp[β0(εj − µ0)] + 1, +(9) +where µ0 is the chemical potential of and β0 = 1/kBT0 +denotes the temperature of the bath, respectively. The +assumption of the initial density matrix only having +weight on the bath is useful but it is not strictly nec- +essary. This is because in the limit in which the bath +spectrum becomes a dense continuum, the information +of the initial condition for the component of density ma- +trix on the system will decay over time and only the in- +formation of the initial condition for the density matrix +on the bath will dictate the late time steady state [this +will become more clear in Eq. (16) which is a subsequent +version of Eq. (5)]. Notice also that we have equated the +evolution of the single particle density matrix with that +of the many-body one-particle density matrix (equal time +Greens function), which is possible thanks to the fact that +the system is non-interacting. +With assumptions (i) and (ii), by evolving the initial +condition in Eq. (8) under Eqs. (3) and (4), one finds that +the one-body density matrix projected onto the system +at time t is given by +ρS(t) = +� +n,j +f0(εj) |ψ(j) +n (t)⟩ ⟨ψ(j) +n (t)| , +(10) +where |ψ(j) +n (t)⟩ is the component within system Hilbert +space that evolves out of the initial state |ψB(t0)⟩ = +|ϕn,j⟩ in the bath at t0. Eq. (10) states that the density +matrix for the system is the weighted sum of contribu- +tions from all bath states with their corresponding initial +occupations. +Using Eqs. (5), (6), and (7), we obtain the open-system +Schrödinger equation for |ψ(j) +n (t)⟩: +i∂t |ψ(j) +n (t)⟩ = HS(t) |ψ(j) +n (t)⟩ + λ exp[−iεj(t − t0)] |χn⟩ +− i +� ∞ +t0 +dt′ γ(t − t′) |ψ(j) +n (t′)⟩ . +(11) +system +DoS +energy +bath +(a) +(b) +(c) +FIG. 2. (a) Schematic of the system-bath coupling HSB. (b) +The bath’s density of states is much wider than that of the +system, and we simplify it to be flat in the energy range of +interest. (c) Schematic of the bath acting as a source as well +as a sink for the system [see Eq. (27)]. + +4 +Here, λ exp[−iεj(t−t0)] |χn⟩ is a source term for |ψ(j) +n (t)⟩ +arising from the bath, while the memory function in the +second line is given by, +γ(t) = λ2Θ(t) +� +j +exp(−iεjt), +(12) +which encodes the memory of decay for |ψ(j) +n (t)⟩ due to +the bath. This memory function makes the Schrödinger +equation for open systems non-local in time, and in gen- +eral it incorporates decay and renormalizations of the +system energies due to their coupling to the bath [see +Fig. 2 for a depiction]. +(iii) To remove the finite memory delay, we impose one +further property defining the featureless fermionic bath, +namely that it has an infinitely broad and flat spectrum +[see Fig. 2(b)], i.e., the bath density of state is constant: +νB(ωb) = 2π +� +j +δ(ωb − εj) ≡ ν0. +(13) +With this simplification, the finite delay or non-local +memory of the past time t′ in Eq. (11) is lost, the memory +function becomes: +γ(t) = λ2Θ(t) +� +j +� +∞ +−∞ +dωb δ(ωb − εj) exp(−iωbt) += λ2ν0Θ(t) +� +∞ +−∞ +dωb +2π exp(−iωbt) = δ(t) Γ, +(14) +where we used Eq. (13) to obtain the second equation +and defined +Γ ≡ λ2ν0 +2 +. +(15) +With the above simplification of infinitely broad spec- +trum for the bath, the open-system Schrödinger’s equa- +tion reduces to: +i∂t |ψ(j) +n (t)⟩ = +� +HS(t) − iΓ +� +|ψ(j) +n (t)⟩ ++ λ exp[−iεj(t − t0)] |χn⟩ . +(16) +The above equation is remarkably simple. It is a sim- +ple non-Hermitian version of the Schrödinger equation in +which the system Hamiltonian is dressed by a constant +imaginary part “−iΓ” which captures the decay into the +bath. Many recent studies of open quantum systems have +used non-Hermitian Schrödinger equations that only in- +clude the first line of Eq. (16). However, we see that the +influence of the bath is not merely to induce decay, but +it also produces the second term that acts a source and +makes the equation inhomogeneous. The balance of these +two terms is what allows the existence of non-trivial late +time steady states (see Fig. 2 for depiction). +B. +Ideal fermionic bath +To illustrate that our bath leads to the expected equi- +librium when the system is not driven in time, we first +consider the the special case in which HS(t) is time in- +dependent, +HS(t) → H0 = +� +n +ϵn |χn⟩ ⟨χn| , +(17) +Eq. (16) can be equivalently expressed as +i∂ts(j) +n += [ϵn − iΓ]s(j) +n + λ exp[−iεj(t − t0)], +(18) +where +s(j) +n += ⟨χn|ψ(j) +n (t)⟩ , +(19) +is the amplitude for the system state |χn⟩. Solving the +above Eq. (18) gives +s(j) +n += −iλ exp +� +− i +� t +t0 +dt′ (ϵn − iΓ) +� +× +� t +t0 +dt′ exp +� +i +� t′ +t0 +dt′′(ϵn − iΓ − εj) +� += +λ +ϵn − iΓ − εj +� +e−(Γ+iϵn)(t−t0) − e−iεj(t−t0)� +. (20) +Then using Eq. (10), we obtain the steady state, diagonal +density matrix for the system: +ρS(t → +∞) = +� +n +fΓ(ϵn) |χn⟩ ⟨χn| , +(21) +in which fΓ(ϵn) = limt→+∞ +� +j f0(εj)|s(j) +n |2 and reads +explicitly as +fΓ(ϵn) = +� +j +f0(εj) +λ2 +(ϵn − iΓ − εj)(ϵn + iΓ − εj) += +� +∞ +−∞ +dωb f0(ωb) +λ2 � +j δ(ωb − εj) +(ϵn − iΓ − ωb)(ϵn + iΓ − ωb) += +� +∞ +−∞ +dωb +π f0(ωb) +Γ +(ϵn − ωb)2 + Γ2 , +(22) +where we used Eqs. (13) and (15) in obtaining the last +equation. The above distribution fΓ(ϵn) shows that when +HS(t) is time independent, the system “thermalizes” by +approaching a time independent steady state dictated by +the initial condition of the bath, f0(ωb), while a finite Γ +accounts for the broadening of the energy levels of the +system due to its coupling to the bath. +Importantly, taking the limit in which the coupling to +the bath vanishes from Eq. (22), we obtain +lim +Γ→0 fΓ(ϵn) = f0(ϵn), +(23) +i.e., fΓ(ϵn) reduces to the ideal Fermi-Dirac distribution +in the limit of Γ → 0. We will then call this Γ → 0 limit +of the “featureless fermionic bath” an “ideal fermionic +bath”. The fact that the ideal Fermi-Dirac distribution + +5 +appears only when the coupling to the bath is vanish- +ingly weak is consistent with general considerations of +statistical physics. +However, Eq. (22) still allow us to obtain analytically +the modified occupation at finite coupling to the bath, +which will be used in subsequent manipulations. By in- +tegrating over ωb in Eq. (22) using Cauchy’s residue the- +orem, we find that: +fΓ(ϵ) = 1 +2 +� +f+(ϵ) + f−(ϵ) +� +, +(24) +where f+(ϵ) = [f−(ϵ)]∗ and they are given by: +f±(ϵ) = 1 +2 ± i +π Ψ(0) +�1 +2 ± iβ ϵ ∓ iΓ − µ +2π +� +, +(25) +with Ψ(0) the 0-th order Polygamma function (or the +digamma function). f±(ϵ) will also appear repeatedly in +more general cases. +C. +Diagonal and time-periodic Hamiltonians +1. +Diagonal system Hamiltonian +In this work, we will develop the above general for- +malism to the special case where the system Hamilto- +nian HS(t) is time dependent but diagonal in the system +states. Let us then take the following form for the system +Hamiltonian: +⟨χn|HS(t)|χm⟩ = δnm[ϵn + Vn(t)] = δnmϵn(t). +(26) +With this, Eq. (16) then reduces to +i∂ts(j) +n += [ϵn(t) − iΓ]s(j) +n + λ exp[−iεj(t − t0)]. +(27) +Solving the above Eq. (27) gives +s(j) +n (t) = −iλ exp +� +− i +� t +t0 +dt′ [ϵn(t′) − iΓ] +� +× +� t +t0 +dt′ exp +� +i +� t′ +t0 +dt′′[ϵn(t′′) − iΓ − εj] +� +, +(28) +and then with Eq. (10), we obtain the diagonal density +matrix for the system: +ρS(t) = +� +n +pn(t) |χn⟩ ⟨χn| , +(29) +pn(t) = +� +j +f0(εj)|s(j) +n (t)|2. +(30) +2. +Periodic system Hamiltonian +Now we consider a periodically driven system. Namely, +we take the diagonal elements of the Hamiltonian to be +periodic in time: +ϵn(t + T) = ϵn(t) = ++∞ +� +l=−∞ +ϵ(l) +n exp[−ilω(t − t0)], +(31) +where T is the period and ω = 2π/T is the frequency, +and +ϵ(l) +n = +� T +0 +dt +T ϵn(t) exp[ilω(t − t0)] +(32) +is the l-th Fourier coefficient for ϵn(t). In particular, +¯ϵn ≡ ϵ(0) +n += +� T +0 +dt +T ϵn(t), +(33) +is the time-average of the diagonal element of the Hamil- +tonian, which as we will show next, coincides with the +Floquet energy of state n. To see this, notice that the +wavefunction that would solve the system Schrödinger’s +equation in the absence of the bath, can be expressed as +follows: +exp +� +− i +� t +t0 +dt′ ϵn(t′) +� += exp +� +− i +� t +t0 +dt′[ϵn(t′) − ¯ϵn] +� +× exp +� +− i +� t +t0 +¯ϵn +� +≡ φn(t) × exp +� +− i¯ϵn(t − t0) +� +. +(34) +The periodicity of the first factor denoted by φn(t) can +be shown explicitly: +φn(t + T) = φn(t) × exp +� +− i +� t+T +t +dt′[ϵn(t′) − ¯ϵn] +� += φn(t), +(35) +where we used Eq. (33) in obtaining the second equation. +Therefore we see from second factor in the last line of +Eq. (35), that the time-average of the diagonal element +of the Hamiltonian is the Floquet energy itself. +Let us now consider the Fourier expansion of the peri- +odic part of the Floquet wavefunction: +φn(t) = exp +� +− i +� t +t0 +dt′[ϵn(t′) − ¯ϵn] +� += ++∞ +� +l=−∞ +φ(l) +n exp[−ilω(t − t0)], +(36) +or equivalently, +φ(l) +n = 1 +T +� t0+T +t0 +dt +� +exp[ilω(t − t0)] +× exp +� +− i +� t +t0 +dt′[ϵn(t′) − ¯ϵn] +�� +. (37) +The above expression makes clear that the amplitude of +the harmonics of the wavefunction, φ(l) +n , are functions + +6 +of the full time dependence of the instantaneous energy +ϵn(t), and are independent of the Floquet energy ¯ϵn. This +property will be crucial later on for purposes of under- +standing why there is in-gap rectification. In other words, +Eq. (37) defines φ(l) +n as a function of all the harmonics of +the time dependent energy from Eq. (32) as follows: +φ(l) +n = φ(l) +n (ϵ(±1) +n +, ϵ(±2) +n +, · · · ). +(38) +Also from Eq. (36) it can be shown that these amplitudes +satisfy the following normalization condition: ++∞ +� +l=−∞ +��φ(l) +n +��2 = 1. +(39) +With Eqs. (28), (30), (36), and (13), and by taking +the late-time limit that allows to neglect transient terms +of the form exp[−Γ(t − t0)] → 0, we obtain the system +steady state occupation: +pn(t) = +� +∞ +−∞ +dωb +π f0(ωb) +× Γ +����� ++∞ +� +l=−∞ +φ(l) +n +exp[−ilω(t − t0)] +¯ϵn − ωb − lω − iΓ +����� +2 +. +(40) +Similar to Eq. (22), by integrating over ωb in Eq. (40), +we find that: +pn(t) = ++∞ +� +l,m=−∞ +� +φ(m) +n +�∗φ(l) +n exp +� +i(m − l)ω(t − t0) +� +× +Γ +2Γ + i(m − l)ω +� +f+(¯ϵn − lω) + f−(¯ϵn − mω) +� +, +(41) +where f±(ϵ) is given in Eq. (25). The Eq. (41) is one of +the central formulas of our work because it allows to com- +pute expectation values of any equal-time system observ- +ables, even at a finite coupling Γ to featureless fermionic +bath. +The expression in Eq. (41) captures the steady state +occupation of the n-th state in the case of featureless +fermionic bath, and thus it replaces what would be the +Fermi-Dirac distribution in equilibrium. One important +feature of this steady state is that it displays “synchro- +nization”, namely, it is strictly periodic in the drive: +pn(t + T) = pn(t). +(42) +Remarkably, in the limit of an “ideal bath” (Γ → 0) the +above distribution becomes time independent and it is +given by: +lim +Γ→0 pn = ++∞ +� +l=−∞ +|φ(l) +n |2f0(¯ϵn − lω). +(43) +Here ¯ϵn is the Floquet energy of n-th state, and φ(l) +n are +the Harmonics of the periodic part of the wave-functions +defined in Eq. (37). The reader is encouraged to contrast +this occupation function with that in Eq. (23) obtained +when the Hamiltonian was time independent. Notice also +that because the occupation function becomes time inde- +pendent in this limit, there are no time fluctuations of the +average fermion occupation of each state n. +Thus the distribution is an infinite sum of sev- +eral Fermi-Dirac distributions with chemical potentials +shifted by the various harmonics of the driving frequency +lω and weighed by amplitudes of the harmonics of the +Floquet wavefunctions |φ(l) +n |2. It is therefore clear that +the occupation of the state is completely different from +how the state is filled in equilibrium [see Fig. 1(a) for an +illustration of the non-equilibrium occupation function]. +One recovers an occupation similar to equilibrium when +one neglects all the higher harmonics of φ(l) +n +with l ̸= 0 +and forces by hand the amplitude of the l = 0 term to +be φ(0) +n +→ 1, but this is not justified in general (not even +perturbatively as we will illustrate in Sec. II B). We note +that the idea that Floquet states are not filled in the +same way as equilibrium states has been emphasized in +several studies, by using a variety of models for the re- +laxation when the system is coupled to a heat bath [36– +39, 41, 63]. In fact the expression for the non-equilibrium +time independent steady states we find in Eq. (43) has +been reported before, and is in particular the same kind +of expression shown in Eq. (12) of Ref. [38]. +3. +Harmonic time dependent driving +Computing analytically the integral in Eq. (37) that +relates the harmonics of the Floquet wavefunction to the +harmonics of the energy is in general involved. There is +a simple case where these integrals can be computed in +a simple closed analytical form, which is when the time +dependent part Vn(t) of the Hamiltonian has a single +harmonic: +Vn(t) = Vn cos[ω(t − t0)]. +(44) +In this case the coefficients φ(l) +n from Eq. (37) correspond +to the l-th Bessel function: +φ(l) +n = Jl +� +Vn/ω +� +. +(45) +Substitution of Eq. (45) into Eq. (41) leads to the fol- +lowing non-perturbative expression for the occupation of +the states in the limit of Γ → 0: +lim +Γ→0 pn = ++∞ +� +l=−∞ +J2 +l +� +Vn/ω +� +f0(¯ϵn − lω). +(46) +We therefore see that the occupation in the case of the +ideal fermionic bath becomes a sum of several Fermi- +Dirac distributions boosted by the different harmonics of +the Floquet quasi-energies ¯ϵn − lω (l ∈ Z). It is interest- +ing to note that this ladder-like behavior is analogous to + +7 +the Tien-Gordon effect that arises in nanostructures that +are simultaneously subjected to AC and DC drives [64]. +Similarly as in that case, the ladder behaviour becomes +more pronounced as the driving becomes stronger [see +Fig. 1(a)]. +II. +SINGLE BAND MODEL UNDER +MONOCHROMATIC LIGHT +In this section we will use the formalism developed in +the previous ones to determine the self-consistent occupa- +tion of an electronic band driven by an oscillating electric +field and demonstrate the existence of in-gap rectifica- +tion. Because we are primarily interested here in proving +and clarifying the origin of in-gap rectification, we will +focus on a simple model of a Bloch band that has vanish- +ing Berry connections. These bands can display however +the in-gap Jerk current effect that arises from the energy +band dispersions [43]. However, other mechanisms driven +by the Berry phases, such as the non-linear Hall effect, +can also lead to in-gap rectification as we have recently +demonstrated [43]. +Let us now consider our system Hamiltonian to be a +tight-binding model with a single site per unit cell and +a trivial single Bloch band (with no Berry connections) +coupled to a uniform monochromatic electric field. The +time dependent system Hamiltonian is: +HS(t) = +� +k +ϵk(t) |χk⟩ ⟨χk| , +(47) +ϵk(t) ≡ ϵ(k − A(t)), +� +k +≡ +� +BZ +dk +(2π)d . +(48) +The system states are now labelled by the wave vector k +and ϵ(k) is the unperturbed band dispersion. We assume +a monochromatic electric field which leads to the periodic +vector potential using E(t) = −∂tA(t): +A(t) = − i +ω Eω exp(−iωt) + c. c. +(49) +A. +Electric current in the steady state +Since the system Hamiltonian is diagonal in crystal +momenta k, we can apply the formalism of Sec. I C to +compute the steady state occupation of each momenta +k, by replacing the label in previous sections n → k. If +we denote the occupation of each state by pk(t), then the +system’s electric current reads as follows: +j(t) = +� +k +pk(t)∇kϵk(t) += ++∞ +� +s=−∞ +j(s) exp[−isω(t − t0)], +(50) +where we set e = ℏ = 1 throughout the paper. By com- +bining Eqs. (32) and (41), the weight of each oscillating +mode of the electric current can be written as: +j(s) = +� +k ++∞ +� +m,l=−∞ +Γ +2Γ + i(l − s)ω +� +φ(m) +k +�∗φ(s+m−l) +k +× +� +f+(¯ϵk − (s + m − l)ω) + f−(¯ϵk − mω) +� +∇kϵ(l) +k . (51) +Interestingly, as discussed in Sec. I C, in the limit of an +ideal heat bath Γ → 0, the distribution function pk(t) be- +comes time independent, and therefore the time averaged +electric current (also referred to as rectified current), is +given by: +¯j = +� T +0 +dt +T j(t) = +� +k +pk∇k¯ϵk, +(52) +where ¯ϵk is the Floquet energy of the band and in our +current simple single-band model, and is given by the +time averaged band energy (l = 0 component): +¯ϵk ≡ ϵ(0) +k += +� T +0 +dt +T ϵ(k − A(t)). +(53) +Therefore, we see that Eq. (52) has a resemblance to +how one would compute the current in a time indepen- +dent equilibrium system, but with the equilibrium Fermi- +Dirac distribution replaced by occupation function pk, +and the bare band dispersion replaced by the dressed +Floquet band energy ¯ϵk. +At first glance, this point of +view might suggest that the time averaged rectified cur- +rent vanishes in the ideal limit of ω ≫ Γ → 0, just in +the same way it is expected to vanish in a time inde- +pendent equilibrium system. In fact, several classic and +more recent works have incorrectly taken this point of +view that the non-equilibrium steady state occupation pk +is a Fermi-Dirac distribution of the dressed Floquet band +energy [45, 46, 48–50] (see Appendix E for detailed com- +ments on previous works). However, as we have shown +in Sec. I C, the correct occupation of the states in the +non-equilibrium steady state is not a simple Fermi-Dirac +distribution, but it is given by the following expression +[see Eqs. (38) and (43)]: +pk(¯ϵk, ϵ(±1) +k +, · · · ) ≡ ++∞ +� +l=−∞ +|φ(l) +k |2f0(¯ϵk − lω). +(54) +In the argument of pk in the above expression, we have +emphasized that pk is not only a function of the Flo- +quet band energy ¯ϵk, but also of all the higher har- +monics ϵ(±1) +k +, ϵ(±2) +k +· · · of the time dependent energy +ϵk(t) through its dependence on the amplitudes φ(l) +k [see +Eqs. (37) and (38)]. Precisely because of this, the recti- +fication current ¯j can not be expressed as an integral of +a total derivative over the Brillouin zone and generally +does not vanish, i.e., +¯j ̸= +� +k +∇k ˜P(¯ϵk) = 0, +(55) + +8 +where ˜P(¯ϵk) would be defined through +∂ ˜P(¯ϵk) +∂¯ϵk +≡ ˜pk(¯ϵk), +(56) +which would be possible if the occupation depended only +on the dressed Floquet energy pk → ˜pk(¯ϵk) [but this is +not the case for Eq. (55)]. +Therefore, we see that in general a non-zero rectified +current is expected in the non-equilibrium steady state, +even in the limit of the ω ≫ Γ → 0. As we will show +in detail in the following section, this finite rectified cur- +rent remains non-zero within the optical gap of a metal, +even within the usual second order of perturbation the- +ory in the amplitude of the electric field for which recti- +fication currents are typically computed. These findings +further substantiate our recent work showing the exis- +tence of in-gap rectification [43] but appear in tension +with some other statements in the literature [44–46, 48– +50]. In Appendix E, we comment in more detail on some +of these other works clarifying some partial agreements +but also pointing out some of their imprecisions and in- +correct statements. +B. +Perturbative results +In this subsection we will compute perturbatively the +electric current in powers of electric field to the currents +at modes [see Eqs. (50) and (51)]: s = 0 representing +rectification conductivity, s = 1 representing linear con- +ductivity. s = 2 representing second harmonic generation +is discussed in Appendix A. We will show explicitly that +even to 2nd order in electric fields, the non-equilibrium +distribution in the steady state for an ideal bath, pk, dif- +fers clearly from the naive Fermi-Dirac distribution eval- +uated in the dressed Floquet bands. This will allow us +to compute analytically the rectification conductivities +and prove rigorously that they remain finite within the +optical gap of the metal. +Although our conclusions and formulae are valid and +can be used for any single band model (with no Berry +connections) in arbitrary dimensions, for simplicity we +will illustrate our results for a simple 1D model with the +following band dispersion: +ϵ(kx) = −t1 cos(a0kx) − t2 sin(2a0kx) + ϵ0, +(57) +where ϵ0 is a constant that we have added for convenience +in order to shift the band energy so that it lies within 0 +and ∆ [See Fig. 3(b)], and a0 is the lattice constant. +Notice that the above band-structure breaks not only +inversion, which is always needed to have rectification, +but also time-reversal symmetry, and therefore it has no +symmetry relating k → −k. As we will see, this is indeed +crucial in order to obtain a non-zero in-gap rectification +conductivities for the models without Berry curvature +that we are considering in this study. More generally, as +discussed in Ref. [43], in the case of bands with non-trivial +Berry connections one can alternatively obtain a non- +zero in-gap rectification, e.g., via the Berry-Dipole effect +by breaking time reversal symmetry only by having a +circularly polarized light instead of having a time-reversal +breaking band-structure. +1. +Occupation function to the second order of electric field +We begin by deriving the explicit perturbative expres- +sions for ϵ(l) +k +and φ(l) +k +discussed in the previous sections +and can be computed from Eqs. (32) and (37) by replac- +ing n → k. Up to the second order in the electric field, +it is sufficient to expand the band dispersion up to the +same second order, namely: +ϵ(k − A(t)) = ¯ϵk + ϵ(1) +k e−iω(t−t0) + ϵ(−1) +k +eiω(t−t0) ++ ϵ(2) +k e−2iω(t−t0) + ϵ(−2) +k +e2iω(t−t0) + · · · , +(58) +Using Eq. (49), this perturbative expansion leads to the +following expressions for ϵ(l) +k : +¯ϵk ≡ ϵ(0) +k += ϵ(k) + 1 +ω2 +� +αβ +∂α∂βϵ(k) Eα +ωEβ +−ω + O(|Eω|4), +ϵ(1) +k += i +ω +� +α +∂αϵ(k) Eα +ω + O(|Eω|3), +ϵ(2) +k += − 1 +2ω2 +� +αβ +∂α∂βϵ(k) Eα +ωEβ +ω + O(|Eω|4), +ϵ(−l) +k += +� +ϵ(l) +k +�∗. +(59) +We can use Eq. (37) to perturbatively evaluate φ(l) +k lead- +ing to: +φ(0) +k += 1 − ϵ(1) +k +− ϵ(−1) +k +ω ++ +� +ϵ(1) +k +�2 + +� +ϵ(−1) +k +�2 − 4ϵ(1) +k ϵ(−1) +k +− ϵ(2) +k ++ ϵ(−2) +k +2ω2 +, +φ(1) +k += −ϵ(1) +k +ω − ϵ(1) +k +� +ϵ(1) +k +− ϵ(−1) +k +� +ω2 +, +φ(−1) +k += ϵ(−1) +k +ω +− ϵ(−1) +k +� +ϵ(−1) +k +− ϵ(1) +k +� +ω2 +, +φ(2) +k += +� +ϵ(1) +k +�2 − ϵ(2) +k +2ω2 +, +φ(−2) +k += +� +ϵ(−1) +k +�2 + ϵ(−2) +k +2ω2 +. +(60) +The other φ(l) +k +with |l| > 2 will scale with higher powers +of electric fields, and therefore can be neglected to second +order. The norm squared of those terms above are: +��φ(0) +k +��2 = 1 − 2 +��ϵ(1) +k +��2 +ω2 ++ O(|Eω|3), +��φ(1) +k +��2 = +��ϵ(1) +k +��2 +ω2 ++ O(|Eω|3), +��φ(2) +k +��2 = O(|Eω|4). +(61) + +9 +Therefore the ideal occupation function pk in the limit +Γ → 0 to second order in electric fields reads as +pk = +� +1 − 2 +��ϵ(1) +k +��2 +ω2 +� +f0(¯ϵk) ++ +��ϵ(1) +k +��2 +ω2 +f0(¯ϵk − ω) + +��ϵ(−1) +k +��2 +ω2 +f0(¯ϵk + ω). +(62) +The above expansion contains all the correct terms to sec- +ond order in electric fields, even though it is not strictly +perturbative, because the Floquet band energy ¯ϵk also in- +cludes implicitly a correction of order |Eω|2 [see Eq. (59)]. +In other words, if one wants to obtain a strictly pertur- +bative expansion to order |Eω|2 one simply needs to Tay- +lor expand the Fermi-Dirac distribution f0(¯ϵk) above as +well. However we find it convenient to keep the above +form, with the understanding that we can only trust its +predictions to order |Eω|2. +Let us now comment on the significance of Eq. (62). +We see above that even to second order, the non- +equilibrium distribution, pk, contains not only the Fermi- +Dirac distribution evaluated for the Floquet bands, +f0(¯ϵk), but also several other terms that make it clearly +deviate from f0(¯ϵk). +As we will see these additional +terms, are precisely the ones that lead to a finite in- +gap rectification in the clean limit Γ → 0. +In Ap- +pendix D, we also demonstrate that the above occupa- +tion function agrees with the one obtained from a sim- +pler Boltzmann/relaxation-time description in the limit +ω ≪ ¯ϵk. Notice also that the above occupation differs +even to up second order |Eω|2 from the naive Fermi-Dirac +occupation of the Floquet band, f0(¯ϵk), that was pres- +sumed in Refs. [45, 46, 48–50] (see Appendix Appendix E +for further comments on previous studies). +2. +Linear conductivity +The linear conductivity is defined from: +j(1) +α += σαβ +Γ (ω)Eβ +ω + O(|Eω|3), +(63) +where the sub-index Γ emphasizes a finite coupling of the +system to the bath. Using Eqs. (51), (37), (32), the exact +conductivity of our model at finite coupling to the bath +is found to be: +σαβ +Γ (ω) = i +ω +� +k +fΓ(¯ϵk)∂α∂β¯ϵk ++ +� +k +∂α¯ϵk∂β¯ϵk +ω2 +iΓ +2Γ − iω L1(¯ϵk, ω), +(64) +where ∂γ ≡ ∂/∂kγ, and +L1(¯ϵk, ω) = f+(¯ϵk) + f−(¯ϵk + ω) +− f+(¯ϵk − ω) − f−(¯ϵk), +(65) +where f± are defined in Eq. (25). Just as for Eq. (62), +we have kept the dressed Floquet band energy, ¯ϵk, in the +integrands of Eq. (64), and therefore this is not a strictly +perturbative expression. But if desired, the strictly per- +turbative expression can simply be obtained from the +one above by replacing the dressed Floquet band en- +ergy dispersion by the bare unperturbed band dispersion: +¯ϵk → ϵ(k). This also applies to the subsequent formulas +of this section. +In the clean limit (ω ̸= 0 and Γ → 0), the above ex- +pression reduces to the standard Drude form: +lim +Γ→0 σαβ +Γ (ω) = i +ω +� +k +f0(¯ϵk)∂α∂β¯ϵk. +(66) +Therefore, we see that the real part of the linear con- +ductivity at finite frequency vanishes when Γ → 0. In +Fig. 3(c) we illustrate this in detail for the simple model +1D from Eq. (57). The above Drude form follows from +the fact that to the linear order of the electric field, the +ideal occupation function pk in the limit Γ → 0 is the +same with the equilibrium Fermi-Dirac distribution [see +Eq. (62)]. +In the DC limit ω → 0 the linear conductivity ap- +proaches a finite Drude-like value (see Appendix A for +details): +lim +ω→0 σαβ +Γ (ω) = 1 +2 +� +k +∂α∂β¯ϵk +�fΓ(¯ϵk) +Γ +− ∂gΓ(¯ϵk) +∂¯ϵk +� +≈ 1 +2 +� +k +∂α∂β¯ϵk +�f0(¯ϵk) +Γ ++ O(Γ) +� +, +(67) +in which +gΓ(ϵ) = 1 +2i +� +f+(ϵ) − f−(ϵ) +� +(68) +is the imaginary part of f+(ϵ) defined in Eq. (25). There- +fore the clean limit of the DC conductivity resembles the +prediction of the classic Drude theory for τ ≡ 1/(2Γ), and +has a Drude peak in the DC limit when the chemical po- +tential of the bath is within the bandwidth of the system +µ ∈ [0, ∆] [see Fig. 3(c)]. The fact that the conductivity +is finite when ω → 0 and has the expected Drude behav- +ior, evidences that our simple bath produces the correct +behavior for the relaxation of currents. +In the limit in which the frequency is small compared +to the bandwidth but much larger than Γ, we obtain +the usual decay power 1/ω2 associated with the Drude +behavior [see Fig. 3(e), left-hand side region]: +lim +Γ≪ω≪∆ Re +� +σαβ +Γ (ω) +� += −2Γ +ω2 +� +k +(∂α¯ϵk)(∂β¯ϵk)∂f(¯ϵk) +∂¯ϵk +. +(69) +On the other hand, in the ultra-large frequency regime +when the frequency greatly exceeds even the bandwidth, +the real part of the linear conductivity has a different +scaling from that of Drude theory: +lim +ω≫∆ Re +� +σαβ +Γ (ω) +� += Γ +ω3 +� +k +(∂α¯ϵk)(∂β¯ϵk), +(70) +decaying as 1/ω3 [see Fig. 3(e), right-hand side region]. + +10 +0.2 +0.4 +0 +(c) +0 +1 +2 +(d) +0 +10 +20 +0.2 +0.4 +0 +(a) +0 +(b) +0 +1 +2 +3 +0 +1 +2 +-1 +-2 +(e) +-10 +0 +(f) +-5 +0 +5 +0 +-1 +-2 +1 +FIG. 3. +(a) The 1D tight binding model whose inversion +and time-reversal symmetries are broken by the next-nearest- +neighbour hopping ±it2/2, and its (b) dispersion relation with +0 the band bottom and ∆ the band top. (c) Real part of the +dimensionless linear conductivity Re σxx +Γ (ω)/σ(1) +0 +illustrating +how it vanishes at finite frequency as Γ → 0 (which defines the +optical transparency region), and (d) dimensionless rectifica- +tion conductivity σxxx +Γ +(ω, −ω)/σ(2) +0 +for different Γ illustrating +the existence of in-gap rectification in the metal, namely that +it approaches a finite non-zero value in the limit of Γ → 0 at +finite ω. The characteristic linear and second order conductiv- +ities in 1D used here are σ(1) +0 += a0·e2/ℏ and σ(2) +0 += a2 +0τ0·e3/ℏ2 +with τ0 = ℏ/t1. (e) and (f) Log-log plots of Re σxx +Γ (ω)/σ(1) +0 +and σxxx +Γ +(ω, −ω)/σ(2) +0 +for different Γ illustrating their power +dependencies over ω in different frequency ranges. Parame- +ters used: a0 = 1, t1/t2 = 2, µ = 5t1/7, β0 = 109/t1. +3. +Rectification conductivity +The rectification conductivity is a three-index tensor +that relates the time averaged current [namely the aver- +age DC current corresponding to s = 0 in Eq. (50)] to the +bilinears of electric field amplitudes. Without loss of gen- +erality, we define it by choosing the following symmetry +convention for indices of the electric field bilinears: +j(0) +γ += σγαβ +Γ +(ω, −ω)Eα +ω(Eβ +ω)∗ ++ σγαβ +Γ +(−ω, ω)(Eα +ω)∗Eβ +ω + O(|Eω|4). +(71) +The exact rectification conductivity of our model at finite +coupling to the bath, Γ, is given by: +σγαβ +Γ +(ω, −ω) += +� +k +∂γ¯ϵk∂α¯ϵk∂β¯ϵk +2ω4 +[fΓ(¯ϵk + ω) + fΓ(¯ϵk − ω) − 2fΓ(¯ϵk)] ++ +Γ +2Γ − iω +� +k +∂α¯ϵk∂γ∂β¯ϵk +2ω3 +L1(¯ϵk, ω) ++ +Γ +2Γ + iω +� +k +∂β¯ϵk∂γ∂α¯ϵk +2ω3 +L∗ +1(¯ϵk, ω). +(72) +The DC limit of the rectification conductivity can be +shown to be (see Appendix B for details): +lim +ω→0 σγαβ +Γ +(ω, −ω) += 1 +4 +� +k +∂α∂β∂γ¯ϵk +�fΓ(¯ϵk) +Γ2 +− 1 +Γ +∂gΓ(¯ϵk) +∂¯ϵk +− 1 +3 +∂2fΓ(¯ϵk) +∂¯ϵ2 +k +� +≈ 1 +4 +� +k +∂α∂β∂γ¯ϵk +�f0(¯ϵk) +Γ2 ++ O(Γ0) +� +. +(73) +The leading term of the above expression in the sec- +ond line coincides with the Jerk conductivity predicted +within the relaxation time approximation from a simple +Boltzmann-relaxation-time formalism [43, 47, 51]. +For +an illustration see Fig. 3(d). We have also verified that +the above ω → 0 limit of the rectification conductivity +is identical to the ω → 0 limit of the second-harmonic +generation conductivity σγαβ +Γ +(ω, ω) (see Appendix C for +details). +Let us now focus on the main regime of our interest, +which is the “clean-limit” in which the relaxation rate +vanishes (Γ → 0) while the frequency remains finite. The +exact expression for the rectification conductivity in this +limit is given by: +lim +Γ→0σγαβ +Γ +(ω, −ω) = +1 +2ω4 +� +k +(∂γ¯ϵk)(∂α¯ϵk)(∂β¯ϵk) +× +� +f0(¯ϵk + ω) + f0(¯ϵk − ω) − 2f0(¯ϵk) +� +. +(74) +Notice that the above rectification conductivity would +vanish under any symmetry that enforces ¯ϵk = ¯ϵ−k, such +as time reversal or inversion symmetry. +Therefore, the +above expression proves one of our central claims, namely +that the rectification conductivity remains finite at finite +frequency within the optical transparency region of the +metal. The “transparency” here refers to the fact that +the real part of the linear conductivity vanishes in this +same limit ω ≫ Γ → 0. We illustrate this behavior in +Fig. 3(d) for our toy 1D model, confirming that the in +gap rectification is possible. The origin of this finite rec- +tification conductivity can be traced back to the fact that + +11 +(b) +0 +1 +10 +15 +5 +0 +0 +(a) +0 +-2 +2 +4 +(c) +-8 +-4 +0 +0 +1 +-1 +-2 +2 +FIG. 4. (a) Schematic of the original band (denoted by solid +line l = 0) and the boosted Floquet bands (denoted by dashed +lines l = ±1). +Here the chemical potential µ is below the +original band. The threshold frequency ωt is the minimum +frequency for boosted Floquet bands to cross the chemical +potential. (b) and (c) dimensionless rectification conductivity +σxxx +Γ +(ω, −ω)/σ(2) +0 +and its Log-log plots for different Γ, show- +ing that rectification conductivity is non-zero when ω > ωt. +Parameters used are the same with those in Fig. 3. +to the second order of the electric field, the ideal occupa- +tion function pk in the limit Γ → 0 is different from the +equilibrium Fermi-Dirac distribution [see Eq. (62)]. +While the expression of Eq. (74) is the exact clean limit +of the rectification conductivity in our model, it can be +shown that this expression reduces to the more famil- +iar expression for the Jerk current prediction of the sim- +ple Boltzmann-relaxation-time expression in the limit in +which the frequency is small compared to the bandwidth, +namely Γ ≪ ω ≪ ∆, and it is given by: +lim +ω→0 lim +Γ→0 σγαβ +Γ +(ω, −ω) = 1 +ω2 +� +k +f0(¯ϵk)∂α∂β∂γ¯ϵk, +(75) +which coincides with Eq. (21) of Ref. [43] for the +Jerk mechanism which has a 1/ω2 decaying power [see +Fig. 3(f), left-hand side region]. +More details of this +agreement with the simpler Boltzmann approach are dis- +cussed in Appendix D. +Interestingly, in the “ultra-high” frequency limit, when +the frequency is much larger than the bandwidth ω ≫ ∆, +the clean rectification conductivity transits to a different +scaling and decays much faster [see Fig. 3(f), right-hand +side region]: +lim +ω≫∆ lim +Γ→0 σγαβ +Γ +(ω, −ω) += +1 +2ω4 +� +k +� +1 − 2f0(¯ϵk) +� +(∂α¯ϵk)(∂β¯ϵk)(∂γ¯ϵk). +(76) +In contrast to the Boltzmann-relaxation-time result +where the large frequency regime is controlled by the +third momentum derivative of the band dispersion, here, +the large frequency response is controlled by the third +power of band velocity, which is a different intrinsic prop- +erty of the band. +It is interesting to note that the expression in Eq. (76) +remains finite even when the unperturbed band is either +fully occupied [f0(¯ϵk) = 1] or fully empty [f0(¯ϵk) = 0], +namely the system would be nominally an insulator with- +out a Fermi surface. This behavior is possible because our +bath does not conserve the total particle number of the +system, and therefore, there appears a finite occupation +of the bands when they are driven by the electric field, +even if the bands were initially empty in the distant past +before turning on the time dependent drive. +In other +words, all our calculations are performed strictly for a +bath with fixed chemical potential but not fixed density. +The appearance of a finite occupation of the bands to +second order of perturbation theory occurs when the fre- +quency exceeds the threshold so that one of the copies of +the Floquet bands boosted by ±ω crosses the chemical +potential, as depicted in Fig. 4. +III. +SUMMARY AND DISCUSSION +We have shown rigorously that the occupation of states +in a periodically driven fermionic system coupled to a fea- +tureless fermionic heat bath approaches a time indepen- +dent occupation function in the limit in which the cou- +pling to this bath is vanishingly small. This occupation +function can be computed analytically and differs from +the naive Fermi-Dirac occupation of the dressed Floquet +energies. This non-equilibrium steady state occupation +instead resembles a staircase version of the Fermi-Dirac +distribution [see Fig. 1(a) for an illustration], and also +cannot be expressed as a function of the Floquet energy +alone, but in general contains information on all the har- +monics encoding the full time dependence of the Hamil- +tonian. +We applied these results to the case in which the +fermionic system has a Hamiltonian corresponding to a +single Bloch band without Berry connections (e.g. aris- +ing from a tight-binding model with a single site per unit +cell) driven by a monochromatic electric field. We showed +that this staircase Fermi-Dirac distribution leads to a fi- +nite rectification conductivity within the optical trans- +parency region of a metal, which at small frequencies +compared to the bandwidth agrees exactly with the pre- +diction of the Jerk current effect expected from a simpler +Boltzmann-relaxation-time description [43, 47]. Because + +12 +the oscillating electric field is monochromatic, this rec- +tification conductivity does not arise because of the fre- +quency difference effect of Ref. [44] or the Raman-like +scattering effect of Refs. [45, 46]. +Our results validate our recent findings [43] that in- +gap rectification within the optical transparency region +of metals are indeed possible, even in the limit in which +carrier relaxation rates vanish, and clarify a discussion +surrounding this matter [44–46, 48–50]. 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Blanter, Quantum transport: +introduction to nanoscience (Cambridge university press, +2009). + +14 +Appendix A: Linear conductivity in the DC limit +In this appendix we show additional details of the linear conductivity in the DC limit discussed in the main text. +In the DC limit ω → 0 the linear conductivity [see Eq. (64) in the main text] becomes: +lim +ω→0 σαβ +Γ (ω) = 1 +2 +� +k +∂α∂β¯ϵk +�fΓ(¯ϵk) +Γ +− ∂gΓ(¯ϵk) +∂¯ϵk +� += 1 +2 +� +k +∂α∂β¯ϵk +�f0(¯ϵk) +Γ ++ Γ +2 +∂3f0(¯ϵk) +∂¯ϵ3 +k ++ Γ2 +3 +∂3g0(¯ϵk) +∂¯ϵ3 +k ++ O(Γ3) +� +, +(A-1) +in which +gΓ(ϵ) = 1 +2i +� +f+(ϵ) − f−(ϵ) +� +, +g0(ϵ) ≡ lim +Γ→0 gΓ(ϵ), +(A-2) +where gΓ(ϵ) is the imaginary part of f+(ϵ) defined in Eq. (25) in the main text, and we used the Cauchy–Riemann +equations satisfied by fΓ(ϵ) and gΓ(ϵ) +∂fΓ(ϵ) +∂Γ += ∂gΓ(ϵ) +∂ϵ +, +∂fΓ(ϵ) +∂ϵ += −∂gΓ(ϵ) +∂Γ +, +(A-3) +and the resulting relation +fΓ(ϵ) = f0(ϵ) + Γ∂g0(ϵ) +∂ϵ ++ O(Γ2), +(A-4) +to obtaining the second equation of Eq. (A-1). +Therefore the clean limit of the DC conductivity resembles the +prediction of the classic Drude theory for τ ≡ 1/(2Γ): +lim +Γ→0 lim +ω→0 σαβ +Γ (ω) = 1 +2Γ +� +k +f0(¯ϵk)∂α∂β¯ϵk, +(A-5) +and linear conductivity has a Drude peak in the DC limit when the chemical potential of the bath is within the +bandwidth of the system µ ∈ [0, ∆]. The system can still have a finite linear DC conductivity even if the band is +nominally fully empty or occupied at finite Γ, namely, +lim +ω→0 σαβ +Γ (ω) = 1 +2 +� +k +∂α∂β¯ϵk +�Γ2 +3 +∂3g0(¯ϵk) +∂¯ϵ3 +k ++ O(Γ3) +� +∝ Γ2 + O(Γ3), +� +T0 → 0, µ /∈ [0, ∆] +� +. +(A-6) +This conductance vanishes when Γ → 0. +Appendix B: Rectification conductivity in the DC limit +In this appendix we show more details of the rectification conductivity in the DC limit discussed in the main text. +In the DC limit, the rectification conductivity [see Eq. (72) in the main text] is: +lim +ω→0 σγαβ +Γ +(ω, −ω) = 1 +4 +� +k +∂α∂β∂γ¯ϵk +�fΓ(¯ϵk) +Γ2 +− 1 +Γ +∂gΓ(¯ϵk) +∂¯ϵk +− 1 +3 +∂2fΓ(¯ϵk) +∂¯ϵ2 +k +� += 1 +4 +� +k +∂α∂β∂γ¯ϵk +�f0(¯ϵk) +Γ2 ++ 1 +6 +∂2f0(¯ϵk) +∂¯ϵ2 +k ++ Γ2 +24 +∂4f0(¯ϵk) +∂¯ϵ4 +k ++ Γ3 +45 +∂5g0(¯ϵk) +∂¯ϵ5 +k ++ O(Γ4) +� +, +(B-1) +where we again used Eq. (A-4) in arriving at the second equation. In the clean limit Γ → 0, this coincides with the +Jerk conductivity predicted within the relaxation time approximation, but here we also present the sub-leading in Γ +correction: +lim +Γ→0 lim +ω→0σγαβ +Γ +(ω, −ω) = 1 +4 +� +k +∂α∂β∂γ¯ϵk +�f0(¯ϵk) +Γ2 ++ 1 +6 +∂2f0(¯ϵk) +∂¯ϵ2 +k +� +. +(B-2) +Therefore, similarly to the linear conductivity, second order rectification conductivity has a Jerk peak at DC limit +when the chemical potential is within the bandwidth of the system µ ∈ [0, ∆]. When the band is nominally fully +empty or occupied, for the rectification conductivity we now have +lim +ω→0σγαβ +Γ +(ω, −ω) = 1 +4 +� +k +∂α∂β∂γ¯ϵk +�Γ3 +45 +∂5g0(¯ϵk) +∂¯ϵ5 +k ++ O(Γ4) +� +∝ Γ3 + O(Γ4), +� +T0 → 0, µ /∈ [0, ∆] +� +. +(B-3) +This finite DC rectification conductivity again vanishes in the clean limit Γ → 0. + +15 +Appendix C: Second harmonic generation +In this appendix we show the second harmonic conductivity mentioned in the main text. The second harmonic +conductivity is the one that controls the response oscillating at the double frequency of the drive (s = 2), we define +it as: +j(2) +γ += σγαβ +Γ +(ω, ω)Eα +ωEβ +ω + O(|Eω|3), +(C-1) +and it is given by the following expression: +σγαβ +Γ +(ω, ω) = − 1 +2ω2 +� +k +fΓ∂α∂β∂γ¯ϵk − 1 +ω3 +Γ +2Γ − iω +� +k +(∂α¯ϵk)(∂β∂γ¯ϵk)L1(¯ϵk, ω) +− +1 +2ω4 +Γ +2Γ − 2iω +� +k +(∂γ¯ϵk) +� +(∂α¯ϵk)(∂β¯ϵk)L2(¯ϵk, ω) + ω +2 ∂α∂β¯ϵkL1(¯ϵk, 2ω) +� +, +(C-2) +where +L2(¯ϵk, ω) = f+(¯ϵk − 2ω) − 2f+(¯ϵk − ω) + f+(¯ϵk) + f−(¯ϵk) − 2f−(¯ϵk + ω) + f−(¯ϵk + 2ω). +(C-3) +The low frequency limit of second harmonic conductivity coincides with the low frequency limit of rectification +conductivity from Eq. (73) in the main text: +lim +ω→0 σγαβ +Γ +(ω, ω) = 1 +4 +� +k +∂α∂β∂γ¯ϵk +�fΓ(¯ϵn) +Γ2 +− 1 +Γ +∂gΓ(¯ϵn) +∂¯ϵn +− 1 +3 +∂2fΓ(¯ϵn) +∂¯ϵ2n +� +. +(C-4) +Interestingly, at large frequencies ω ≫ ∆ the real part of the second harmonic conductivity decays as 1/ω2 in contrast +to 1/ω4 power decay of the rectification conductivity. +Appendix D: Relation to the Boltzmann theory +In this appendix we discuss the relation between our result and that from a simpler Boltzmann/relaxation-time +approach. We begin by writing a Boltzmann equation for a single band system in the relaxation time approximation: +∂tf(k, t) + E(t) · ∇kf(k, t) = −[f(k, t) − f0(ϵk)]/τ, +(D-1) +where E(t) = Eωe−iωt + c. c. is a monochromatic electric field. +The above equations are written in a different gauge with respect to the main text: here k is viewed as a gauge +invariant mechanical crystal momentum, which corresponds to k−A(t) in the main text. In order to obtain expressions +for occupation functions in the same gauge as in the main text, we convert to a gauge in which we keep track of the +occupation of canonical crystal momenta, using the following relation: +p(k, t) ≡ f(k − A(t), t). +(D-2) +The occupation function p(k, t) satisfies the following equation +∂tp(k, t) = ∂tf(k − A(t), t) − ∂tA(t) · ∇kf(k − A(t), t) += ∂tf(k − A(t), t) + E(t) · ∇kf(k − A(t), t) += −[f(k − A(t), t) − f0(ϵk−A(t))]/τ, +(D-3) +where we used Eq. (D-1) in obtaining the last equation. Therefore we see that the distribution function p(k, t) satisfies +an equation without explicit electric field derivative term: +∂tp(k, t) = −[p(k, t) − f0(ϵk−A(t))]/τ. +(D-4) +Using the fact that the late-time steady state distribution is periodic, we perform Fourier series expansions for both +p(k, t) and f0(ϵk−A(t)): +p(k, t) = ++∞ +� +l=−∞ +p(l)(k) exp(−ilωt), +p(l)(k) = +� T +0 +dt +T p(k, t) exp(ilωt); +f0(ϵk−A(t)) = ++∞ +� +l=−∞ +f (l) +0 (k) exp(−ilωt), +f (l) +0 (k) = +� T +0 +dt +T f0(ϵk−A(t)) exp(ilωt). +(D-5) + +16 +With the above expansions, Eq. (D-4) becomes +−ilωp(l)(k) = −p(l)(k)/τ + f (l) +0 (k)/τ, +(D-6) +and leads to +p(l)(k) = +1 +1 − ilωτ f (l) +0 (k). +(D-7) +The above solution in general requires an explicit calculation of the following mixed harmonics of the distribution: +f (l) +0 (k) = +� T +0 +dt +T f0(ϵ(0) +k ++ ϵ(1) +k e−iωt + ϵ(−1) +k +eiωt + · · · ) exp(ilωt), +(D-8) +where +ϵk−A(t) = ++∞ +� +l=−∞ +ϵ(l) +k exp(−ilωt), +ϵ(l) +k = +� T +0 +dt +T ϵk−A(t) exp(ilωt). +(D-9) +Let us consider however the clean limit τ → +∞. +Notice that f (l) +0 (k) is independent of τ, therefore for l ̸= 0 +components we have +lim +τ→+∞ p(l̸=0)(k) = +lim +τ→+∞ +1 +1 − ilωτ f (l) +0 (k) = 0. +(D-10) +However the l = 0 component, or time averaged component, which is independent of τ and therefore remains finite +as τ → +∞, is given by: +p(0)(k) = f (0) +0 (k) = +� T +0 +dt +T f0(ϵ(0) +k ++ ϵ(1) +k e−iωt + ϵ(−1) +k +eiωt + · · · ). +(D-11) +Therefore, similarly to Eq. (54) obtained from the full formalism with the bath, the distribution from the Boltzmann +theory becomes time independent in the canonical crystal momentum, but not in the mechanical physical momentum, +in the analogous ideal limit of τ → +∞. Notice, however, that the above result has to be viewed as a limit of τ → +∞, +and not as a situation in which there is no relaxation. This is because in the strict absence of relaxation mechanisms +there is no unique late-time steady state, namely by taking 1/τ = 0 and neglecting altogether the relaxations in the +right hand side of Eq. (D-4) any time-independent distribution of the canonical momenta would be a solution. +If we expand up to the second order of electric fields Eq. (D-11) we obtain: +p(0)(k) = +� T +0 +dt +T +� +f0(¯ϵk) + +� +ϵ(1) +k e−iωt + ϵ(−1) +k +eiωt + ϵ(2) +k e−i2ωt + ϵ(−2) +k +ei2ωt� +f ′ +0(¯ϵk) ++ 1 +2 +� +ϵ(1) +k e−iωt + ϵ(−1) +k +eiωt�2f ′′ +0 (¯ϵk) + O(|Eω|3) +� += f0(¯ϵk) + |ϵ(1) +k |2f ′′ +0 (¯ϵk) + O(|Eω|3). +(D-12) +Interestingly the above distribution function coincides with the asymptotic behavior of the staircase distribution +function discussed in the main text [see e.g., Eq. (62)] in limit of ∆ ≫ ω ≫ Γ → 0: +lim +ω→0 lim +Γ→0 pk = lim +ω→0 +�� +1 − 2 +��ϵ(1) +k +��2 +ω2 +� +f0(¯ϵk) + +��ϵ(1) +k +��2 +ω2 +f0(¯ϵk − ω) + +��ϵ(−1) +k +��2 +ω2 +f0(¯ϵk + ω) +� += f0(¯ϵk) + |ϵ(1) +k |2f ′′ +0 (¯ϵk). +(D-13) +Therefore the expectation value of all equal time observables, such as the electric current, coincide with those of the +more microscopic Floquet-bath theory of the main text, at least to second order in electric fields. In particular one +obtains the same rectification conductivity in the above limit as that in Eq. (75) of the main text, that we refer to as +Jerk effect. + +17 +Appendix E: Comments and connections to other works in the literature +There has been a long-standing debate in the literature about the possibility of in-gap rectification which has been +clouded by previous imprecise and incorrect statements. In this section we will try to clarify some of this. We begin by +defining precisely what do we mean by in-gap rectification. The optical gap is defined as the region in the frequency +domain in which the the hermitian symmetric part of the conductivity tensor vanishes in the limit of low temperatures +and small scattering rates (see Ref. [43] for a review). We then say that a system has in-gap rectification if any of the +elements of the rectification conductivity tensor that lead to finite DC currents generated by a monochromatic AC +electric field with a frequency within the optical gap remain non-zero in that same limit. More specifically: +Definition of “optical gap” : +lim +T0→0 lim +Γ→0 +� +σαβ(ω) + [σβα(ω)]∗� +→ 0, +when ω ∈ optical gap. +(E-1) +Definition of “in-gap rectification” : +lim +T0→0 lim +Γ→0 σγαβ(ω, −ω) ̸= 0, +when ω ∈ optical gap. +(E-2) +Therefore our current manuscript and our previous work in Ref. [43], demonstrate rigorously that in-gap rectification +in the above sense is indeed possible. +Nevertheless, some confusion in the literature appears to have originated from different interpretations of the work +of Belinicher, Ivchenko, and Pikus (BIP) in Ref. [48]. That paper contained statements such as “The conclusion +that a steady-state photocurrent may appear on illumination in the transparency range of a crystal, reached in earlier +publications, is shown to be in error”. This statement could be read as implying the impossibility of in-gap rectification +in the sense we defined above. In fact, this reading of the BIP paper appears to have been made in several references +claiming that in-gap rectification in the above sense is impossible [44, 45, 50]. Even us in our recent work of Ref. [43], +read the BIP paper as trying to prove that in-gap rectification is impossible in the above sense. +However, part of the issue with reading the aforementioned BIP paper, is that it left several crucial gaps in its +discussion and its derivations that can make it hard to know in a precise way what exactly BIP implied at various +places and the precise framework that BIP used for reaching such conclusions. For example, a crucial point that can +lead to a different readings of the BIP paper relates to the definition of the term “gn” that appears in the right hand +side of their Eq. (8) in Ref. [48], which is a central equation from which various conclusions are derived. Unfortunately +BIP never spelled out an explicit form for this term, but simply wrote that “gn is the generation function, i.e., the +rate of change of the distribution function due to optical transitions.”. This leaves open to interpretation what exactly +they had in mind for “optical transitions”. For example, one could read this by interpreting “gn” as associated only +with inter-band optical transitions, and in this case, one would be lead to read the BIP paper as trying to imply that +in-gap rectification in the above sense is impossible. +There is however an alternative way to interpret “gn” and the notion of “optical transitions” in Ref. [48] as a +more general notion of irreversible “transitions” that can take place even within what would nominally be the optical +gap defined in the above sense. This more nuanced way of interpreting the BIP paper has indeed been recently +emphasized by Glazov and Golub in Ref. [46]. For example Golub and Glazov write in Ref. [46] that “... even for +transparent media, real electronic transitions should occur to enable the photocurrent.” and that “We reiterate that +in the absence of any real electronic transitions DC current is forbidden. It is obvious from general reasons: If a DC +current is generated then this current results in a Joule heat in the sample or in the external circuit connected to the +sample. It is forbidden by the energy conservation law in the absence of real transitions.”. What Golub and Glazov +are trying to explain there is in line with our recent thermodynamic analysis in Ref. [43], where we emphasized that in +order to guarantee the positivity of entropy production, specially when the system is connected to an external circuit, +it is always important to view the scattering rate Γ as possibly arbitrarily small but not strictly zero. This requirement +means that physically it is important to have always a non-zero absorption within the nominal optical gap of the +material. In fact Golub, Ivchenko himself and Spivak, have also emphasized a related aspect of this in Ref. [55] where +they demonstrated that the CPGE effect associated with the Berry dipole term remains finite within the optical in +the limit of Γ → 0, but also coexists the other contributions that originate from impurity scattering mechanisms that +scale in the same way with frequency and remain finite inside of the gap in the limit of Γ → 0. One way to state this +state of affairs, that has been emphasized by Golub and Glazov to us in private communications, is that while the +real transitions associated with scattering lead to a vanishingly small linear dissipative conductivity in the limit of +Γ → 0, there are cancellations of the scattering rate that lead to finite rectification conductivity in this limit but the +“real electronic transitions” are still taking place. These “real electronic transitions” are therefore the more general +notion of “optical transitions” that can contribute to the term “gn” in the BIP reference. Therefore, within this point +of view, one can say that the BIP should not be read as implying that in-gap rectification is impossible in the sense +we defined above. We are in agreement with the physics of this point of view broadly speaking. +There is however another crucial aspect of the BIP work in Ref. [48] with which we still find ourselves in disagreement +and that we believe our current paper provides good evidence to be incorrect in general. BIP stated that “... in + +18 +the case of continuous illumination the steady-state distribution function is f0(¯ϵk) irrespective of how weak is the +interaction of electrons with phonons.” In this statement f0 is the “equilibrium distribution function” (the Fermi- +Dirac occupation function) and ¯ϵk is the Floquet energy of the band. These statement has been echoed in several +subsequent works [45, 46, 49, 50]. However, our current work demonstrates that in the limit of Γ → 0 the distribution +function is sharply different from the naive Fermi-Dirac occupation, but becomes instead the non-trivial Fermi-Dirac +staircase discussed in the main text, even to the leading order |Eω|2 in the driving monochromatic field. Crucially +the resulting occupation function cannot be expressed as a function of the Floquet energy alone [see Fig. 1, Eq. (54), +and Eq. (62) of the main text]. Notice that in order to have a unique and well defined steady state at late times, we +must necessarily view the relaxation rate Γ as being arbitrarily small but not strictly zero. Therefore, the notion of +the ideal occupation in the steady state has to be necessarily interpreted as the limit of Γ → 0 of the occupation of +systems with a finite Γ. This is because systems with strictly zero relaxation rate (Γ = 0) do not have a way to erase +the memory of their initial conditions and therefore their steady state in the presence of the monochromatic light is +not uniquely defined. +We have demonstrated rigorously that at least for an ideal fermionic bath the occupation of states in the limit of +Γ → 0 is not f0(¯ϵk) as Refs. [45, 46, 48–50] presumed. We would like to emphasize that while the fermionic bath +might appear to be a somewhat artificial approximation to the true mechanisms of relaxation for certain realistic +physical situations, it behaves as an ideal thermal bath in the limit Γ → 0. In particular, the particle number becomes +effectively conserved in such limit since the self-consistent occupations at each momentum become a time independent +function as we have shown. We have in particular demonstrated that in equilibrium this bath leads to the expected +Fermi-Dirac occupation of the system. +More generally speaking, in equilibrium one expects a universality of all +intensive thermodynamic physical properties of the system of interest for a large class of baths regardless of their +details, which essentially defines the class of “ideal thermodynamic baths”. However, how this universality carries over +to non-equilibrium settings is still unclear to us. Therefore whether other baths or other relaxation mechanisms such +as coupling to phonons, impurities or self-thermalization via electron-electron interactions lead to a similar stair-case +occupation to the one we have found in the limit of vanishing relaxation rates Γ → 0, remains an interesting open +problem. We note however that none of the aforementioned Refs. [45, 46, 48–50] has provided a rigorous and controlled +derivation of the self-consistent occupation of Floquet bands based on any microscopically explicit mechanism of +relaxation, like the one we have provided. Therefore we do not see any rigorous substance to their claim that the +occupation is f0(¯ϵk) even for other microscopic relaxation mechanisms such as phonons. Moreover, it has become +abundantly clear in the study of thermalization of Floquet systems in recent years that the self-consistent occupation +of Floquet bands coupled to baths that are also bosonic differs clearly from the naive Fermi-Dirac distribution of the +Floquet bands f0(¯ϵk) [36–41]. + diff --git a/IdAyT4oBgHgl3EQf5vqw/content/tmp_files/load_file.txt b/IdAyT4oBgHgl3EQf5vqw/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c8e02894ceff0e02ec3154bdf1e21dbee12245ba --- /dev/null +++ b/IdAyT4oBgHgl3EQf5vqw/content/tmp_files/load_file.txt @@ -0,0 +1,914 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf,len=913 +page_content='The Fermi-Dirac staircase occupation of Floquet bands and current rectification inside the optical gap of metals: a rigorous perspective Oles Matsyshyn,1, 2 Justin C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' Song,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content='2 Inti Sodemann Villadiego,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' ∗ and Li-kun Shi1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' † 1Max-Planck-Institut für Physik komplexer Systeme,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' Nöthnitzer Straße 38,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' 01187 Dresden,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' Germany 2Division of Physics and Applied Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' School of Physical and Mathematical Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' Nanyang Technological University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' Singapore 637371,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' Republic of Singapore 3Institut für Theoretische Physik,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' Universität Leipzig,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' Brüderstraße 16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' 04103,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' Leipzig,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' Germany (Dated: January 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' 2023) We consider a model of a Bloch band subjected to an oscillating electric field and coupled to a featureless fermionic heat bath,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' which can be solved exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' We demonstrate rigorously that in the limit of vanishing coupling to this bath (so that it acts as an ideal thermodynamic bath) the occu- pation of the Floquet band is not a simple Fermi-Dirac distribution function of the Floquet energy, but instead it becomes a “staircase” version of this distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' We show that this distribution generically leads to a finite rectified electric current within the optical gap of a metal even in the limit of vanishing carrier relaxation rates, providing a rigorous demonstration that such rectification is generically possible and clarifying previous statements in the optoelectronics literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' We show that this current remains non-zero even up to the leading perturbative second order in the amplitude of electric fields, and that it approaches the standard perturbative expression of the Jerk current obtained from a simpler Boltzmann description within a relaxation time approximation when the frequencies are small compared to the bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' INTRODUCTION Quantum many body systems that are periodically driven in time have garnered attention over the last decades as rich platforms to realize novel collective phe- nomena and non-equilibrium states beyond those realized in equilibrium [1–25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' A phenomenon that can arise in such periodically driven systems and which is forbidden in equilibrium, is the existence of an average net recti- fied particle flow or ratchet effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' In particular for the case of electrons in crystals driven by oscillating electric fields, these effects have enjoyed recent renewed attention due to their interesting interplay with the dispersions and Berry phases of band structures and their potential for novel opto-electronic devices [26–32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' Despite their nat- ural connection, only a handful of studies have studied current rectification effects in Bloch bands through the lens of Floquet theory [33–35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' This is in part due to the difficulty that even for non-interacting systems, there is no simple general formula dictating the occupation of Floquet bands analogous to the Fermi-Dirac distribution that dictates the occupation of bands in equilibrium [36– 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' One of the central goals of our study is to provide simple analytical formulae for the occupation of a single Floquet band coupled to a “featureless fermionic bath” (which is a commonly used model of bath that for exam- ple was employed in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' [33, 35, 42]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' This featureless bath is a physical system that has a finite coupling to the fermionic system of interest and is characterized by a single relaxation scale Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' The dynamics of the system ∗ sodemann@itp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content='uni-leipzig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content='de † shi@itp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content='uni-leipzig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content='de 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content='5 1 0 2 0 1 1 2 3 4 (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (a) Time independent limΓ→0 pn as a function of ¯ϵn/ω calculated using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (46) showing a ladder-like occupa- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' Parameters used: β/ω = 50, µ/ω = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (b) Schematic of the ladder-like occupation for a Bloch band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' coupled to this bath can be described in an exact man- ner thanks to the fact that we will take the combined system plus bath as a non-interacting fermionic system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' As we will see, this featureless fermionic bath behaves as an ideal thermodynamic bath in the limit in which its coupling to system is vanishingly small (Γ → 0), and in particular in this limit it relaxes the system towards the equilibrium Fermi-Dirac occupation of the bands in arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content='00811v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content='mes-hall] 2 Jan 2023 2 the absence of an external periodic drive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' As we will show, however, when the system is periodically driven in time, this bath leads to a self-consistent occupation the Floquet bands that is sharply different from that of the equilibrium Fermi-Dirac distribution, but which we can determine analytically with no approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' This oc- cupation is instead a staircase version of the Fermi-Dirac distribution with several jumps that occur at copies of the chemical potential shifted by all the harmonics of the driving frequency [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' 1(a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' In the limit of an ideal bath (Γ → 0), we will show that this distribution coincides with the distribution that has been previously obtained within the Boltzmann approach to Floquet sys- tems (see in particular Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (12) of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' [38]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' Another central purpose of our study is to exploit the Floquet formalism to further elaborate on our recent find- ing [43] that it is indeed possible for time dependent os- cillating electric fields with a frequency that lies within the optical gap of a metal, to induce a net rectified DC electric current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' We will see that this is true even when the electric field has a single monochromatic frequency ω that is much larger than the relaxation rates and this current remains finite in the limit when these rates van- ish (Γ → 0) (and therefore does not rely on the fre- quency difference effect [44] or in the Raman scattering effect [45, 46]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' We will demonstrate that this is possi- ble by choosing a simple model containing a single Bloch band with no Berry curvature, which in a simpler relax- ation time Boltzmann description would give rise to the so-called “Jerk” effect described in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' [43, 47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' Our aim is to use this simple model because it will allow us to carry out calculations of its response coupled to the fermionic bath in a clear and exact analytical manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' We are motivated to do this rigorously in order to clarify a series of misconceptions that originated from the work of Belinicher, Ivchenko and Pikus [48] and that have propagated into some of the subsequent literature [43, 45, 46, 49, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' In Appendix E we comment in more detail about some of these previous works and point out more specifically some of their imprecisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' One of the central messages of our study is that it is in- deed possible to have a net rectified current in response to a monochromatic oscillating electric field whose fre- quency lies within the optical gap of a system, in the limit of vanishing carrier relaxation rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' We will demonstrate this within a self-consistent picture of the occupation of Floquet Bloch band in the steady state of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' More specifically we will show that in the limit of an ideal bath (Γ → 0) the average rectified current in the non-equilibrium steady state of the system is given by: ¯j = � k pk∇k¯ϵk, (1) where ¯ϵk is the Floquet energy of the band, pk is the occu- pation of the Floquet band, ¯j is the current averaged over one period, and the integral is over the crystal momen- tum in the Brillouin zone with the usual normalization of 1/(2π)d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' The crucial difference between the above expression and that for the average current in an equilibrium sys- tem, is that the occupation function pk is not simply the Fermi-Dirac distribution associated with ¯ϵk, but instead it is precisely the stair-case occupation function depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' Crucially, we will show that generically this stair-case occupation is a function that depends on all the information of the time dependence of the Hamiltonian, and cannot be expressed as a function of only ¯ϵk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' We will then show that as a consequence of this, the rectified current is in fact generically non-zero in the optical gap of a metal that breaks inversion and time-reversal sym- metries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' This result remains true even to second order in the amplitude of the oscillating electric field, which is the leading order at which rectification currents appear, and therefore implies a non-vanishing rectification con- ductivity within the optical gap of metals, in agreement with our previous results [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' For other previous dis- cussions of the possibility of in-gap rectification see also Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' [46, 51–55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' Our paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' In Section I, we setup the approach to open quantum systems, obtain ex- act occupation functions for diagonal and time-periodic Hamiltonians coupled to a featureless fermionic bath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' In Section II, based on the exact occupation functions, we calculate exact linear and rectification conductivities and show that there is a net rectified current in response to a monochromatic oscillating electric field whose frequency lies within the optical gap of a metal, in the limit of van- ishing carrier relaxation rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' THE OPEN-SYSTEM SCHRÖDINGER EQUATION APPROACH TO OPEN QUANTUM SYSTEMS In descriptions of quantum open systems it is typically natural to view the combined Hilbert space of the “sys- tem” and the “bath” as a tensor product of their Hilbert spaces in isolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' There are situations, however, where it is possible to alternatively cast this separation of sys- tem and bath as a direct sum of their individual Hilbert spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' As we will show, such separation into sums of Hilbert spaces is extremely powerful and convenient, be- cause it allows to integrate out the dynamics of the “bath” in an exact manner and to obtain a simple non-Hermitian generalization of Schrödinger’s equation for the system which captures its coupling to the bath without any ap- proximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' One example of the class of models which admits such direct sum separation into system and bath are those of non-interacting particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' To see this let us imagine that the system and the bath as a whole can be described by a non-interacting model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' For concreteness we can imagine this to be a tight biding model of particles hopping on a lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' Because the problem is non-interacting, then the dynamics can be analyzed by computing the trajecto- ries of single individual particles and then adding them 3 up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' However, for a single particle the Hilbert space of the “system” and the “bath” can be naturally viewed as a direct sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' For example, in the case of a tight-binding model, some sites can be viewed as belonging to the sys- tem and the remainder sites as belonging to the bath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' Let us then consider that the Hilbert space of the sys- tem and the bath can be decomposed into a direct sum, namely their Hamiltonian and states have block form as follows: H(t) = � HS(t) HSB(t) HBS(t) HB(t) � , |ψ(t)⟩ = � |ψS(t)⟩ |ψB(t)⟩ � , (2) where HBS(t) = H† SB(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (2), the coupled Schrödinger equations for system and bath states then read: i∂t |ψS(t)⟩ = HS(t) |ψS(t)⟩ + HSB(t) |ψB(t)⟩ , (3) i∂t |ψB(t)⟩ = HBS(t) |ψS(t)⟩ + HB(t) |ψB(t)⟩ , (4) where we set ℏ = 1 throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' By integrating Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (4) over time and inserting it into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (3) allows to formally eliminate the bath state dynamics |ψB(t)⟩ and to obtain the open-system Schrödinger equation for |ψS(t)⟩: i∂t |ψS(t)⟩ = HS(t) |ψS(t)⟩ + HSB(t)UB(t, t0) |ψB(t0)⟩ − iHSB(t) � t t0 dt′ UB(t, t′)HBS(t′) |ψS(t′)⟩ , (5) where UB(t, t′) is the bath (intrinsic) evolution operator satisfying i∂tUB(t, t0) = HB(t)UB(t, t0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' This procedure is often carried within the Schwinger-Keldysh formalism by integrating out part of the action describing the de- grees of freedom of the bath (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' [33, 35, 42]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' But this is easier and more physically transparent in our first quantization notation and the final results would be identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' Featureless fermionic bath We will now specialize the above equation to a model of a “featureless fermionic bath”, which we define as having the following characteristics: (i) In a featureless fermionic bath every state of the system is coupled to a collection of identical sites with the same energy spectrum and the same coupling λ [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' 2(a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' If the system states (basis) are denoted by |χn⟩ and the bath states (basis) by |ϕn,j⟩, the bath and the system-bath coupling are HB = � n,j εj |ϕn,j⟩ ⟨ϕn,j| , (6) HSB = λ � n,j |χn⟩ ⟨ϕn,j| , (7) where εj is the energy for the bath state |ϕn,j⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' This model of the bath is identical to that employed in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' [35, 43, 56–62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (ii) The featureless fermionic bath is prepared in an ini- tial condition at t0 with a Fermi-Dirac distribution that only has weight on the bath sites, described by ρS(t0) = 0, ρB(t0) = � n,jf0(εj) |ϕn,j⟩ ⟨ϕn,j| , (8) f0(εj) = 1 exp[β0(εj − µ0)] + 1, (9) where µ0 is the chemical potential of and β0 = 1/kBT0 denotes the temperature of the bath, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' The assumption of the initial density matrix only having weight on the bath is useful but it is not strictly nec- essary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' This is because in the limit in which the bath spectrum becomes a dense continuum, the information of the initial condition for the component of density ma- trix on the system will decay over time and only the in- formation of the initial condition for the density matrix on the bath will dictate the late time steady state [this will become more clear in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (16) which is a subsequent version of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (5)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' Notice also that we have equated the evolution of the single particle density matrix with that of the many-body one-particle density matrix (equal time Greens function), which is possible thanks to the fact that the system is non-interacting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' With assumptions (i) and (ii), by evolving the initial condition in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (8) under Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (3) and (4), one finds that the one-body density matrix projected onto the system at time t is given by ρS(t) = � n,j f0(εj) |ψ(j) n (t)⟩ ⟨ψ(j) n (t)| , (10) where |ψ(j) n (t)⟩ is the component within system Hilbert space that evolves out of the initial state |ψB(t0)⟩ = |ϕn,j⟩ in the bath at t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (10) states that the density matrix for the system is the weighted sum of contribu- tions from all bath states with their corresponding initial occupations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' Using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (5), (6), and (7), we obtain the open-system Schrödinger equation for |ψ(j) n (t)⟩: i∂t |ψ(j) n (t)⟩ = HS(t) |ψ(j) n (t)⟩ + λ exp[−iεj(t − t0)] |χn⟩ − i � ∞ t0 dt′ γ(t − t′) |ψ(j) n (t′)⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (11) system DoS energy bath (a) (b) (c) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (a) Schematic of the system-bath coupling HSB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (b) The bath’s density of states is much wider than that of the system, and we simplify it to be flat in the energy range of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (c) Schematic of the bath acting as a source as well as a sink for the system [see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (27)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' 4 Here, λ exp[−iεj(t−t0)] |χn⟩ is a source term for |ψ(j) n (t)⟩ arising from the bath, while the memory function in the second line is given by, γ(t) = λ2Θ(t) � j exp(−iεjt), (12) which encodes the memory of decay for |ψ(j) n (t)⟩ due to the bath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' This memory function makes the Schrödinger equation for open systems non-local in time, and in gen- eral it incorporates decay and renormalizations of the system energies due to their coupling to the bath [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' 2 for a depiction].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (iii) To remove the finite memory delay, we impose one further property defining the featureless fermionic bath, namely that it has an infinitely broad and flat spectrum [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' 2(b)], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=', the bath density of state is constant: νB(ωb) = 2π � j δ(ωb − εj) ≡ ν0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (13) With this simplification, the finite delay or non-local memory of the past time t′ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (11) is lost, the memory function becomes: γ(t) = λ2Θ(t) � j � +∞ −∞ dωb δ(ωb − εj) exp(−iωbt) = λ2ν0Θ(t) � +∞ −∞ dωb 2π exp(−iωbt) = δ(t) Γ, (14) where we used Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (13) to obtain the second equation and defined Γ ≡ λ2ν0 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (15) With the above simplification of infinitely broad spec- trum for the bath, the open-system Schrödinger’s equa- tion reduces to: i∂t |ψ(j) n (t)⟩ = � HS(t) − iΓ � |ψ(j) n (t)⟩ + λ exp[−iεj(t − t0)] |χn⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (16) The above equation is remarkably simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' It is a sim- ple non-Hermitian version of the Schrödinger equation in which the system Hamiltonian is dressed by a constant imaginary part “−iΓ” which captures the decay into the bath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' Many recent studies of open quantum systems have used non-Hermitian Schrödinger equations that only in- clude the first line of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' However, we see that the influence of the bath is not merely to induce decay, but it also produces the second term that acts a source and makes the equation inhomogeneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' The balance of these two terms is what allows the existence of non-trivial late time steady states (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' 2 for depiction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' Ideal fermionic bath To illustrate that our bath leads to the expected equi- librium when the system is not driven in time, we first consider the the special case in which HS(t) is time in- dependent, HS(t) → H0 = � n ϵn |χn⟩ ⟨χn| , (17) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (16) can be equivalently expressed as i∂ts(j) n = [ϵn − iΓ]s(j) n + λ exp[−iεj(t − t0)], (18) where s(j) n = ⟨χn|ψ(j) n (t)⟩ , (19) is the amplitude for the system state |χn⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' Solving the above Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (18) gives s(j) n = −iλ exp � − i � t t0 dt′ (ϵn − iΓ) � × � t t0 dt′ exp � i � t′ t0 dt′′(ϵn − iΓ − εj) � = λ ϵn − iΓ − εj � e−(Γ+iϵn)(t−t0) − e−iεj(t−t0)� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (20) Then using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (10), we obtain the steady state, diagonal density matrix for the system: ρS(t → +∞) = � n fΓ(ϵn) |χn⟩ ⟨χn| , (21) in which fΓ(ϵn) = limt→+∞ � j f0(εj)|s(j) n |2 and reads explicitly as fΓ(ϵn) = � j f0(εj) λ2 (ϵn − iΓ − εj)(ϵn + iΓ − εj) = � +∞ −∞ dωb f0(ωb) λ2 � j δ(ωb − εj) (ϵn − iΓ − ωb)(ϵn + iΓ − ωb) = � +∞ −∞ dωb π f0(ωb) Γ (ϵn − ωb)2 + Γ2 , (22) where we used Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (13) and (15) in obtaining the last equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' The above distribution fΓ(ϵn) shows that when HS(t) is time independent, the system “thermalizes” by approaching a time independent steady state dictated by the initial condition of the bath, f0(ωb), while a finite Γ accounts for the broadening of the energy levels of the system due to its coupling to the bath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' Importantly, taking the limit in which the coupling to the bath vanishes from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (22), we obtain lim Γ→0 fΓ(ϵn) = f0(ϵn), (23) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=', fΓ(ϵn) reduces to the ideal Fermi-Dirac distribution in the limit of Γ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' We will then call this Γ → 0 limit of the “featureless fermionic bath” an “ideal fermionic bath”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' The fact that the ideal Fermi-Dirac distribution 5 appears only when the coupling to the bath is vanish- ingly weak is consistent with general considerations of statistical physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' However, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (22) still allow us to obtain analytically the modified occupation at finite coupling to the bath, which will be used in subsequent manipulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' By in- tegrating over ωb in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (22) using Cauchy’s residue the- orem, we find that: fΓ(ϵ) = 1 2 � f+(ϵ) + f−(ϵ) � , (24) where f+(ϵ) = [f−(ϵ)]∗ and they are given by: f±(ϵ) = 1 2 ± i π Ψ(0) �1 2 ± iβ ϵ ∓ iΓ − µ 2π � , (25) with Ψ(0) the 0-th order Polygamma function (or the digamma function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' f±(ϵ) will also appear repeatedly in more general cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' Diagonal and time-periodic Hamiltonians 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' Diagonal system Hamiltonian In this work, we will develop the above general for- malism to the special case where the system Hamilto- nian HS(t) is time dependent but diagonal in the system states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' Let us then take the following form for the system Hamiltonian: ⟨χn|HS(t)|χm⟩ = δnm[ϵn + Vn(t)] = δnmϵn(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (26) With this, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (16) then reduces to i∂ts(j) n = [ϵn(t) − iΓ]s(j) n + λ exp[−iεj(t − t0)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (27) Solving the above Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (27) gives s(j) n (t) = −iλ exp � − i � t t0 dt′ [ϵn(t′) − iΓ] � × � t t0 dt′ exp � i � t′ t0 dt′′[ϵn(t′′) − iΓ − εj] � , (28) and then with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (10), we obtain the diagonal density matrix for the system: ρS(t) = � n pn(t) |χn⟩ ⟨χn| , (29) pn(t) = � j f0(εj)|s(j) n (t)|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (30) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' Periodic system Hamiltonian Now we consider a periodically driven system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' Namely, we take the diagonal elements of the Hamiltonian to be periodic in time: ϵn(t + T) = ϵn(t) = +∞ � l=−∞ ϵ(l) n exp[−ilω(t − t0)], (31) where T is the period and ω = 2π/T is the frequency, and ϵ(l) n = � T 0 dt T ϵn(t) exp[ilω(t − t0)] (32) is the l-th Fourier coefficient for ϵn(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' In particular, ¯ϵn ≡ ϵ(0) n = � T 0 dt T ϵn(t), (33) is the time-average of the diagonal element of the Hamil- tonian, which as we will show next, coincides with the Floquet energy of state n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' To see this, notice that the wavefunction that would solve the system Schrödinger’s equation in the absence of the bath, can be expressed as follows: exp � − i � t t0 dt′ ϵn(t′) � = exp � − i � t t0 dt′[ϵn(t′) − ¯ϵn] � × exp � − i � t t0 ¯ϵn � ≡ φn(t) × exp � − i¯ϵn(t − t0) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (34) The periodicity of the first factor denoted by φn(t) can be shown explicitly: φn(t + T) = φn(t) × exp � − i � t+T t dt′[ϵn(t′) − ¯ϵn] � = φn(t), (35) where we used Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (33) in obtaining the second equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' Therefore we see from second factor in the last line of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (35), that the time-average of the diagonal element of the Hamiltonian is the Floquet energy itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' Let us now consider the Fourier expansion of the peri- odic part of the Floquet wavefunction: φn(t) = exp � − i � t t0 dt′[ϵn(t′) − ¯ϵn] � = +∞ � l=−∞ φ(l) n exp[−ilω(t − t0)], (36) or equivalently, φ(l) n = 1 T � t0+T t0 dt � exp[ilω(t − t0)] × exp � − i � t t0 dt′[ϵn(t′) − ¯ϵn] �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (37) The above expression makes clear that the amplitude of the harmonics of the wavefunction, φ(l) n , are functions 6 of the full time dependence of the instantaneous energy ϵn(t), and are independent of the Floquet energy ¯ϵn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' This property will be crucial later on for purposes of under- standing why there is in-gap rectification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' In other words, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (37) defines φ(l) n as a function of all the harmonics of the time dependent energy from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (32) as follows: φ(l) n = φ(l) n (ϵ(±1) n , ϵ(±2) n , · · · ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (38) Also from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (36) it can be shown that these amplitudes satisfy the following normalization condition: +∞ � l=−∞ ��φ(l) n ��2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (39) With Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (28), (30), (36), and (13), and by taking the late-time limit that allows to neglect transient terms of the form exp[−Γ(t − t0)] → 0, we obtain the system steady state occupation: pn(t) = � +∞ −∞ dωb π f0(ωb) × Γ ����� +∞ � l=−∞ φ(l) n exp[−ilω(t − t0)] ¯ϵn − ωb − lω − iΓ ����� 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (40) Similar to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (22), by integrating over ωb in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (40), we find that: pn(t) = +∞ � l,m=−∞ � φ(m) n �∗φ(l) n exp � i(m − l)ω(t − t0) � × Γ 2Γ + i(m − l)ω � f+(¯ϵn − lω) + f−(¯ϵn − mω) � , (41) where f±(ϵ) is given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' The Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (41) is one of the central formulas of our work because it allows to com- pute expectation values of any equal-time system observ- ables, even at a finite coupling Γ to featureless fermionic bath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' The expression in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (41) captures the steady state occupation of the n-th state in the case of featureless fermionic bath, and thus it replaces what would be the Fermi-Dirac distribution in equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' One important feature of this steady state is that it displays “synchro- nization”, namely, it is strictly periodic in the drive: pn(t + T) = pn(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (42) Remarkably, in the limit of an “ideal bath” (Γ → 0) the above distribution becomes time independent and it is given by: lim Γ→0 pn = +∞ � l=−∞ |φ(l) n |2f0(¯ϵn − lω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (43) Here ¯ϵn is the Floquet energy of n-th state, and φ(l) n are the Harmonics of the periodic part of the wave-functions defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' The reader is encouraged to contrast this occupation function with that in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (23) obtained when the Hamiltonian was time independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' Notice also that because the occupation function becomes time inde- pendent in this limit, there are no time fluctuations of the average fermion occupation of each state n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' Thus the distribution is an infinite sum of sev- eral Fermi-Dirac distributions with chemical potentials shifted by the various harmonics of the driving frequency lω and weighed by amplitudes of the harmonics of the Floquet wavefunctions |φ(l) n |2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' It is therefore clear that the occupation of the state is completely different from how the state is filled in equilibrium [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' 1(a) for an illustration of the non-equilibrium occupation function].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' One recovers an occupation similar to equilibrium when one neglects all the higher harmonics of φ(l) n with l ̸= 0 and forces by hand the amplitude of the l = 0 term to be φ(0) n → 1, but this is not justified in general (not even perturbatively as we will illustrate in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' II B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' We note that the idea that Floquet states are not filled in the same way as equilibrium states has been emphasized in several studies, by using a variety of models for the re- laxation when the system is coupled to a heat bath [36– 39, 41, 63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' In fact the expression for the non-equilibrium time independent steady states we find in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (43) has been reported before, and is in particular the same kind of expression shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (12) of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' Harmonic time dependent driving Computing analytically the integral in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (37) that relates the harmonics of the Floquet wavefunction to the harmonics of the energy is in general involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' There is a simple case where these integrals can be computed in a simple closed analytical form, which is when the time dependent part Vn(t) of the Hamiltonian has a single harmonic: Vn(t) = Vn cos[ω(t − t0)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (44) In this case the coefficients φ(l) n from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (37) correspond to the l-th Bessel function: φ(l) n = Jl � Vn/ω � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (45) Substitution of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (45) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (41) leads to the fol- lowing non-perturbative expression for the occupation of the states in the limit of Γ → 0: lim Γ→0 pn = +∞ � l=−∞ J2 l � Vn/ω � f0(¯ϵn − lω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (46) We therefore see that the occupation in the case of the ideal fermionic bath becomes a sum of several Fermi- Dirac distributions boosted by the different harmonics of the Floquet quasi-energies ¯ϵn − lω (l ∈ Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' It is interest- ing to note that this ladder-like behavior is analogous to 7 the Tien-Gordon effect that arises in nanostructures that are simultaneously subjected to AC and DC drives [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' Similarly as in that case, the ladder behaviour becomes more pronounced as the driving becomes stronger [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' 1(a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' SINGLE BAND MODEL UNDER MONOCHROMATIC LIGHT In this section we will use the formalism developed in the previous ones to determine the self-consistent occupa- tion of an electronic band driven by an oscillating electric field and demonstrate the existence of in-gap rectifica- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' Because we are primarily interested here in proving and clarifying the origin of in-gap rectification, we will focus on a simple model of a Bloch band that has vanish- ing Berry connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' These bands can display however the in-gap Jerk current effect that arises from the energy band dispersions [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' However, other mechanisms driven by the Berry phases, such as the non-linear Hall effect, can also lead to in-gap rectification as we have recently demonstrated [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' Let us now consider our system Hamiltonian to be a tight-binding model with a single site per unit cell and a trivial single Bloch band (with no Berry connections) coupled to a uniform monochromatic electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' The time dependent system Hamiltonian is: HS(t) = � k ϵk(t) |χk⟩ ⟨χk| , (47) ϵk(t) ≡ ϵ(k − A(t)), � k ≡ � BZ dk (2π)d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (48) The system states are now labelled by the wave vector k and ϵ(k) is the unperturbed band dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' We assume a monochromatic electric field which leads to the periodic vector potential using E(t) = −∂tA(t): A(t) = − i ω Eω exp(−iωt) + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (49) A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' Electric current in the steady state Since the system Hamiltonian is diagonal in crystal momenta k, we can apply the formalism of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' I C to compute the steady state occupation of each momenta k, by replacing the label in previous sections n → k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' If we denote the occupation of each state by pk(t), then the system’s electric current reads as follows: j(t) = � k pk(t)∇kϵk(t) = +∞ � s=−∞ j(s) exp[−isω(t − t0)], (50) where we set e = ℏ = 1 throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' By com- bining Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (32) and (41), the weight of each oscillating mode of the electric current can be written as: j(s) = � k +∞ � m,l=−∞ Γ 2Γ + i(l − s)ω � φ(m) k �∗φ(s+m−l) k × � f+(¯ϵk − (s + m − l)ω) + f−(¯ϵk − mω) � ∇kϵ(l) k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (51) Interestingly, as discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' I C, in the limit of an ideal heat bath Γ → 0, the distribution function pk(t) be- comes time independent, and therefore the time averaged electric current (also referred to as rectified current), is given by: ¯j = � T 0 dt T j(t) = � k pk∇k¯ϵk, (52) where ¯ϵk is the Floquet energy of the band and in our current simple single-band model, and is given by the time averaged band energy (l = 0 component): ¯ϵk ≡ ϵ(0) k = � T 0 dt T ϵ(k − A(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (53) Therefore, we see that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (52) has a resemblance to how one would compute the current in a time indepen- dent equilibrium system, but with the equilibrium Fermi- Dirac distribution replaced by occupation function pk, and the bare band dispersion replaced by the dressed Floquet band energy ¯ϵk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' At first glance, this point of view might suggest that the time averaged rectified cur- rent vanishes in the ideal limit of ω ≫ Γ → 0, just in the same way it is expected to vanish in a time inde- pendent equilibrium system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' In fact, several classic and more recent works have incorrectly taken this point of view that the non-equilibrium steady state occupation pk is a Fermi-Dirac distribution of the dressed Floquet band energy [45, 46, 48–50] (see Appendix E for detailed com- ments on previous works).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' However, as we have shown in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' I C, the correct occupation of the states in the non-equilibrium steady state is not a simple Fermi-Dirac distribution, but it is given by the following expression [see Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (38) and (43)]: pk(¯ϵk, ϵ(±1) k , · · · ) ≡ +∞ � l=−∞ |φ(l) k |2f0(¯ϵk − lω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (54) In the argument of pk in the above expression, we have emphasized that pk is not only a function of the Flo- quet band energy ¯ϵk, but also of all the higher har- monics ϵ(±1) k , ϵ(±2) k · · of the time dependent energy ϵk(t) through its dependence on the amplitudes φ(l) k [see Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (37) and (38)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' Precisely because of this, the recti- fication current ¯j can not be expressed as an integral of a total derivative over the Brillouin zone and generally does not vanish, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=', ¯j ̸= � k ∇k ˜P(¯ϵk) = 0, (55) 8 where ˜P(¯ϵk) would be defined through ∂ ˜P(¯ϵk) ∂¯ϵk ≡ ˜pk(¯ϵk), (56) which would be possible if the occupation depended only on the dressed Floquet energy pk → ˜pk(¯ϵk) [but this is not the case for Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (55)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' Therefore, we see that in general a non-zero rectified current is expected in the non-equilibrium steady state, even in the limit of the ω ≫ Γ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' As we will show in detail in the following section, this finite rectified cur- rent remains non-zero within the optical gap of a metal, even within the usual second order of perturbation the- ory in the amplitude of the electric field for which recti- fication currents are typically computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' These findings further substantiate our recent work showing the exis- tence of in-gap rectification [43] but appear in tension with some other statements in the literature [44–46, 48– 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' In Appendix E, we comment in more detail on some of these other works clarifying some partial agreements but also pointing out some of their imprecisions and in- correct statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' Perturbative results In this subsection we will compute perturbatively the electric current in powers of electric field to the currents at modes [see Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (50) and (51)]: s = 0 representing rectification conductivity, s = 1 representing linear con- ductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' s = 2 representing second harmonic generation is discussed in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' We will show explicitly that even to 2nd order in electric fields, the non-equilibrium distribution in the steady state for an ideal bath, pk, dif- fers clearly from the naive Fermi-Dirac distribution eval- uated in the dressed Floquet bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' This will allow us to compute analytically the rectification conductivities and prove rigorously that they remain finite within the optical gap of the metal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' Although our conclusions and formulae are valid and can be used for any single band model (with no Berry connections) in arbitrary dimensions, for simplicity we will illustrate our results for a simple 1D model with the following band dispersion: ϵ(kx) = −t1 cos(a0kx) − t2 sin(2a0kx) + ϵ0, (57) where ϵ0 is a constant that we have added for convenience in order to shift the band energy so that it lies within 0 and ∆ [See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' 3(b)], and a0 is the lattice constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' Notice that the above band-structure breaks not only inversion, which is always needed to have rectification, but also time-reversal symmetry, and therefore it has no symmetry relating k → −k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' As we will see, this is indeed crucial in order to obtain a non-zero in-gap rectification conductivities for the models without Berry curvature that we are considering in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' More generally, as discussed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' [43], in the case of bands with non-trivial Berry connections one can alternatively obtain a non- zero in-gap rectification, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=', via the Berry-Dipole effect by breaking time reversal symmetry only by having a circularly polarized light instead of having a time-reversal breaking band-structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' Occupation function to the second order of electric field We begin by deriving the explicit perturbative expres- sions for ϵ(l) k and φ(l) k discussed in the previous sections and can be computed from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (32) and (37) by replac- ing n → k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' Up to the second order in the electric field, it is sufficient to expand the band dispersion up to the same second order, namely: ϵ(k − A(t)) = ¯ϵk + ϵ(1) k e−iω(t−t0) + ϵ(−1) k eiω(t−t0) + ϵ(2) k e−2iω(t−t0) + ϵ(−2) k e2iω(t−t0) + · · · , (58) Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (49), this perturbative expansion leads to the following expressions for ϵ(l) k : ¯ϵk ≡ ϵ(0) k = ϵ(k) + 1 ω2 � αβ ∂α∂βϵ(k) Eα ωEβ −ω + O(|Eω|4), ϵ(1) k = i ω � α ∂αϵ(k) Eα ω + O(|Eω|3), ϵ(2) k = − 1 2ω2 � αβ ∂α∂βϵ(k) Eα ωEβ ω + O(|Eω|4), ϵ(−l) k = � ϵ(l) k �∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (59) We can use Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (37) to perturbatively evaluate φ(l) k lead- ing to: φ(0) k = 1 − ϵ(1) k − ϵ(−1) k ω + � ϵ(1) k �2 + � ϵ(−1) k �2 − 4ϵ(1) k ϵ(−1) k − ϵ(2) k + ϵ(−2) k 2ω2 , φ(1) k = −ϵ(1) k ω − ϵ(1) k � ϵ(1) k − ϵ(−1) k � ω2 , φ(−1) k = ϵ(−1) k ω − ϵ(−1) k � ϵ(−1) k − ϵ(1) k � ω2 , φ(2) k = � ϵ(1) k �2 − ϵ(2) k 2ω2 , φ(−2) k = � ϵ(−1) k �2 + ϵ(−2) k 2ω2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (60) The other φ(l) k with |l| > 2 will scale with higher powers of electric fields, and therefore can be neglected to second order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' The norm squared of those terms above are: ��φ(0) k ��2 = 1 − 2 ��ϵ(1) k ��2 ω2 + O(|Eω|3), ��φ(1) k ��2 = ��ϵ(1) k ��2 ω2 + O(|Eω|3), ��φ(2) k ��2 = O(|Eω|4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (61) 9 Therefore the ideal occupation function pk in the limit Γ → 0 to second order in electric fields reads as pk = � 1 − 2 ��ϵ(1) k ��2 ω2 � f0(¯ϵk) + ��ϵ(1) k ��2 ω2 f0(¯ϵk − ω) + ��ϵ(−1) k ��2 ω2 f0(¯ϵk + ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (62) The above expansion contains all the correct terms to sec- ond order in electric fields, even though it is not strictly perturbative, because the Floquet band energy ¯ϵk also in- cludes implicitly a correction of order |Eω|2 [see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (59)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' In other words, if one wants to obtain a strictly pertur- bative expansion to order |Eω|2 one simply needs to Tay- lor expand the Fermi-Dirac distribution f0(¯ϵk) above as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' However we find it convenient to keep the above form, with the understanding that we can only trust its predictions to order |Eω|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' Let us now comment on the significance of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (62).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' We see above that even to second order, the non- equilibrium distribution, pk, contains not only the Fermi- Dirac distribution evaluated for the Floquet bands, f0(¯ϵk), but also several other terms that make it clearly deviate from f0(¯ϵk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' As we will see these additional terms, are precisely the ones that lead to a finite in- gap rectification in the clean limit Γ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' In Ap- pendix D, we also demonstrate that the above occupa- tion function agrees with the one obtained from a sim- pler Boltzmann/relaxation-time description in the limit ω ≪ ¯ϵk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' Notice also that the above occupation differs even to up second order |Eω|2 from the naive Fermi-Dirac occupation of the Floquet band, f0(¯ϵk), that was pres- sumed in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' [45, 46, 48–50] (see Appendix Appendix E for further comments on previous studies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' Linear conductivity The linear conductivity is defined from: j(1) α = σαβ Γ (ω)Eβ ω + O(|Eω|3), (63) where the sub-index Γ emphasizes a finite coupling of the system to the bath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' Using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (51), (37), (32), the exact conductivity of our model at finite coupling to the bath is found to be: σαβ Γ (ω) = i ω � k fΓ(¯ϵk)∂α∂β¯ϵk + � k ∂α¯ϵk∂β¯ϵk ω2 iΓ 2Γ − iω L1(¯ϵk, ω), (64) where ∂γ ≡ ∂/∂kγ, and L1(¯ϵk, ω) = f+(¯ϵk) + f−(¯ϵk + ω) − f+(¯ϵk − ω) − f−(¯ϵk), (65) where f± are defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' Just as for Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (62), we have kept the dressed Floquet band energy, ¯ϵk, in the integrands of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (64), and therefore this is not a strictly perturbative expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' But if desired, the strictly per- turbative expression can simply be obtained from the one above by replacing the dressed Floquet band en- ergy dispersion by the bare unperturbed band dispersion: ¯ϵk → ϵ(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' This also applies to the subsequent formulas of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' In the clean limit (ω ̸= 0 and Γ → 0), the above ex- pression reduces to the standard Drude form: lim Γ→0 σαβ Γ (ω) = i ω � k f0(¯ϵk)∂α∂β¯ϵk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (66) Therefore, we see that the real part of the linear con- ductivity at finite frequency vanishes when Γ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' 3(c) we illustrate this in detail for the simple model 1D from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (57).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' The above Drude form follows from the fact that to the linear order of the electric field, the ideal occupation function pk in the limit Γ → 0 is the same with the equilibrium Fermi-Dirac distribution [see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (62)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' In the DC limit ω → 0 the linear conductivity ap- proaches a finite Drude-like value (see Appendix A for details): lim ω→0 σαβ Γ (ω) = 1 2 � k ∂α∂β¯ϵk �fΓ(¯ϵk) Γ − ∂gΓ(¯ϵk) ∂¯ϵk � ≈ 1 2 � k ∂α∂β¯ϵk �f0(¯ϵk) Γ + O(Γ) � , (67) in which gΓ(ϵ) = 1 2i � f+(ϵ) − f−(ϵ) � (68) is the imaginary part of f+(ϵ) defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' There- fore the clean limit of the DC conductivity resembles the prediction of the classic Drude theory for τ ≡ 1/(2Γ), and has a Drude peak in the DC limit when the chemical po- tential of the bath is within the bandwidth of the system µ ∈ [0, ∆] [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' 3(c)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' The fact that the conductivity is finite when ω → 0 and has the expected Drude behav- ior, evidences that our simple bath produces the correct behavior for the relaxation of currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' In the limit in which the frequency is small compared to the bandwidth but much larger than Γ, we obtain the usual decay power 1/ω2 associated with the Drude behavior [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' 3(e), left-hand side region]: lim Γ≪ω≪∆ Re � σαβ Γ (ω) � = −2Γ ω2 � k (∂α¯ϵk)(∂β¯ϵk)∂f(¯ϵk) ∂¯ϵk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (69) On the other hand, in the ultra-large frequency regime when the frequency greatly exceeds even the bandwidth, the real part of the linear conductivity has a different scaling from that of Drude theory: lim ω≫∆ Re � σαβ Γ (ω) � = Γ ω3 � k (∂α¯ϵk)(∂β¯ϵk), (70) decaying as 1/ω3 [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' 3(e), right-hand side region].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content='4 0 (c) 0 1 2 (d) 0 10 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content='4 0 (a) 0 (b) 0 1 2 3 0 1 2 1 2 (e) 10 0 (f) 5 0 5 0 1 2 1 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (a) The 1D tight binding model whose inversion and time-reversal symmetries are broken by the next-nearest- neighbour hopping ±it2/2, and its (b) dispersion relation with 0 the band bottom and ∆ the band top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (c) Real part of the dimensionless linear conductivity Re σxx Γ (ω)/σ(1) 0 illustrating how it vanishes at finite frequency as Γ → 0 (which defines the optical transparency region), and (d) dimensionless rectifica- tion conductivity σxxx Γ (ω, −ω)/σ(2) 0 for different Γ illustrating the existence of in-gap rectification in the metal, namely that it approaches a finite non-zero value in the limit of Γ → 0 at finite ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' The characteristic linear and second order conductiv- ities in 1D used here are σ(1) 0 = a0·e2/ℏ and σ(2) 0 = a2 0τ0·e3/ℏ2 with τ0 = ℏ/t1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (e) and (f) Log-log plots of Re σxx Γ (ω)/σ(1) 0 and σxxx Γ (ω, −ω)/σ(2) 0 for different Γ illustrating their power dependencies over ω in different frequency ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' Parame- ters used: a0 = 1, t1/t2 = 2, µ = 5t1/7, β0 = 109/t1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' Rectification conductivity The rectification conductivity is a three-index tensor that relates the time averaged current [namely the aver- age DC current corresponding to s = 0 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (50)] to the bilinears of electric field amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' Without loss of gen- erality, we define it by choosing the following symmetry convention for indices of the electric field bilinears: j(0) γ = σγαβ Γ (ω, −ω)Eα ω(Eβ ω)∗ + σγαβ Γ (−ω, ω)(Eα ω)∗Eβ ω + O(|Eω|4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (71) The exact rectification conductivity of our model at finite coupling to the bath, Γ, is given by: σγαβ Γ (ω, −ω) = � k ∂γ¯ϵk∂α¯ϵk∂β¯ϵk 2ω4 [fΓ(¯ϵk + ω) + fΓ(¯ϵk − ω) − 2fΓ(¯ϵk)] + Γ 2Γ − iω � k ∂α¯ϵk∂γ∂β¯ϵk 2ω3 L1(¯ϵk, ω) + Γ 2Γ + iω � k ∂β¯ϵk∂γ∂α¯ϵk 2ω3 L∗ 1(¯ϵk, ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (72) The DC limit of the rectification conductivity can be shown to be (see Appendix B for details): lim ω→0 σγαβ Γ (ω, −ω) = 1 4 � k ∂α∂β∂γ¯ϵk �fΓ(¯ϵk) Γ2 − 1 Γ ∂gΓ(¯ϵk) ∂¯ϵk − 1 3 ∂2fΓ(¯ϵk) ∂¯ϵ2 k � ≈ 1 4 � k ∂α∂β∂γ¯ϵk �f0(¯ϵk) Γ2 + O(Γ0) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (73) The leading term of the above expression in the sec- ond line coincides with the Jerk conductivity predicted within the relaxation time approximation from a simple Boltzmann-relaxation-time formalism [43, 47, 51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' For an illustration see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' 3(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' We have also verified that the above ω → 0 limit of the rectification conductivity is identical to the ω → 0 limit of the second-harmonic generation conductivity σγαβ Γ (ω, ω) (see Appendix C for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' Let us now focus on the main regime of our interest, which is the “clean-limit” in which the relaxation rate vanishes (Γ → 0) while the frequency remains finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' The exact expression for the rectification conductivity in this limit is given by: lim Γ→0σγαβ Γ (ω, −ω) = 1 2ω4 � k (∂γ¯ϵk)(∂α¯ϵk)(∂β¯ϵk) × � f0(¯ϵk + ω) + f0(¯ϵk − ω) − 2f0(¯ϵk) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (74) Notice that the above rectification conductivity would vanish under any symmetry that enforces ¯ϵk = ¯ϵ−k, such as time reversal or inversion symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' Therefore, the above expression proves one of our central claims, namely that the rectification conductivity remains finite at finite frequency within the optical transparency region of the metal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' The “transparency” here refers to the fact that the real part of the linear conductivity vanishes in this same limit ω ≫ Γ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' We illustrate this behavior in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' 3(d) for our toy 1D model, confirming that the in gap rectification is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' The origin of this finite rec- tification conductivity can be traced back to the fact that 11 (b) 0 1 10 15 5 0 0 (a) 0 2 2 4 (c) 8 4 0 0 1 1 2 2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (a) Schematic of the original band (denoted by solid line l = 0) and the boosted Floquet bands (denoted by dashed lines l = ±1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' Here the chemical potential µ is below the original band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' The threshold frequency ωt is the minimum frequency for boosted Floquet bands to cross the chemical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (b) and (c) dimensionless rectification conductivity σxxx Γ (ω, −ω)/σ(2) 0 and its Log-log plots for different Γ, show- ing that rectification conductivity is non-zero when ω > ωt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' Parameters used are the same with those in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' to the second order of the electric field, the ideal occupa- tion function pk in the limit Γ → 0 is different from the equilibrium Fermi-Dirac distribution [see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (62)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' While the expression of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (74) is the exact clean limit of the rectification conductivity in our model, it can be shown that this expression reduces to the more famil- iar expression for the Jerk current prediction of the sim- ple Boltzmann-relaxation-time expression in the limit in which the frequency is small compared to the bandwidth, namely Γ ≪ ω ≪ ∆, and it is given by: lim ω→0 lim Γ→0 σγαβ Γ (ω, −ω) = 1 ω2 � k f0(¯ϵk)∂α∂β∂γ¯ϵk, (75) which coincides with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (21) of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' [43] for the Jerk mechanism which has a 1/ω2 decaying power [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' 3(f), left-hand side region].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' More details of this agreement with the simpler Boltzmann approach are dis- cussed in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' Interestingly, in the “ultra-high” frequency limit, when the frequency is much larger than the bandwidth ω ≫ ∆, the clean rectification conductivity transits to a different scaling and decays much faster [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' 3(f), right-hand side region]: lim ω≫∆ lim Γ→0 σγαβ Γ (ω, −ω) = 1 2ω4 � k � 1 − 2f0(¯ϵk) � (∂α¯ϵk)(∂β¯ϵk)(∂γ¯ϵk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (76) In contrast to the Boltzmann-relaxation-time result where the large frequency regime is controlled by the third momentum derivative of the band dispersion, here, the large frequency response is controlled by the third power of band velocity, which is a different intrinsic prop- erty of the band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' It is interesting to note that the expression in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (76) remains finite even when the unperturbed band is either fully occupied [f0(¯ϵk) = 1] or fully empty [f0(¯ϵk) = 0], namely the system would be nominally an insulator with- out a Fermi surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' This behavior is possible because our bath does not conserve the total particle number of the system, and therefore, there appears a finite occupation of the bands when they are driven by the electric field, even if the bands were initially empty in the distant past before turning on the time dependent drive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' In other words, all our calculations are performed strictly for a bath with fixed chemical potential but not fixed density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' The appearance of a finite occupation of the bands to second order of perturbation theory occurs when the fre- quency exceeds the threshold so that one of the copies of the Floquet bands boosted by ±ω crosses the chemical potential, as depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' SUMMARY AND DISCUSSION We have shown rigorously that the occupation of states in a periodically driven fermionic system coupled to a fea- tureless fermionic heat bath approaches a time indepen- dent occupation function in the limit in which the cou- pling to this bath is vanishingly small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' This occupation function can be computed analytically and differs from the naive Fermi-Dirac occupation of the dressed Floquet energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' This non-equilibrium steady state occupation instead resembles a staircase version of the Fermi-Dirac distribution [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' 1(a) for an illustration], and also cannot be expressed as a function of the Floquet energy alone, but in general contains information on all the har- monics encoding the full time dependence of the Hamil- tonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' We applied these results to the case in which the fermionic system has a Hamiltonian corresponding to a single Bloch band without Berry connections (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' aris- ing from a tight-binding model with a single site per unit cell) driven by a monochromatic electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' We showed that this staircase Fermi-Dirac distribution leads to a fi- nite rectification conductivity within the optical trans- parency region of a metal, which at small frequencies compared to the bandwidth agrees exactly with the pre- diction of the Jerk current effect expected from a simpler Boltzmann-relaxation-time description [43, 47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' Because 12 the oscillating electric field is monochromatic, this rec- tification conductivity does not arise because of the fre- quency difference effect of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' [44] or the Raman-like scattering effect of Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' [45, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' Our results validate our recent findings [43] that in- gap rectification within the optical transparency region of metals are indeed possible, even in the limit in which carrier relaxation rates vanish, and clarify a discussion surrounding this matter [44–46, 48–50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' More details of the partial agreement with some of these references but also the corrections of imprecisions and incorrect state- ments in some of them can be found in Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' ACKNOWLEDGMENTS We would like to thank Yugo Onishi, Naoto Nagaosa, Liang Fu, Victor Yakovenko, Sergey Ganichev, Bing- hai Yan, and Fernando de Juan for stimulating discus- sions and correspondence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' We are particularly thankful to Mikhail Glazov and Leonid Golub for patiently and openly discussing with us their views on Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' [48] and also some of the subtle aspects of the physics of in-gap rectification, from which 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' Lindner, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' Rudner, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' Refael, Steady states of interacting flo- quet insulators, Physical Review B 99, 014307 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' [64] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' Nazarov and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' Blanter, Quantum transport: introduction to nanoscience (Cambridge university press, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' 14 Appendix A: Linear conductivity in the DC limit In this appendix we show additional details of the linear conductivity in the DC limit discussed in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' In the DC limit ω → 0 the linear conductivity [see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (64) in the main text] becomes: lim ω→0 σαβ Γ (ω) = 1 2 � k ∂α∂β¯ϵk �fΓ(¯ϵk) Γ − ∂gΓ(¯ϵk) ∂¯ϵk � = 1 2 � k ∂α∂β¯ϵk �f0(¯ϵk) Γ + Γ 2 ∂3f0(¯ϵk) ∂¯ϵ3 k + Γ2 3 ∂3g0(¯ϵk) ∂¯ϵ3 k + O(Γ3) � , (A-1) in which gΓ(ϵ) = 1 2i � f+(ϵ) − f−(ϵ) � , g0(ϵ) ≡ lim Γ→0 gΓ(ϵ), (A-2) where gΓ(ϵ) is the imaginary part of f+(ϵ) defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (25) in the main text, and we used the Cauchy–Riemann equations satisfied by fΓ(ϵ) and gΓ(ϵ) ∂fΓ(ϵ) ∂Γ = ∂gΓ(ϵ) ∂ϵ , ∂fΓ(ϵ) ∂ϵ = −∂gΓ(ϵ) ∂Γ , (A-3) and the resulting relation fΓ(ϵ) = f0(ϵ) + Γ∂g0(ϵ) ∂ϵ + O(Γ2), (A-4) to obtaining the second equation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (A-1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' Therefore the clean limit of the DC conductivity resembles the prediction of the classic Drude theory for τ ≡ 1/(2Γ): lim Γ→0 lim ω→0 σαβ Γ (ω) = 1 2Γ � k f0(¯ϵk)∂α∂β¯ϵk, (A-5) and linear conductivity has a Drude peak in the DC limit when the chemical potential of the bath is within the bandwidth of the system µ ∈ [0, ∆].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' The system can still have a finite linear DC conductivity even if the band is nominally fully empty or occupied at finite Γ, namely, lim ω→0 σαβ Γ (ω) = 1 2 � k ∂α∂β¯ϵk �Γ2 3 ∂3g0(¯ϵk) ∂¯ϵ3 k + O(Γ3) � ∝ Γ2 + O(Γ3), � T0 → 0, µ /∈ [0, ∆] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (A-6) This conductance vanishes when Γ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' Appendix B: Rectification conductivity in the DC limit In this appendix we show more details of the rectification conductivity in the DC limit discussed in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' In the DC limit, the rectification conductivity [see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (72) in the main text] is: lim ω→0 σγαβ Γ (ω, −ω) = 1 4 � k ∂α∂β∂γ¯ϵk �fΓ(¯ϵk) Γ2 − 1 Γ ∂gΓ(¯ϵk) ∂¯ϵk − 1 3 ∂2fΓ(¯ϵk) ∂¯ϵ2 k � = 1 4 � k ∂α∂β∂γ¯ϵk �f0(¯ϵk) Γ2 + 1 6 ∂2f0(¯ϵk) ∂¯ϵ2 k + Γ2 24 ∂4f0(¯ϵk) ∂¯ϵ4 k + Γ3 45 ∂5g0(¯ϵk) ∂¯ϵ5 k + O(Γ4) � , (B-1) where we again used Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (A-4) in arriving at the second equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' In the clean limit Γ → 0, this coincides with the Jerk conductivity predicted within the relaxation time approximation, but here we also present the sub-leading in Γ correction: lim Γ→0 lim ω→0σγαβ Γ (ω, −ω) = 1 4 � k ∂α∂β∂γ¯ϵk �f0(¯ϵk) Γ2 + 1 6 ∂2f0(¯ϵk) ∂¯ϵ2 k � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (B-2) Therefore, similarly to the linear conductivity, second order rectification conductivity has a Jerk peak at DC limit when the chemical potential is within the bandwidth of the system µ ∈ [0, ∆].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' When the band is nominally fully empty or occupied, for the rectification conductivity we now have lim ω→0σγαβ Γ (ω, −ω) = 1 4 � k ∂α∂β∂γ¯ϵk �Γ3 45 ∂5g0(¯ϵk) ∂¯ϵ5 k + O(Γ4) � ∝ Γ3 + O(Γ4), � T0 → 0, µ /∈ [0, ∆] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (B-3) This finite DC rectification conductivity again vanishes in the clean limit Γ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' 15 Appendix C: Second harmonic generation In this appendix we show the second harmonic conductivity mentioned in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' The second harmonic conductivity is the one that controls the response oscillating at the double frequency of the drive (s = 2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' we define it as: j(2) γ = σγαβ Γ (ω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' ω)Eα ωEβ ω + O(|Eω|3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (C-1) and it is given by the following expression: σγαβ Γ (ω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' ω) = − 1 2ω2 � k fΓ∂α∂β∂γ¯ϵk − 1 ω3 Γ 2Γ − iω � k (∂α¯ϵk)(∂β∂γ¯ϵk)L1(¯ϵk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' ω) − 1 2ω4 Γ 2Γ − 2iω � k (∂γ¯ϵk) � (∂α¯ϵk)(∂β¯ϵk)L2(¯ϵk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' ω) + ω 2 ∂α∂β¯ϵkL1(¯ϵk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' 2ω) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (C-2) where L2(¯ϵk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' ω) = f+(¯ϵk − 2ω) − 2f+(¯ϵk − ω) + f+(¯ϵk) + f−(¯ϵk) − 2f−(¯ϵk + ω) + f−(¯ϵk + 2ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (C-3) The low frequency limit of second harmonic conductivity coincides with the low frequency limit of rectification conductivity from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (73) in the main text: lim ω→0 σγαβ Γ (ω, ω) = 1 4 � k ∂α∂β∂γ¯ϵk �fΓ(¯ϵn) Γ2 − 1 Γ ∂gΓ(¯ϵn) ∂¯ϵn − 1 3 ∂2fΓ(¯ϵn) ∂¯ϵ2n � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (C-4) Interestingly, at large frequencies ω ≫ ∆ the real part of the second harmonic conductivity decays as 1/ω2 in contrast to 1/ω4 power decay of the rectification conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' Appendix D: Relation to the Boltzmann theory In this appendix we discuss the relation between our result and that from a simpler Boltzmann/relaxation-time approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' We begin by writing a Boltzmann equation for a single band system in the relaxation time approximation: ∂tf(k, t) + E(t) · ∇kf(k, t) = −[f(k, t) − f0(ϵk)]/τ, (D-1) where E(t) = Eωe−iωt + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' is a monochromatic electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' The above equations are written in a different gauge with respect to the main text: here k is viewed as a gauge invariant mechanical crystal momentum, which corresponds to k−A(t) in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' In order to obtain expressions for occupation functions in the same gauge as in the main text, we convert to a gauge in which we keep track of the occupation of canonical crystal momenta, using the following relation: p(k, t) ≡ f(k − A(t), t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (D-2) The occupation function p(k, t) satisfies the following equation ∂tp(k, t) = ∂tf(k − A(t), t) − ∂tA(t) · ∇kf(k − A(t), t) = ∂tf(k − A(t), t) + E(t) · ∇kf(k − A(t), t) = −[f(k − A(t), t) − f0(ϵk−A(t))]/τ, (D-3) where we used Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (D-1) in obtaining the last equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' Therefore we see that the distribution function p(k, t) satisfies an equation without explicit electric field derivative term: ∂tp(k, t) = −[p(k, t) − f0(ϵk−A(t))]/τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (D-4) Using the fact that the late-time steady state distribution is periodic, we perform Fourier series expansions for both p(k, t) and f0(ϵk−A(t)): p(k, t) = +∞ � l=−∞ p(l)(k) exp(−ilωt), p(l)(k) = � T 0 dt T p(k, t) exp(ilωt);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' f0(ϵk−A(t)) = +∞ � l=−∞ f (l) 0 (k) exp(−ilωt), f (l) 0 (k) = � T 0 dt T f0(ϵk−A(t)) exp(ilωt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (D-5) 16 With the above expansions, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (D-4) becomes −ilωp(l)(k) = −p(l)(k)/τ + f (l) 0 (k)/τ, (D-6) and leads to p(l)(k) = 1 1 − ilωτ f (l) 0 (k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (D-7) The above solution in general requires an explicit calculation of the following mixed harmonics of the distribution: f (l) 0 (k) = � T 0 dt T f0(ϵ(0) k + ϵ(1) k e−iωt + ϵ(−1) k eiωt + · · · ) exp(ilωt), (D-8) where ϵk−A(t) = +∞ � l=−∞ ϵ(l) k exp(−ilωt), ϵ(l) k = � T 0 dt T ϵk−A(t) exp(ilωt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (D-9) Let us consider however the clean limit τ → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' Notice that f (l) 0 (k) is independent of τ, therefore for l ̸= 0 components we have lim τ→+∞ p(l̸=0)(k) = lim τ→+∞ 1 1 − ilωτ f (l) 0 (k) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (D-10) However the l = 0 component, or time averaged component, which is independent of τ and therefore remains finite as τ → +∞, is given by: p(0)(k) = f (0) 0 (k) = � T 0 dt T f0(ϵ(0) k + ϵ(1) k e−iωt + ϵ(−1) k eiωt + · · · ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (D-11) Therefore, similarly to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (54) obtained from the full formalism with the bath, the distribution from the Boltzmann theory becomes time independent in the canonical crystal momentum, but not in the mechanical physical momentum, in the analogous ideal limit of τ → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' Notice, however, that the above result has to be viewed as a limit of τ → +∞, and not as a situation in which there is no relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' This is because in the strict absence of relaxation mechanisms there is no unique late-time steady state, namely by taking 1/τ = 0 and neglecting altogether the relaxations in the right hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (D-4) any time-independent distribution of the canonical momenta would be a solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' If we expand up to the second order of electric fields Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (D-11) we obtain: p(0)(k) = � T 0 dt T � f0(¯ϵk) + � ϵ(1) k e−iωt + ϵ(−1) k eiωt + ϵ(2) k e−i2ωt + ϵ(−2) k ei2ωt� f ′ 0(¯ϵk) + 1 2 � ϵ(1) k e−iωt + ϵ(−1) k eiωt�2f ′′ 0 (¯ϵk) + O(|Eω|3) � = f0(¯ϵk) + |ϵ(1) k |2f ′′ 0 (¯ϵk) + O(|Eω|3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (D-12) Interestingly the above distribution function coincides with the asymptotic behavior of the staircase distribution function discussed in the main text [see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=', Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (62)] in limit of ∆ ≫ ω ≫ Γ → 0: lim ω→0 lim Γ→0 pk = lim ω→0 �� 1 − 2 ��ϵ(1) k ��2 ω2 � f0(¯ϵk) + ��ϵ(1) k ��2 ω2 f0(¯ϵk − ω) + ��ϵ(−1) k ��2 ω2 f0(¯ϵk + ω) � = f0(¯ϵk) + |ϵ(1) k |2f ′′ 0 (¯ϵk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (D-13) Therefore the expectation value of all equal time observables, such as the electric current, coincide with those of the more microscopic Floquet-bath theory of the main text, at least to second order in electric fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' In particular one obtains the same rectification conductivity in the above limit as that in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (75) of the main text, that we refer to as Jerk effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' 17 Appendix E: Comments and connections to other works in the literature There has been a long-standing debate in the literature about the possibility of in-gap rectification which has been clouded by previous imprecise and incorrect statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' In this section we will try to clarify some of this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' We begin by defining precisely what do we mean by in-gap rectification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' The optical gap is defined as the region in the frequency domain in which the the hermitian symmetric part of the conductivity tensor vanishes in the limit of low temperatures and small scattering rates (see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' [43] for a review).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' We then say that a system has in-gap rectification if any of the elements of the rectification conductivity tensor that lead to finite DC currents generated by a monochromatic AC electric field with a frequency within the optical gap remain non-zero in that same limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' More specifically: Definition of “optical gap” : lim T0→0 lim Γ→0 � σαβ(ω) + [σβα(ω)]∗� → 0, when ω ∈ optical gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (E-1) Definition of “in-gap rectification” : lim T0→0 lim Γ→0 σγαβ(ω, −ω) ̸= 0, when ω ∈ optical gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (E-2) Therefore our current manuscript and our previous work in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' [43], demonstrate rigorously that in-gap rectification in the above sense is indeed possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' Nevertheless, some confusion in the literature appears to have originated from different interpretations of the work of Belinicher, Ivchenko, and Pikus (BIP) in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' That paper contained statements such as “The conclusion that a steady-state photocurrent may appear on illumination in the transparency range of a crystal, reached in earlier publications, is shown to be in error”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' This statement could be read as implying the impossibility of in-gap rectification in the sense we defined above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' In fact, this reading of the BIP paper appears to have been made in several references claiming that in-gap rectification in the above sense is impossible [44, 45, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' Even us in our recent work of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' [43], read the BIP paper as trying to prove that in-gap rectification is impossible in the above sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' However, part of the issue with reading the aforementioned BIP paper, is that it left several crucial gaps in its discussion and its derivations that can make it hard to know in a precise way what exactly BIP implied at various places and the precise framework that BIP used for reaching such conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' For example, a crucial point that can lead to a different readings of the BIP paper relates to the definition of the term “gn” that appears in the right hand side of their Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (8) in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' [48], which is a central equation from which various conclusions are derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' Unfortunately BIP never spelled out an explicit form for this term, but simply wrote that “gn is the generation function, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=', the rate of change of the distribution function due to optical transitions.”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' This leaves open to interpretation what exactly they had in mind for “optical transitions”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' For example, one could read this by interpreting “gn” as associated only with inter-band optical transitions, and in this case, one would be lead to read the BIP paper as trying to imply that in-gap rectification in the above sense is impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' There is however an alternative way to interpret “gn” and the notion of “optical transitions” in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' [48] as a more general notion of irreversible “transitions” that can take place even within what would nominally be the optical gap defined in the above sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' This more nuanced way of interpreting the BIP paper has indeed been recently emphasized by Glazov and Golub in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' For example Golub and Glazov write in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' [46] that “.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' even for transparent media, real electronic transitions should occur to enable the photocurrent.” and that “We reiterate that in the absence of any real electronic transitions DC current is forbidden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' It is obvious from general reasons: If a DC current is generated then this current results in a Joule heat in the sample or in the external circuit connected to the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' It is forbidden by the energy conservation law in the absence of real transitions.”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' What Golub and Glazov are trying to explain there is in line with our recent thermodynamic analysis in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' [43], where we emphasized that in order to guarantee the positivity of entropy production, specially when the system is connected to an external circuit, it is always important to view the scattering rate Γ as possibly arbitrarily small but not strictly zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' This requirement means that physically it is important to have always a non-zero absorption within the nominal optical gap of the material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' In fact Golub, Ivchenko himself and Spivak, have also emphasized a related aspect of this in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' [55] where they demonstrated that the CPGE effect associated with the Berry dipole term remains finite within the optical in the limit of Γ → 0, but also coexists the other contributions that originate from impurity scattering mechanisms that scale in the same way with frequency and remain finite inside of the gap in the limit of Γ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' One way to state this state of affairs, that has been emphasized by Golub and Glazov to us in private communications, is that while the real transitions associated with scattering lead to a vanishingly small linear dissipative conductivity in the limit of Γ → 0, there are cancellations of the scattering rate that lead to finite rectification conductivity in this limit but the “real electronic transitions” are still taking place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' These “real electronic transitions” are therefore the more general notion of “optical transitions” that can contribute to the term “gn” in the BIP reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' Therefore, within this point of view, one can say that the BIP should not be read as implying that in-gap rectification is impossible in the sense we defined above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' We are in agreement with the physics of this point of view broadly speaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' There is however another crucial aspect of the BIP work in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' [48] with which we still find ourselves in disagreement and that we believe our current paper provides good evidence to be incorrect in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' BIP stated that “.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' in 18 the case of continuous illumination the steady-state distribution function is f0(¯ϵk) irrespective of how weak is the interaction of electrons with phonons.” In this statement f0 is the “equilibrium distribution function” (the Fermi- Dirac occupation function) and ¯ϵk is the Floquet energy of the band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' These statement has been echoed in several subsequent works [45, 46, 49, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' However, our current work demonstrates that in the limit of Γ → 0 the distribution function is sharply different from the naive Fermi-Dirac occupation, but becomes instead the non-trivial Fermi-Dirac staircase discussed in the main text, even to the leading order |Eω|2 in the driving monochromatic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' Crucially the resulting occupation function cannot be expressed as a function of the Floquet energy alone [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' 1, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (54), and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' (62) of the main text].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' Notice that in order to have a unique and well defined steady state at late times, we must necessarily view the relaxation rate Γ as being arbitrarily small but not strictly zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' Therefore, the notion of the ideal occupation in the steady state has to be necessarily interpreted as the limit of Γ → 0 of the occupation of systems with a finite Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' This is because systems with strictly zero relaxation rate (Γ = 0) do not have a way to erase the memory of their initial conditions and therefore their steady state in the presence of the monochromatic light is not uniquely defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' We have demonstrated rigorously that at least for an ideal fermionic bath the occupation of states in the limit of Γ → 0 is not f0(¯ϵk) as Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' [45, 46, 48–50] presumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' We would like to emphasize that while the fermionic bath might appear to be a somewhat artificial approximation to the true mechanisms of relaxation for certain realistic physical situations, it behaves as an ideal thermal bath in the limit Γ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' In particular, the particle number becomes effectively conserved in such limit since the self-consistent occupations at each momentum become a time independent function as we have shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' We have in particular demonstrated that in equilibrium this bath leads to the expected Fermi-Dirac occupation of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' More generally speaking, in equilibrium one expects a universality of all intensive thermodynamic physical properties of the system of interest for a large class of baths regardless of their details, which essentially defines the class of “ideal thermodynamic baths”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' However, how this universality carries over to non-equilibrium settings is still unclear to us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' Therefore whether other baths or other relaxation mechanisms such as coupling to phonons, impurities or self-thermalization via electron-electron interactions lead to a similar stair-case occupation to the one we have found in the limit of vanishing relaxation rates Γ → 0, remains an interesting open problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' We note however that none of the aforementioned Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' [45, 46, 48–50] has provided a rigorous and controlled derivation of the self-consistent occupation of Floquet bands based on any microscopically explicit mechanism of relaxation, like the one we have provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' Therefore we do not see any rigorous substance to their claim that the occupation is f0(¯ϵk) even for other microscopic relaxation mechanisms such as phonons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} +page_content=' Moreover, it has become abundantly clear in the study of thermalization of Floquet systems in recent years that the self-consistent occupation of Floquet bands coupled to baths that are also bosonic differs clearly from the naive Fermi-Dirac distribution of the Floquet bands f0(¯ϵk) [36–41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf'} diff --git a/IdFAT4oBgHgl3EQfuh5k/content/2301.08670v1.pdf b/IdFAT4oBgHgl3EQfuh5k/content/2301.08670v1.pdf new file mode 100644 index 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instances. Cost-sensitive learning adapts classification algorithms to account for differences in misclassifica- +tion costs. Stacking is an ensemble method that uses predictions from several classifiers as the training data for another +classifier, which in turn makes the final classification decision. +While a large body of empirical work exists where stacking is applied in various domains, very few of these works +take the misclassification costs into account. In fact, there is no consensus in the literature as to what cost-sensitive +stacking is. In this paper we perform extensive experiments with the aim of establishing what the appropriate setup +for a cost-sensitive stacking ensemble is. Our experiments, conducted on twelve datasets from a number of application +domains, using real, instance-dependent misclassification costs, show that for best performance, both levels of stacking +require cost-sensitive classification decision. +Keywords +Cost-sensitive learning, classification, ensemble learning, stacked generalization, stacking, blending +1 +Introduction +Cost-sensitive learning is relevant in many real-world classification problems, where different misclassification errors incur +different costs. A prominent example is the field of medicine, where misdiagnosing an ill patient for a healthy one (a false +negative) entails delayed treatment and potentially life-threatening consequences, while an error in the opposite direction (a +false positive) would incur unnecessary medical examination costs and stress for the patient. Cost-sensitive classifiers can +account for the differences in costs not only between different classes, but also between data instances, making instance- +dependent cost-sensitive classification decisions. +Many cost-sensitive classifiers employ ensemble methods, which combine predictions from several classifiers to obtain +better generalisation performance. Superiority of ensembles over individual classifiers is very well known and has been +extensively studied ([8, 37]). Most cost-sensitive classification ensembles are homogeneous in nature, meaning their +components are instantiated using the same learning algorithm. +Stacked generalization or stacking [31] is a well known and widely applied heterogeneous ensemble, where the pre- +dictions of classifiers produced by different learning algorithms (the base-learners) are used as training inputs to another +learning algorithm (the meta-learner) to produce a meta classifier, which makes the final classification decision. In the +literature, the base- and meta- levels of stacking are also referred to level-0 and level-1. +Homogeneous cost-sensitive ensembles such as cost-sensitive boosting and bagging are widely studied and have been +shown very successful [25]. Examples of cost-sensitive stacking, on the other hand, are scarce and unsystematic, represent- +ing for the most part applications to single domains, where the classifiers are trained on synthetic, class-dependent costs +and are evaluated with cost-insensitive performance metrics. For a discussion on the importance of real costs for a proper +evaluation see the work by [25]. In fact, there is currently no consensus as to how a cost-sensitive stacking ensemble is to +be composed and at what stage (level-0 or level-1) cost-sensitive decision-making should be used. This can be clearly seen +in Table 1, which gives an overview of existing cost-sensitive stacking literature. Stacking is typically made cost-sensitive +simply through the application of a cost-sensitive classifier either at level-0 (CS-CiS), level-1 (CiS-CS) or at both levels of +*Email: natalie.lawrance@vub.be +1 +arXiv:2301.01748v1 [cs.LG] 4 Jan 2023 + +the the ensemble (CS-CS), resulting in a total of three possible stacking setups. To the best of our knowledge, no compari- +son of all three setups on multiple domains with appropriate evaluation exists in the literature. Previous related work used +arbitrary artificial costs in model training and evaluated cost-sensitive models using performance metrics that are either +cost-invariant or that focus on the performance of only the positive class. +In this work we aim to fill this gap by providing a thorough comparison of various cost-sensitive stacking ensembles on +multiple domains using real, instance-dependent costs and performance metrics appropriate for cost-sensitive problems. +1.1 +Our contributions +• The main contribution of this work is a rigorous empirical comparison of different setups of cost-sensitive stacking +ensembles over multiple domains. We evaluate using appropriate performance metrics and attempt to establish best +practice. +• Secondly, we introduce a novel cost-sensitive classifier combination method, inspired by MEC-voting and stacking, +which we call MEC-weighted-stacking. +• Finally, we present a list of publicly available datasets with clearly defined instance-dependent misclassification +costs. The costs are based either on the literature, or are defined by us based on both the literature and expert +knowledge of the data providers. We also define instance-dependent costs for a well known ‘credit-g’ dataset from +the UCI Machine learning repository, for which only class-dependent costs were available to date. +1.2 +Outline +The remainder of the paper is structured as follows. Section 2 presents an overview of the relevant literature. MEC- +weighted stacking is introduced in Section 3. Our hypotheses to be tested, the experimental setup and the datasets used in +the study are discussed in Section 4. Section 5 details the results of our extensive experiments, while the main outcomes +and limitations are discussed in Section 6. Section 7 concludes the paper. +2 +Related work +While stacking has been widely used in machine learning applications (the interested reader is invited to peruse the survey +on stacking literature by [27]), few works are dedicated to the study of cost-sensitive stacking. +We identified in the literature three different cost-sensitive stacking setups: CiS-CS, CS-CiS or CS-CS, where the +ensemble was made cost-sensitive simply through the application of a cost-sensitive classifier either at level-0, level-1 +or at both levels of the ensemble. In most cases, the method used to make the classification cost-sensitive is the direct +cost-sensitive decision as introduced by [35], also called DMECC [25]. +One of the first papers to discuss stacking in a cost-sensitive context was [6]. The authors propose cost-insensitive +level-0 and cost-sensitive level-1 stacking setup (CiS-CS setup), which was compared to a number of different classifier +combinations schemes on 16 classification problems. The misclassification costs they used were artificially generated by +randomly and uniformally sampling costs from on the interval [1,10]. Several other studies followed adopting the same +CiS-CS stacking setup, however none of the studies explicitly reasoned or justified this choice. +Several more papers demonstrated similar examples of multiple-domain studies of CiS-CS stacking with arbitrary costs +( [7, 33, 34]). These mainly differ in the type and the number of algorithms that are employed in the ensemble. We note +that all of them used cost-insensitive metrics for classifier evaluation. +[19] considers a stacking setup, where level-0 classifiers were cost-sensitive while level-1 was cost-insensitive (CS- +CiS setup). The misclassification costs were assumed to be equal to the inverse of the class priors. This approach is very +commonly adopted in the absence of information about real misclassification costs. It is, however, not appropriate, see [25] +for a discussion. The resulting stacking classifier was compared to known ensemble methods using classification accuracy, +a metric that by design assumes equal misclassification costs. +Most examples of stacking use different learning algorithms in level-0, however in his original work Wolpert suggested +that this must not be the case and the technique can also be applied when a single algorithm is considered. [5] propose a +cost-sensitive variant of bag-stacking, a method originally proposed by [29], using bagged cost-sensitive decision trees in +level-0 and using cost-sensitive logistic regression in level-1, thus implicitly proposing a CS-CS stacking setup. To the best +2 + +Table 1: Summary of cost-sensitive stacking literature +Publication +Stacking +Level-0 +Level-1 +Real +Costs +CS +setup +algorithm +algorithm +costs +type +evaluation +[6] +CiS-CS +DT, KNN, NB +LR +c +✓ +[19] +CS-CiS +DT, KNN, NB +MT +c +[5] +CS-CS +DT +LR +✓ +i +✓ +[34] +CiS-CS +DT, KNN, NB +LR +c +[7] +CiS-CS +ExT, GBDT, LDA, LR, RF +LR +c +[33] +CiS-CS +DT, KNN, RF, SVM +DT, KNN, NB, SVM +c +[13] +CiS-CS, CS-CiS, CS-CS +DT, NB, KNN, SVM +LR, ExT +c +✓ +this paper +CiS-CS, CS-CiS, CS-CS +DT, KNN, LR, SVM +Adab, DT, KNN, LR, RF, SVM +✓ +i +✓ +Costs type: +c: class-dependent, i: instance-dependent +Algorithms: +Adab: Adaboost, DT:decision tree, ExT: extremely randomised trees, GBDT: gradient boosted trees, KNN: k-nearest neighbour, +LDA: linear discriminant, LR: logistic regression, MT: Meta Decision Trees, NB: naive bayes, RF: random forest, SVM: support vector machines +of our knowledge, this study is the only example where real instance-dependent costs were used in model training. Models +were evaluated using a cost-sensitive metric called the savings score, proposed in [2]. +The only study to date that considers all three different cost-sensitive stacking setups is one by [13] on the application +domain of software defect prediction. The misclassification costs were selected based on a literature however the authors +emphasised that they treated costs as one of the hyperparameters of the classifier, which, we must note, is incorrect, as was +previously discussed in [15]. The experiments are run on 15 datasets using the same class-dependent cost matrix on all. +Balanced error-based metrics were used for evaluation together with cost-based evaluation metrics. +Identifying real misclassification costs is a complex task, which for many applications may prove too difficult to de- +fine and compute. Most studies resort to artificially generated misclassification costs (see [25] for a discussion on why +this is inappropriate) and error-based evaluation metrics are typically employed to assess generalisation performance of +cost-sensitive stacking. Examples of metrics used include the AUC, the arithmetic or geometric mean of class-specific +accuracies, the F-measure, and the Matthew’s correlation coefficient (MCC). All of these metrics assume equal misclassi- +fication costs, and the F-measure does not incorporate the performance on the negative class, so using these metrics is not +compatible with cost-sensitive learning [17]. +One of the challenges of stacking is the choice of the learning algorithms for the ensemble. Earlier studies proposed to +use linear regression to combine level-0 inputs [30], however Wolpert does not impose any particular restrictions on which +algorithm to use in level-1, and he believed that his famous ‘No Free Lunch Theorem’ [32] applies to the meta-learner as +well. For the overview of which learning algorithms were used in cost-sensitive stacking ensembles to date we refer our +reader to the summary Table 1. +3 +MEC-weighted stacked generalization +In the typical supervised classification framework, a learning algorithm A is presented with a set S of data instances +(xi,yi), each describing some object i. We call xi a feature vector, and yi the class label of that object, drawn from a finite, +discrete set of classes {1,...,K}. In this paper we will consider the binary classification problem, where yi ∈ {0,1}. The +learning algorithm A, given S as input, after a process called training, produces a classifier C, whose task is to predict the +correict class label ˆyC(xj) ∈ {0,1} for a previously unseen feature vector xj. +Training any number L of learning algorithms on the same set of data instances S , we obtain a set of classifiers +C = {C1 ...CL}, and for each feature vector xi the corresponding set of predictions +ˆ +Y (xi) = {ˆyC1(xi),..., ˆyCL(xi)}. C is +called an ensemble of classifiers if the predictions from +ˆ +Y (xi) are combined, in some way, into a single prediction of the +class label for the data instance xi. +Stacking differs from other classifier ensembles in that the predictions from the set +ˆ +Y (xi) are combined with the +original class label yi to form the set Smeta = {(ˆyC1(xi),..., ˆyCL(xi)),yi} of meta level data instances subsequently used in +another round of algorithm training to produce a new classifier, which is used to obtain the final predictions. +The novel method we propose in this paper is inspired by the cost-sensitive weights for model votes paradigm described +in [25], and consequently called MEC-weighted stacking. To each classifier C, we can assign a weight wC based on that +classifier’s cost-performance on the validation set: wC = f(ε), where ε is the sum of the misclassification costs of all data +3 + +Table 2: Characteristics of the datasets used in our experiments +Application +Dataset alias +# instances +# Attr +% positives +Instance-dependent +domain +costs source +1 +Bankruptcy +bankruptcy (private) +404999 +221 +3.31 +this publication +2 +Churn +churn kgl (Kaggle*) +7043 +21 +26.54 +[25] +3 +Churn +churn AB [3] +9410 +45 +4.83 +[3] +4 +Credit risk +credit kgl (Kaggle*) +112915 +15 +11.70 +[2] +5 +Credit risk +credit de uci [12] +1000 +20 +30.00 +this publication +6 +Credit risk +credit kdd09 [28] +38938 +39 +19.89 +[2] +7 +Credit risk +credit ro vub [24] +18918 +24 +16.95 +[24] +8 +Direct marketing +dm pt uci [12, 22] +45211 +17 +11.27 +[4] +9 +Direct marketing +dm kdd98 [12] +95412 (train) +479 +5.08 +[25] +96367 (test) +5.06 +10 +Fraud detection +fraud ulb kgl [21] +284807 +31 +0.17 +[25] +11 +Fraud detection +fraud ieee kgl (Kaggle*) +590540 +432 +3.50 +[25] +12 +HR analytics +absenteeism be (private) +36853 (train) +71 +14.50 +[20] +35884 (test) +10.76 +* Kaggle: https://www.kaggle.com/ +instances incorrectly classified by C on a validation set and f(ε) is a transformation function, which for example can take +one of the following forms: f(ε) = ln((1−ε)/ε), f(ε) = 1−ε, f(ε) = exp((1−ε)/ε), and f(ε) = ((1−ε)/ε)2. +The general stacking procedure is thus modified with the additional step of collecting the MEC-weights for each of +the predictions from the set ˆ +Y (xi), yielding the weighted set of predictions ˆ +YMEC(xi) = {(wC1 ˆyC1(xi),...,wCL ˆyCL(xi)),yi}, +which is used in meta classifier training instead of ˆ +Y (xi). +4 +Experimental setup +4.1 +Data +In this study we use a collection of 10 publicly available datasets and 2 private datasets, for which misclassification costs +have either already been defined or will be defined here. This collection of datasets represents a number of application +domains: credit scoring, customer churn prediction, direct marketing, credit card fraud detection, and HR analytics. +4.2 +Misclassification costs +Table 2 presents the references both to the datasets and to relevant publications where the instance-dependent misclassifi- +cation costs for a given domain were introduced. Most of the datasets are large, the number of instances ranging between +1000 and almost 600000. The number of input features ranges from 15 to 479. All of these datasets demonstrate a large +degree of class imbalance, where the percentage of positives reaches at most 30%, and in dataset fraud ulb kgl less than +1%. +In this work we propose instance-dependent costs for these two datasets, for which no costs were previously defined. +The German credit dataset is well known and is referred to as credit de uci in Table 2. Only class-dependent costs +were available for this data set, where the prediction task is to identify customers that will default on their loan. We define +instance-dependent costs using the conceptual framework proposed by [2]. For any data instance i, the cost of a false +negative Ci +FN is defined as loss given default and constitutes 75% of the credit line, while the cost of a false positive Ci +FP is +the loss of the potential profit from rejecting a good customer, plus the sum of the average expected loss and the average +expected profit estimated on the training sample. We define profits as simply the interests earned on the credit line in the +current year. The profits are calculated using historic interest rates for the year 2000 in Germany, which we apply randomly +and uniformly to the whole sample. +The bankruptcy dataset was provided by the credit risk department of a European utilities-provider, who was interested +4 + +in predicting the risk of corporate bankruptcy for new customers. With minor modifications, it readily transfers to the same +credit risk model described above. Here the credit line is equivalent 90 days of utilities usage by the customer, which, in +case of default, the provider loses in full, so Ci +FP equals the credit amount. The profit margins were provided to us and +are calculated per customer based on the assumption of a 12-month contract. Thus, the Ci +FP then equals the annual profit +margins for the potentially good customer plus the expected average loss and expected average profit calculated on the +given sample. +4.2.1 +Data preprocessing +We take care to employ the same preprocessing steps for each of the datasets in the sample, as recommended by the works +that first published them. +In addition to that, we apply the following preprocessing steps to all datasets. All numeric variables are rescaled +using the quantile statistics, which are robust to outliers. Missing values of numeric variables are imputed with sample +median, and of categorical variables are encoded as a separate category. All categorical variables are transformed using +weight-of-evidence coding [1]. +4.2.2 +Data partitioning +The classifier performance estimates are obtained by means of repeated stratified k-fold cross-validation. The 5×2 cross- +validation suggested by [10] is used to train and evaluate stacking ensembles. This resampling is repeated 5 times using +different random seeds, and the results are averaged across folds and across iterations. Large datasets with more than +100000 observations, to keep training times manageable, were split into five disjoint subsets, uniformly at random. +We note that two datasets in our sample are provided with a separate test set, used to evaluate model performance. In +this case, for fairness of comparison, we perform the split into folds on each of the training and test datasets using the +same seed, we then proceed using the training partition of the training set and the test partition of the test set. The training +partition of the test set remains unused in evaluation. When training and test data sets contain the same observations at +different time periods (e.g. in bankruptcy prediction) we ensure that training and test datasets are disjoint and do not contain +overlapping data instances. +4.3 +Learning algorithms +The choice of the algorithms for the base- and meta-level of stacking remains one of the challenges of stacked generaliza- +tion. To the best of our knowledge, no study exists that demonstrates the necessity to use a specific algorithm combination +in either base- or meta-level of stacking. The main requirement for the base classifiers of any ensemble is that they are +sufficiently accurate (meaning they predict better than a random guess) and sufficiently diverse (meaning their errors are +uncorrelated) [11]. In a heterogeneous ensemble, where the decisions of different learning algorithms are combined, the +number of base-learners need not be large [26]. All algorithms below have previously been described and discussed in +detail in a number of machine learning textbooks, for example [16], so we refrain from repeating these descriptions here. +4.3.1 +Base-learners +The base learners in our experiments are four well known classification algorithms, which are: CART Decision Tree +(DT), K-Nearest Neighbors (KNN), Support Vector Machines (SVM) and Logistic Regression (LR). Unlike [7] and [33] +before us, we choose not to use ensembles such as Random Forest or Extremely Randomised Trees in the base level of +stacking. The reasons for this are two-fold. Firstly, ensembles in general, and stacking in particular are typically built +on weak base-learners, which these very powerful models, which are themselves ensembles, certainly are not. Secondly +these methods are based on decision trees and their errors will be correlated with DT. In our choice we also considered +the recommendations of [8], one of the largest empirical studies known to date comparing algorithm performance on +121 datasets. Their results on binary problems (55 UCI datasets) demonstrate that Random Forest, SVM, Bagging and +Decision Trees have the highest probability of obtaining more than 95% of accuracy, while classifiers of the Naive Bayes +(NB) family are not competitive in comparison. We therefore do not include NB in our experiments, unlike some previous +studies in cost-sensitive stacking. +5 + +4.3.2 +Meta-learners +The choice of the meta-learner constitutes a challenge as well, as was called ’the black art’ by the original author of stacked +generalization [31]. To keep the scale of our experiments manageable and to allow for statistical comparison between +stacking and base classifiers, we use the same four algorithms that were used in the level-0 of stacking. In addition to that +we also use two homogeneous ensemble methods that, according to [8], perform well on most problems, namely Adaboost +(Ada) and Random Forest (RF). +4.3.3 +Cost-sensitive learners +While many variants of cost-sensitive learning algorithms exist that can incorporate the misclassification costs during +classifier training [25], in this study we are not interested in comparing cost-sensitive learning algorithms, but in ways of +combining cost-sensitive and cost-insensitive learners in a single ensemble. For our purposes it is important that the two +classifiers we compare are different in all but one thing, that is the composition of the ensemble. We therefore choose +to turn known cost-insensitive classifiers cost sensitive by applying a cost-sensitive threshold adjustment method called +DMECC [25]. In this method, each data instance is classified according to its individual cost-sensitive decision threshold, +which is based on the ratio of misclassification costs of that particular data instance. The threshold is calculated as follows: +T i +cs = +Ci +FP−Ci +TN +Ci +FP−Ci +TN+Ci +FN−Ci +TP , where Ci +TN and Ci +TP refer to the costs of correct classification, and Ci +FN and Ci +FP refer to the +misclassification costs of the positive and negative data instances respectively. A given record is classified as positive +when its estimated probability of being positive exceeds its individual cost-sensitive threshold T i +cs [35]. +Since some learning algorithms (such as DT or SVM) are known not to produce reliable probability estimates, we +applied isotonic calibration [36] to all base-learners. +4.3.4 +Cost-sensitive stacking +To the best of our knowledge no definition exists of what constitutes cost-sensitive stacking. Based on the insights from +the literature earlier discussed in Section 2, we see three main possibilities of introducing cost-sensitivity into the ensemble +structure. +1. Level-0 classifiers are cost-sensitive, level-1 classifiers are cost-insensitive. +2. Level-0 classifiers are cost-insensitive, level-1 classifiers are cost-sensitive. +3. Both level-0 and level-1 classifiers are cost-sensitive. +We consider 4 functional forms for the MEC-weights as introduced in Section 3, which resulted in a total of 15 stacking +setups to be compared. The complete list of ensemble compositions is presented in Table 3. +MEC-weighted stacking renders the level-1 classifier cost sensitive through manipulation of the training data in a +cost-sensitive way by applying MEC-weights to the training data of the level-1 classifier. We consider it an alternative to +obtaining an ensemble where both training levels are cost-sensitive, which is the third stacking setup stated above. +4.3.5 +Software used +All of our experiments were performed using the Python programming language (version 3.8). Cost-insensitive algorithm +implementations came from the scikit-learn (version 1.1.1) Python library [23], while the cost-sensitive implementations +are our own. +4.4 +Evaluation +4.4.1 +Evaluation metrics +Contrary to previous studies in cost-sensitive stacking, we would like to emphasise the importance of using appropriate +evaluation metrics for cost-sensitive classifiers. Most authors use traditional evaluation metrics such as ROC AUC, Preci- +sion or F1 metric. ROC AUC is known to be cost-invariant, since it is a measure that aggregates classifier performance over +all possible class-dependent thresholds, thus implicitly averaging performance over multiple class-dependent costs, which +6 + +Table 3: The complete list of stacking setups compared in our study. +Stacking setup +Level-0 +Level-1 +Level-1 +alias +algorithm type +input weights f(ε) +algorithm type +1 +type-1 +CS +1 +CiS +2 +type-1 exp +CS +exp((1−ε)/ε) +CiS +3 +type-1 ln +CS +ln((1−ε)/ε) +CiS +4 +type-1 sq +CS +((1−ε)/ε)2 +CiS +5 +type-1 acc +CS +1−ε +CiS +6 +type-2 +CiS +1 +CS +7 +type-2 exp +CiS +exp((1−ε)/ε) +CS +8 +type-2 ln +CiS +ln((1−ε)/ε) +CS +9 +type-2 sq +CiS +((1−ε)/ε)2 +CS +10 +type-2 acc +CiS +1−ε +CS +11 +type-3 +CS +1 +CS +12 +type-3 exp +CS +exp((1−ε)/ε) +CS +13 +type-3 ln +CS +ln((1−ε)/ε) +CS +14 +type-3 sq +CS +((1−ε)/ε)2 +CS +15 +type-3 acc +CS +1−ε +CS +is not appropriate. Other error-based metrics typically assume equal class-dependent costs, which, again, is not appropri- +ate, when instance-dependent costs are known at estimation time. Cost-sensitive learning aims to adapt the classification +decision of a learning algorithm to the differences between misclassification costs assigned to each of the classes. It is +therefore important that the evaluation metrics used to assess the performance of cost-sensitive classifiers is also adapted +to account for the difference in misclassification costs. The typical evaluation metric used in cost-sensitive literature is the +total misclassification cost [14], that simply adds up the errors weighted with their individual misclassification costs, as +defined on the test set. Another option is to normalise the total misclassification cost over some budget constraint, which +will depend on the application domain. A more general way to do this is to use the savings score proposed in [2], where +the total misclassification costs are normalised with the cost of either misclassifying all positives as negatives, or misclas- +sifying all negatives as positives, which ever is smallest. This gives a metric on the interval between 0 and 1, facilitating +comparison across different datasets, when necessary. +Since the majority of comparisons in our study is performed based on average ranks, it requires no commensurability +of the evaluation metrics, so the models are ranked according to their total misclassification costs, which allows for more +precise outcome. +4.4.2 +Multiple classifier comparison +In order to compare multiple classifiers on multiple datasets, we use the standard approach of the combination of the +Friedman omnibus test and post-hoc Nemenyi test [18]. The Friedman test is conducted under the null-hypothesis that +all algorithms in comparison are equivalent in performance. If this null-hypothesis is rejected, the post-hoc test can be +performed to identify pairs of classifiers whose performance is significantly different, which is measured using the critical +difference statistic, and can be visualised using the critical differences diagrams [9]. The non-parametric tests, such as the +Friedman test, are preferred in case where the number of datasets in comparison is less than 30, which is the number of +datasets necessary to satisfy the normality assumptions of parametric statistical tests, such as ANOVA [9]. The post-hoc +test is known to be of low power, not rejecting the null even if the null was rejected for the Friedman test. In this case, +we additionally apply Wilcoxon signed-ranks test, as appropriate, which is used for pairwise comparisons of classifiers on +multiple datasets. This test ranks differences in performances of a given pair of classifiers, under the null hypothesis that +the median difference in ranks is zero. It therefore allows establishing whether the observed differences in performance +between two classifiers are significant. It is considered more powerful than its parametric equivalent, the paired t-test when +the assumptions of the latter cannot be guaranteed. It is also considered more powerful than the Sign test, which counts +the number of wins, losses and ties [9]. +7 + +Ada +DT +KNN +LR +RF +SVM +Level-1 algorithm +type-1 +type-1_acc +type-1_exp +type-1_ln +type-1_sq +type-2 +type-2_acc +type-2_exp +type-2_ln +type-2_sq +type-3 +type-3_acc +type-3_exp +type-3_ln +type-3_sq +Stacking setup +75.10 +72.30 +60.70 +74.60 +71.30 +72.10 +72.40 +69.10 +61.20 +74.60 +70.10 +72.70 +71.90 +71.30 +57.50 +70.70 +71.60 +73.50 +71.50 +70.70 +59.30 +70.60 +70.40 +72.30 +70.70 +70.50 +59.20 +69.00 +70.80 +72.30 +42.90 +45.00 +53.30 +44.40 +44.50 +47.60 +43.60 +45.30 +52.70 +47.20 +44.50 +46.10 +45.10 +46.00 +51.90 +42.40 +46.30 +49.50 +46.50 +49.20 +54.00 +46.20 +43.40 +53.70 +45.20 +44.80 +51.90 +45.40 +45.40 +51.30 +8.00 +6.00 +31.00 +7.00 +5.90 +29.20 +13.40 +16.70 +30.90 +9.00 +14.60 +34.30 +16.60 +13.60 +30.40 +10.40 +14.90 +49.40 +14.30 +14.30 +30.30 +9.60 +12.30 +37.50 +12.80 +13.80 +29.90 +8.00 +14.00 +41.60 +20 +40 +60 +80 +Figure 1: Comparing all classifiers by average rank across 10 datasets. Lower numbers correspond to better rank. +5 +Experimental results +The purpose of our experiments is twofold. Firstly, we would like to compare the performance of the different cost- +sensitive stacking setups in order to determine which of them results in the lowest cost-loss and can be recommended to +practitioners. Secondly, we aim to empirically evaluate MEC-weighted stacking, which is a new cost-sensitive stacking +method we earlier described in Section 3. +Despite our best efforts, not all classifiers trained successfully on all 12 datasets. In particular, we were unable to +collect results for the MEC-weighted stacking where the weights were defined by the logarithmic function on the credit +scoring problem credit ro vub, and MEC-weighted stacking with exponential weights were missing on the fraud detection +dataset fraud ulb kgl. The full results for all 15 stacking setups are thus available on 10 datasets, instead of 12. Unweighted +stacking results, however, are available on all 12 datasets, which we briefly discuss, for completeness. +5.1 +Finding the best cost-senstive stacking setup +5.1.1 +Overall comparison +We begin with an overall comparison, where all classifiers are evaluated and ranked on each of the 10 datasets, and for each +of them an average rank is computed across all datasets. Figure 1 presents the average ranks for all stacking classifiers, +where the vertical axis shows the stacking setup and the horizontal axis shows the corresponding level-1 algorithm. The +comparison consists of a total of 90 classifiers (6 algorithms and 15 stacking setups). For brevity, we adopt the aliases for +each of the stacking setups earlier presented in Table 3. +We notice immediately that the ranking demonstrates clusters with stacking ensembles of type-3 ranking the best, +while type-1 ensembles rank the worst. We note that models built with KNN and SVM algorithms tend to rank lower +than decision tree based models or logistic regression. However, the general picture of type-3 stacking ranking the best +and type-1 ranking the worst remains unchanged for KNN and SVM, although the differences in ranks between the three +groups are smaller than for other algorithms. +Whether these differences in ranks are statistically significant will be discussed in the following subsection, where we +demonstrate the outcomes of statistical tests that compare the performance of various stacking classifiers across multiple +domains. +5.1.2 +Comparing unweighted stacking setups on 12 datasets +We begin by testing the null hypothesis that the three unweighted stacking setups show no difference in performance. The +comparison is performed for each of the six classification algorithms used as level-1 learners. The null hypothesis of the +8 + +Table 4: The outcome of the Friedman multiple hypothesis testing. +Test statistic (χ(k−1)) +Ada +DT +KNN +LR +RF +SVM +Unweighted (k = 3,n = 12) +6.5 +32.69** +32.59** +32.79** +32.69** +32.59** +32.59** +All (k = 15,n = 10) +3.94 +132.44** +132.09** +132.35** +132.29** +132.09** +132.09** +** significant at 0.01 level +k: number of stacking setups in comparison, n: number of datasets in comparison +(a) Adaboost +(b) Decision Tree +(c) K-Nearest Neighbors +(d) Logistic Regression +(e) Random Forest +(f) Support-Vector Machine +Figure 2: Pairwise comparison of the three unweighted stacking setups on 12 datasets using Nemenyi test at 0.05 significance level. +Friedman test was rejected for all 6 comparisons, and the test statistics are presented in row 1 of Table 4. +We proceed with the post-hoc Nemenyi test to evaluate the alternative hypothesis that the performance of three stack- +ings setups is not equal. Figure 2 presents the results of the post-hoc tests at the 0.05 significance level. We find that type-3 +stacking ranks best and is significantly different from both type-2 and type-1 for all algorithms except SVM, where the +difference is only significant for the comparison between type-3 and type-1, but no conclusions can be made regarding the +differences between ensembles of type-3 and type-2. Similarly, no conclusions can be made regarding the differences in +rank between type-2 and type-1 stacking ensembles. +Since the outcome of the post-hoc tests are ambiguous in the case of SVM, we also perform the Wilcoxon rank sum +test under the null hypothesis that the median of the paired differences is zero. For the comparison between type-3 and +type-2 unweighted stacking the null is rejected at 0.05 level. +We conclude from these tests that type-3 stacking performs significantly better than the other two stacking setups. +5.1.3 +Comparing all cost-sensitive stacking setups on 10 datasets +We proceed to compare all 15 stacking classifiers on 10 datasets. The outcome of the Friedman rank sum test can be found +in row 2 of Table 4. The null hypothesis of the Friedman test is rejected for every meta-learner at the 1% significance level, +so we conclude that the performance of all 15 models is not equal and proceed with the post-hoc test. Figure 3 shows the +outcome of the Nemenyi test at 0.05 significance level. +These are for the most part consistent with what we observed in Figure 1, where the classifiers tend to cluster by +stacking setup, type-3 being the leader, type-2 the second-best and type-1 ranking worst. Similar to what we observed +above with unweighted stacking, we can reject the null that type-3 stacking and its MEC-weighted variants are equal in +performance to type-1 stacking and variants. This holds for all algorithms except KNN and SVM. For stacking ensembles +with KNN in level-1 type-3 and type-3 acc classifiers are not significantly different from type-1 exp and type-1 sq, while +for SVM no significant differences were detected between type-3 exp and type-3 sq and other type-1 ensembles. +Since Nemenyi post hoc test is not powerful enough to establish whether the differences between the three stacking +setups are statistically significant, additional testing is required. From the outcomes of the post hoc test we observed that +type-3 stacking generally tends to rank highest, and is therefore of most interest to us. We therefore perform the Wilcoxon +rank sum test for all combinations of pairwise comparisons of stacking algorithms of type-3 vs type-1 and of type-3 vs +type-2 under the null hypothesis that the median of the rank differences between the two groups is equal to zero. The +complete tables with the obtained test statistics and p-values can be found in the Appendix. We find that the null could be +confidently rejected for all comparisons between type-3 and type-1 stacking ensembles, we refer the reader to the Table 7 +9 + +CD +1 +2 +3 +type-3 +type-1 +type-2CD +1 +2 +3 +type-3 +type-1 +type-2CD +H +1 +2 +3 +type-3 +type-1 +type-2CD +H +Y +1 +2 +3 +type-3 +type-1 +type-2CD +1 +2 +3 +type-3 +type-1 +type-2CD +H +2 +type-3 +type-1 +type-2(a) Adaboost +(b) Decision Tree +(c) K-Nearest Neighbors +(d) Logistic Regression +(e) Random Forest +(f) Support-Vector Machine +Figure 3: Comparing all stacking setups on 10 datasets using Nemenyi post-hoc test at 0.05 significance level. +in the Appendix for details. +As for the comparison of stacking type-3 with type-2, the only algorithm where the null could not be rejected was +SVM. We found that the differences between all type-3 MEC-weighted stacking variants and type-2 stacking ensembles +were not significant. However, type-3 unweighted stacking was significantly different from all type-2 stacking variants, +see Table A.2 for details. +We can therefore recommend type-3 stacking, where both levels of stacking are cost-sensitive, as the winner. +5.2 +Evaluating MEC-weighted stacking +Our next research question is whether within the same setup, MEC-weighted stacking offers any improvement over the +unweighted stacking. To determine whether there is a statistically significant difference in performance between the MEC- +weighted stacking models and their unweighted counterparts we perform pairwise comparison using Wilcoxon rank sum +test. The test statistics and corresponding p-values from the 72 comparisons are reported in Table 5. Values that are +significant at 5% level were highlighted with boldface text, weakly significant values at 10% level were highlighted with +italics. As previously, the results are reported per learning algorithm used as the level-1 classifier. +We find almost no significant differences in performance between unweighted and weighted stacking for setups of +type-1 and type-2 with rare exceptions. Surprisingly, only one comparison of stacking type-1, where the level-1 classi- +fier is cost-insensitive shows a significant difference in performance. Namely, the stacking setup type-1 sq learned with +Logistic Regression in level-1. Referring back to the average rankings reported in Figure 1 it happens to be the best +performing Logistic Regression stacking of type-1, so in this instance MEC-weighted stacking is significantly better than +its counterpart with equally weighted meta-inputs. It is, however, an exception, and we must conclude that introducing +cost-sensitivity through MEC-weights into level-1 of stacking has no positive impact on type-1 stacking performance. +Similar conclusions can be drawn for stacking of type-2, where base classifiers are cost-insensitive but the meta- +classifier is made cost-sensitive using DMECC. Here we observe only two statistically significant test outcomes, both of +which have lower average ranks than the unweighted stacking of type-2. +For the setup of type-3 where both level-0 and level-1 of stacking are made cost-sensitive using DMECC, the null +could not be rejected for AdaBoost, Logistic Regression and KNN. Most of the MEC-weighted ensembles built with +Decision Tree, Random Forest and SVM were significantly different from unweighted stacking of type-3. Looking at the +differences in the average ranks, however we note that unweighted stacking of type-3 ranks noticeably better than any of +the MEC-weighted models. +We must therefore conclude that MEC-weighted stacking does not offer a statistically significant improvement over +conventional stacking. +10 + +CD +12345 +6 +7 +89101112131415 +type-3_sq +type-1_acc +type-3_In +type-1_in +type-3_exp +type-i +type-3_acc +type-1_exp +type-3 +type-1_sq +type-2_exp +type-2_ln +type-2_sq +type-2 +type-2_accCD +234567 +8 +9101112131415 +type-3 +type-1_acc +type-3_acc +type-1 +type-3_sq +type-1_exp +type-3_in +type-i_in +type-3_exp +type-1_sq +type-2_exp +type-2_sq +type-2 +type-2_acc +type-2_InCD +234567 +8 +9101112131415 +type-3 +type-1_exp +type-3_In +type-1 +type-3_acc +type-1_sq +type-3_sq +type-i_in +type-3_exp +type-1_acc +type-2_In +type-2_exp +type-2_acc +type-2_sq +type-2CD +2345 +6 +9 101112131415 +type-3 +type-1. +exp +type-3_acc +type-1_acc +type-3_In +type-1_In +type-3_sq +type-1_sq +type-2_acc +type-1 +type-2 +type-3_ +exp +type-2_exp +type-2_sq +type-2_InCD +1 +23456 +57 +8 +9101112131415 +type-3 +type-1 +type-3_sq +type-1_acc +type-3_In +type-1_ln +type-3_acc +type-1_exp +type-3_exp +type-1_sq +type-2 +type-2_in +type-2_acc +type-2_sq +type-2_expCD +1234567 +9101112131415 +type-3 +type-1 +type-3_exp +type-1_sq +type-3_sq +type-1_ln +type-3n +type-1_exp +type-3_acc +type-1_acc +type-2_sq +type-2_in +type-2_acc +type-2 +type-2_expTable 5: Pairwise comparison of unweighted and MEC-weighted stacking using Wilcoxon rank sum test. Statistically significant +values are marked with boldface (significance 0.05) and italics (significance 0.1). +Level-1 +Unweighted stacking type-1 vs +algorithm +type-1 acc +type-1 exp +type-1 ln +type-1 sq +Ada +18.0 (0.3) +18.0 (0.3) +18.0 (0.3) +18.0 (0.3) +DT +23.0 (0.63) +23.0 (0.63) +23.0 (0.63) +23.0 (0.63) +KNN +27.5 (1.0) +21.0 (0.56) +17.0 (0.32) +26.5 (0.92) +LR +27.0 (0.96) +14.0 (0.15) +22.0 (0.56) +10.5 (0.08) +RF +23.0 (0.63) +23.0 (0.63) +23.0 (0.63) +23.0 (0.63) +SVM +18.0 (0.3) +27.0 (0.96) +22.0 (0.56) +27.0 (0.96) +Unweighted stacking type-2 vs +type-2 acc +type-2 exp +type-2 ln +type-2 sq +Ada +22.0 (0.56) +23.0 (0.63) +23.0 (0.63) +23.0 (0.63) +DT +27.5 (1.0) +27.5 (1.0) +26.5 (0.92) +18.5 (0.35) +KNN +25.0 (0.8) +21.0 (0.5) +25.0 (0.8) +21.0 (0.5) +LR +10.5 (0.08) +20.5 (0.47) +24.5 (0.76) +16.5 (0.26) +RF +27.5 (1.0) +22.5 (0.61) +17.5 (0.3) +26.5 (0.92) +SVM +23.5 (0.68) +19.5 (0.41) +18.5 (0.36) +10.5 (0.08) +Unweighted stacking type-3 vs +type-3 acc +type-3 exp +type-3 ln +type-3 sq +Ada +13.0 (0.16) +19.0 (0.43) +19.0 (0.43) +19.0 (0.43) +DT +2.0 (0.01) +11.0 (0.11) +0.0 (0.0) +10.0 (0.08) +KNN +24.0 (0.77) +17.0 (0.32) +24.0 (0.77) +16.0 (0.28) +LR +13.0 (0.16) +19.0 (0.43) +14.0 (0.19) +24.0 (0.77) +RF +4.0 (0.01) +8.0 (0.05) +3.0 (0.01) +9.0 (0.06) +SVM +10.0 (0.08) +0.0 (0.0) +9.0 (0.06) +7.0 (0.04) +5.3 +Comparing cost-sensitive stacking with single cost-sensitive models +Finally, one might ask whether the effort involved in training the level-1 classifier is worth it. To answer this we compare +the best stacking classifier with the corresponding single classifier. Having previously determined the best stacking setup, +where DMECC was applied in both levels, and no MEC-weights are applied, we will omit other classifiers from this +analysis. We average classifier performance across cross-validation folds using the savings metric (see Section 4.4) for +commensurability. We also rank the resulting selection of classifiers by savings and average ranks across 12 datasets. The +results are reported in Table 6, where the winning classifier (per algorithm) is marked with boldface font, the best performer +per dataset is marked with italics. +We note that cost-sensitive stacking always achieves positive savings, meaning its total misclassification costs are lower +than the predetermined budget. Stacking has higher average savings on all algorithms except KNN and Random Forest. In +terms of average ranks, stacking wins for all level-1 algorithms except KNN, which, we note, is one of the worst ranking +algorithms in our study. +6 +Discussion +Outcome 1: using cost-sensitive models in both levels of stacking is recommended +The results presented in this paper, +have demonstrated that there is a statistically significant difference in performance between the three different stacking +setups considered in our experiments, namely CiS-CS, CS-CiS, and CS-CS. Contrary to the majority of cost-sensitive +stacking papers that assumed that one level of cost-sensitive decision-making is sufficient, our experiments demonstrate +that stacking models where the DMECC was applied in both levels of stacking achieved the highest ranking. +While these conclusions hold for this particular post-training method, cost-sensitivity can be introduced either before +or during training of the learning algorithm. Further experiments are required to investigate how different cost-sensitive +methods affect our conclusions. Now that we have established how cost-sensitive stacking should be built, future work +can focus on combining various kinds of cost-sensitive algorithms, including pre-, during- and post-training cost-sensitive +methods [25]. Another interesting avenue for future research would be investigating homogeneous cost-sensitive stacking, +an example of which was proposed in [5] using cost-sensitive decision trees as base classifiers and cost-sensitive logistic +11 + +Table 6: Comparing single classifiers with type-3 unweighted stacking. Savings score is reported for each classifier, higher is better. +Best model per dataset is marked with italics. The winning classifier is marked with boldface letters. +Ada +DT +KNN +LR +RF +SVM +Dataset +Single +Stacking +Single +Stacking +Single +Stacking +Single +Stacking +Single +Stacking +Single +Stacking +absenteeism be 1 +0.188 +0.225 +0.219 +0.242 +0.199 +0.224 +0.188 +0.23 +0.172 +0.243 +0.188 +0.223 +bankruptcy +0.03 +0.123 +0.112 +0.123 +0.105 +0.058 +-0.043 +0.126 +0.31 +0.123 +-0.024 +0.024 +churn AB +0.171 +0.081 +0.052 +0.087 +0.07 +0.019 +0.062 +0.082 +0.1 +0.086 +0.033 +0.04 +churn kgl +0.311 +0.295 +0.0 +0.297 +0.242 +0.031 +0.302 +0.295 +0.273 +0.296 +0.225 +0.066 +credit de uci +0.399 +0.391 +0.293 +0.387 +0.337 +0.351 +0.396 +0.388 +0.424 +0.386 +0.405 +0.354 +credit kdd09 +0.318 +0.303 +0.277 +0.302 +0.287 +0.276 +0.312 +0.303 +0.313 +0.302 +0.289 +0.279 +credit kgl +0.511 +0.411 +0.156 +0.411 +0.408 +0.202 +-0.053 +0.411 +0.499 +0.411 +0.378 +0.175 +credit ro vub +1.793 +1.796 +1.787 +1.795 +1.786 +1.785 +1.727 +1.793 +1.762 +1.797 +1.773 +1.786 +dm kdd98 train +0.108 +0.143 +0.033 +0.147 +0.035 +0.061 +0.122 +0.119 +0.059 +0.147 +0.036 +0.044 +dm pt uci +0.568 +0.558 +0.537 +0.557 +0.551 +0.511 +0.562 +0.558 +0.556 +0.557 +0.551 +0.529 +fraud ieee kgl +0.109 +0.494 +0.444 +0.505 +0.477 +0.438 +0.372 +0.495 +0.584 +0.506 +0.407 +0.444 +fraud ulb kgl +-0.062 +0.714 +0.625 +0.701 +0.679 +0.706 +0.75 +0.72 +0.762 +0.728 +-0.124 +0.688 +Avg Savings +0.37 +0.461 +0.378 +0.463 +0.431 +0.389 +0.391 +0.46 +0.484 +0.465 +0.345 +0.388 +Avg Rank +5.08 +4.17 +9.42 +4.33 +8.17 +9.17 +6.75 +4.08 +4.58 +3.83 +9.25 +9.08 +regression as level-1 classifier. +Outcome 2: cost-insensitive classifiers do not perform well when costs are known, even in stacking +As was pre- +viously shown in [20] cost-insensitive classifiers, having no way to account for differences in misclassification costs, +typically perform worse than cost-sensitive models when evaluated using cost-based performance metrics. In our study, +we observed yet another confirmation to this in the context of heterogeneous ensembles, where base-learners were cost- +sensitive and meta-learners were cost-insensitive. It is, however, surprising that applying MEC-weights has no positive +impact on the performance of these cost-insensitive meta-learners. So we conclude that the transfer of cost information +via cost-sensitive decision-making of the base-classifiers, and via MEC-weights was not sufficient to influence the final +decision of the meta-learner. And even though application of MEC-weights to the meta-inputs makes the meta-level cost- +sensitive, the performance of this method is inferior to unweighted stacking models. We can hypothesise that it may be +different if the meta-learner used misclassification costs internally, but this questions is left for future research. +Limitations +Our current work is not without limitations, which we address below. The choice of the algorithms to be +used in stacking is likely to impact its performance. In order to keep our experiments manageable, we limited ourselves +to algorithms used previously in the cost-sensitive stacking literature. No parameter tuning was performed to preserve +the same base-classifier composition across domains. Those familiar with SVM classifiers could remark that not tuning +this algorithm is a mistake. We are aware of this limitation, which resulted in possibly poor comparative performance of +SVM-based ensembles, however the purpose of the work was to ensure that the ensembles compared differ in only one +thing, which is the inclusion of cost-sensitive decision-making into different levels of the ensemble. We are interested +in relative performance of stacking setups, not in optimal performance of every learning algorithm on every domain. In +order to perform statistical tests, we had to ensure that the classifiers were the same in every ensemble for every dataset, +while parameter tuning will have resulted in different parameter settings on different datasets, which would have prevented +us from performing statistical comparison. In future work we may experiment with homogeneous stacking, where the +diversity of the ensemble will be created by hyperparameter tuning of the base classifiers. +7 +Conclusions +Stacking is a well established state-of-the-art ensemble method, that has been widely applied to many application domains. +In this work we provide insights into ways to make stacking cost-sensitive. We compare 90 stacking models built with +15 different compositions of the stacking ensemble using 6 well known classification algorithms. We evaluate on 12 real- +world cost-sensitive problems with clearly defined, non-synthetic, instance-dependent misclassification costs. 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CRC press, 2012. +A +Additional results +A.1 +Pairwise comparison of stacking setups of type-3 and type-1. +Table 7: Wilcoxon test statistics (p-values). +Level-1 algorithm +type-3 +type-3 acc +type-3 exp +type-3 ln +type-3 sq +Ada +type-1 +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +type-1 acc +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +type-1 exp +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +type-1 ln +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +type-1 sq +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +DT +type-1 +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +type-1 acc +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +type-1 exp +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +type-1 ln +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +type-1 sq +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +KNN +type-1 +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +type-1 acc +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +type-1 exp +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +type-1 ln +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +type-1 sq +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +LR +type-1 +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +type-1 acc +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +type-1 exp +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +type-1 ln +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +type-1 sq +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +RF +type-1 +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +type-1 acc +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +type-1 exp +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +type-1 ln +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +type-1 sq +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +SVM +type-1 +0.0 (0.0) +0.0 (0.0) +8.0 (0.05) +1.0 (0.0) +6.0 (0.03) +type-1 acc +0.0 (0.0) +0.0 (0.0) +8.0 (0.05) +1.0 (0.0) +6.0 (0.03) +type-1 exp +0.0 (0.0) +0.0 (0.0) +8.0 (0.05) +1.0 (0.0) +6.0 (0.03) +type-1 ln +0.0 (0.0) +0.0 (0.0) +8.0 (0.05) +1.0 (0.0) +6.0 (0.03) +type-1 sq +0.0 (0.0) +0.0 (0.0) +8.0 (0.05) +1.0 (0.0) +6.0 (0.03) +15 + +A.2 +Pairwise comparison of stacking setups of type-3 and type-2. +Table 8: Wilcoxon test statistics (p-values). +Level-1 algorithm +type-3 +type-3 acc +type-3 exp +type-3 ln +type-3 sq +Ada +type-2 +0.0 (0.0) +0.0 (0.0) +3.0 (0.01) +3.0 (0.01) +0.0 (0.0) +type-2 acc +0.0 (0.0) +0.0 (0.0) +3.0 (0.01) +3.0 (0.01) +0.0 (0.0) +type-2 exp +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +type-2 ln +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +type-2 sq +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +DT +type-2 +0.0 (0.0) +4.0 (0.01) +3.0 (0.01) +3.0 (0.01) +3.0 (0.01) +type-2 acc +0.0 (0.0) +3.0 (0.01) +3.0 (0.01) +3.0 (0.01) +3.0 (0.01) +type-2 exp +0.0 (0.0) +3.0 (0.01) +2.0 (0.01) +0.0 (0.0) +3.0 (0.01) +type-2 ln +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +type-2 sq +0.0 (0.0) +4.0 (0.01) +3.0 (0.01) +3.0 (0.01) +3.0 (0.01) +KNN +type-2 +7.0 (0.04) +7.0 (0.04) +7.0 (0.04) +7.0 (0.04) +8.0 (0.05) +type-2 acc +7.0 (0.04) +7.0 (0.04) +7.0 (0.04) +7.0 (0.04) +7.0 (0.04) +type-2 exp +7.0 (0.04) +7.0 (0.04) +7.0 (0.04) +7.0 (0.04) +8.0 (0.05) +type-2 ln +5.0 (0.02) +7.0 (0.04) +7.0 (0.04) +7.0 (0.04) +7.0 (0.04) +type-2 sq +7.0 (0.04) +7.0 (0.04) +7.0 (0.04) +7.0 (0.04) +8.0 (0.05) +LR +type-2 +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +type-2 acc +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +type-2 exp +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +type-2 ln +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +type-2 sq +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +0.0 (0.0) +RF +type-2 +0.0 (0.0) +3.0 (0.01) +4.0 (0.01) +3.0 (0.01) +4.0 (0.01) +type-2 acc +0.0 (0.0) +3.0 (0.01) +3.0 (0.01) +3.0 (0.01) +3.0 (0.01) +type-2 exp +0.0 (0.0) +4.0 (0.01) +4.0 (0.01) +3.0 (0.01) +4.0 (0.01) +type-2 ln +0.0 (0.0) +4.0 (0.01) +4.0 (0.01) +3.0 (0.01) +4.0 (0.01) +type-2 sq +0.0 (0.0) +3.0 (0.01) +3.0 (0.01) +3.0 (0.01) +4.0 (0.01) +SVM +type-2 +6.0 (0.03) +13.0 (0.16) +25.0 (0.85) +13.0 (0.16) +20.0 (0.49) +type-2 acc +6.0 (0.03) +14.0 (0.19) +24.0 (0.77) +14.0 (0.19) +20.0 (0.49) +type-2 exp +6.0 (0.03) +13.0 (0.16) +24.0 (0.77) +13.0 (0.16) +19.0 (0.43) +type-2 ln +6.0 (0.03) +13.0 (0.16) +25.0 (0.85) +13.0 (0.16) +20.0 (0.49) +type-2 sq +6.0 (0.03) +13.0 (0.16) +25.0 (0.85) +13.0 (0.16) +19.0 (0.43) +16 + diff --git a/JNAzT4oBgHgl3EQfx_7e/content/tmp_files/load_file.txt b/JNAzT4oBgHgl3EQfx_7e/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5f809674f7249b5fbb851ef1f5082474c656706e --- /dev/null +++ b/JNAzT4oBgHgl3EQfx_7e/content/tmp_files/load_file.txt @@ -0,0 +1,1762 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf,len=1761 +page_content='Cost-Sensitive Stacking: an Empirical Evaluation Natalie Lawrance* ID(�)1, Marie-Anne Guerry ID 1, and George Petrides ID 2 1Department of Business Technology and Operations, Vrije Universiteit Brussel (VUB), Brussels, Belgium 2Department of Mathematics and Statistics, University of Cyprus, Nicosia, Cyprus Abstract Many real-world classification problems are cost-sensitive in nature, such that the misclassification costs vary be- tween data instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' Cost-sensitive learning adapts classification algorithms to account for differences in misclassifica- tion costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' Stacking is an ensemble method that uses predictions from several classifiers as the training data for another classifier, which in turn makes the final classification decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' While a large body of empirical work exists where stacking is applied in various domains, very few of these works take the misclassification costs into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' In fact, there is no consensus in the literature as to what cost-sensitive stacking is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' In this paper we perform extensive experiments with the aim of establishing what the appropriate setup for a cost-sensitive stacking ensemble is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' Our experiments, conducted on twelve datasets from a number of application domains, using real, instance-dependent misclassification costs, show that for best performance, both levels of stacking require cost-sensitive classification decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' Keywords Cost-sensitive learning, classification, ensemble learning, stacked generalization, stacking, blending 1 Introduction Cost-sensitive learning is relevant in many real-world classification problems, where different misclassification errors incur different costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' A prominent example is the field of medicine, where misdiagnosing an ill patient for a healthy one (a false negative) entails delayed treatment and potentially life-threatening consequences, while an error in the opposite direction (a false positive) would incur unnecessary medical examination costs and stress for the patient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' Cost-sensitive classifiers can account for the differences in costs not only between different classes, but also between data instances, making instance- dependent cost-sensitive classification decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' Many cost-sensitive classifiers employ ensemble methods, which combine predictions from several classifiers to obtain better generalisation performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' Superiority of ensembles over individual classifiers is very well known and has been extensively studied ([8, 37]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' Most cost-sensitive classification ensembles are homogeneous in nature, meaning their components are instantiated using the same learning algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' Stacked generalization or stacking [31] is a well known and widely applied heterogeneous ensemble, where the pre- dictions of classifiers produced by different learning algorithms (the base-learners) are used as training inputs to another learning algorithm (the meta-learner) to produce a meta classifier, which makes the final classification decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' In the literature, the base- and meta- levels of stacking are also referred to level-0 and level-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' Homogeneous cost-sensitive ensembles such as cost-sensitive boosting and bagging are widely studied and have been shown very successful [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' Examples of cost-sensitive stacking, on the other hand, are scarce and unsystematic, represent- ing for the most part applications to single domains, where the classifiers are trained on synthetic, class-dependent costs and are evaluated with cost-insensitive performance metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' For a discussion on the importance of real costs for a proper evaluation see the work by [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' In fact, there is currently no consensus as to how a cost-sensitive stacking ensemble is to be composed and at what stage (level-0 or level-1) cost-sensitive decision-making should be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' This can be clearly seen in Table 1, which gives an overview of existing cost-sensitive stacking literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' Stacking is typically made cost-sensitive simply through the application of a cost-sensitive classifier either at level-0 (CS-CiS), level-1 (CiS-CS) or at both levels of Email: natalie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='lawrance@vub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='be 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='01748v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='LG] 4 Jan 2023 the the ensemble (CS-CS), resulting in a total of three possible stacking setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' To the best of our knowledge, no compari- son of all three setups on multiple domains with appropriate evaluation exists in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' Previous related work used arbitrary artificial costs in model training and evaluated cost-sensitive models using performance metrics that are either cost-invariant or that focus on the performance of only the positive class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' In this work we aim to fill this gap by providing a thorough comparison of various cost-sensitive stacking ensembles on multiple domains using real, instance-dependent costs and performance metrics appropriate for cost-sensitive problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='1 Our contributions The main contribution of this work is a rigorous empirical comparison of different setups of cost-sensitive stacking ensembles over multiple domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' We evaluate using appropriate performance metrics and attempt to establish best practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' Secondly, we introduce a novel cost-sensitive classifier combination method, inspired by MEC-voting and stacking, which we call MEC-weighted-stacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' Finally, we present a list of publicly available datasets with clearly defined instance-dependent misclassification costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' The costs are based either on the literature, or are defined by us based on both the literature and expert knowledge of the data providers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' We also define instance-dependent costs for a well known ‘credit-g’ dataset from the UCI Machine learning repository, for which only class-dependent costs were available to date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='2 Outline The remainder of the paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' Section 2 presents an overview of the relevant literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' MEC- weighted stacking is introduced in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' Our hypotheses to be tested, the experimental setup and the datasets used in the study are discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' Section 5 details the results of our extensive experiments, while the main outcomes and limitations are discussed in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' Section 7 concludes the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' 2 Related work While stacking has been widely used in machine learning applications (the interested reader is invited to peruse the survey on stacking literature by [27]), few works are dedicated to the study of cost-sensitive stacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' We identified in the literature three different cost-sensitive stacking setups: CiS-CS, CS-CiS or CS-CS, where the ensemble was made cost-sensitive simply through the application of a cost-sensitive classifier either at level-0, level-1 or at both levels of the ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' In most cases, the method used to make the classification cost-sensitive is the direct cost-sensitive decision as introduced by [35], also called DMECC [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' One of the first papers to discuss stacking in a cost-sensitive context was [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' The authors propose cost-insensitive level-0 and cost-sensitive level-1 stacking setup (CiS-CS setup), which was compared to a number of different classifier combinations schemes on 16 classification problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' The misclassification costs they used were artificially generated by randomly and uniformally sampling costs from on the interval [1,10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' Several other studies followed adopting the same CiS-CS stacking setup, however none of the studies explicitly reasoned or justified this choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' Several more papers demonstrated similar examples of multiple-domain studies of CiS-CS stacking with arbitrary costs ( [7, 33, 34]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' These mainly differ in the type and the number of algorithms that are employed in the ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' We note that all of them used cost-insensitive metrics for classifier evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' [19] considers a stacking setup, where level-0 classifiers were cost-sensitive while level-1 was cost-insensitive (CS- CiS setup).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' The misclassification costs were assumed to be equal to the inverse of the class priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' This approach is very commonly adopted in the absence of information about real misclassification costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' It is, however, not appropriate, see [25] for a discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' The resulting stacking classifier was compared to known ensemble methods using classification accuracy, a metric that by design assumes equal misclassification costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' Most examples of stacking use different learning algorithms in level-0, however in his original work Wolpert suggested that this must not be the case and the technique can also be applied when a single algorithm is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' [5] propose a cost-sensitive variant of bag-stacking, a method originally proposed by [29], using bagged cost-sensitive decision trees in level-0 and using cost-sensitive logistic regression in level-1, thus implicitly proposing a CS-CS stacking setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' To the best 2 Table 1: Summary of cost-sensitive stacking literature Publication Stacking Level-0 Level-1 Real Costs CS setup algorithm algorithm costs type evaluation [6] CiS-CS DT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' KNN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' NB LR c ✓ [19] CS-CiS DT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' KNN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' NB MT c [5] CS-CS DT LR ✓ i ✓ [34] CiS-CS DT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' KNN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' NB LR c [7] CiS-CS ExT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' GBDT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' LDA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' LR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' RF LR c [33] CiS-CS DT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' KNN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' RF,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' SVM DT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' KNN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' NB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' SVM c [13] CiS-CS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' CS-CiS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' CS-CS DT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' NB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' KNN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' SVM LR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' ExT c ✓ this paper CiS-CS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' CS-CiS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' CS-CS DT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' KNN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' LR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' SVM Adab,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' DT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' KNN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' LR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' RF,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' SVM ✓ i ✓ Costs type: c: class-dependent,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' i: instance-dependent Algorithms: Adab: Adaboost,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' DT:decision tree,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' ExT: extremely randomised trees,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' GBDT: gradient boosted trees,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' KNN: k-nearest neighbour,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' LDA: linear discriminant,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' LR: logistic regression,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' MT: Meta Decision Trees,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' NB: naive bayes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' RF: random forest,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' SVM: support vector machines of our knowledge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' this study is the only example where real instance-dependent costs were used in model training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' Models were evaluated using a cost-sensitive metric called the savings score, proposed in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' The only study to date that considers all three different cost-sensitive stacking setups is one by [13] on the application domain of software defect prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' The misclassification costs were selected based on a literature however the authors emphasised that they treated costs as one of the hyperparameters of the classifier, which, we must note, is incorrect, as was previously discussed in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' The experiments are run on 15 datasets using the same class-dependent cost matrix on all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' Balanced error-based metrics were used for evaluation together with cost-based evaluation metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' Identifying real misclassification costs is a complex task, which for many applications may prove too difficult to de- fine and compute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' Most studies resort to artificially generated misclassification costs (see [25] for a discussion on why this is inappropriate) and error-based evaluation metrics are typically employed to assess generalisation performance of cost-sensitive stacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' Examples of metrics used include the AUC, the arithmetic or geometric mean of class-specific accuracies, the F-measure, and the Matthew’s correlation coefficient (MCC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' All of these metrics assume equal misclassi- fication costs, and the F-measure does not incorporate the performance on the negative class, so using these metrics is not compatible with cost-sensitive learning [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' One of the challenges of stacking is the choice of the learning algorithms for the ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' Earlier studies proposed to use linear regression to combine level-0 inputs [30], however Wolpert does not impose any particular restrictions on which algorithm to use in level-1, and he believed that his famous ‘No Free Lunch Theorem’ [32] applies to the meta-learner as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' For the overview of which learning algorithms were used in cost-sensitive stacking ensembles to date we refer our reader to the summary Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' 3 MEC-weighted stacked generalization In the typical supervised classification framework, a learning algorithm A is presented with a set S of data instances (xi,yi), each describing some object i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' We call xi a feature vector, and yi the class label of that object, drawn from a finite, discrete set of classes {1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=',K}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' In this paper we will consider the binary classification problem, where yi ∈ {0,1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' The learning algorithm A, given S as input, after a process called training, produces a classifier C, whose task is to predict the correict class label ˆyC(xj) ∈ {0,1} for a previously unseen feature vector xj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' Training any number L of learning algorithms on the same set of data instances S , we obtain a set of classifiers C = {C1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='CL}, and for each feature vector xi the corresponding set of predictions ˆ Y (xi) = {ˆyC1(xi),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=', ˆyCL(xi)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' C is called an ensemble of classifiers if the predictions from ˆ Y (xi) are combined, in some way, into a single prediction of the class label for the data instance xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' Stacking differs from other classifier ensembles in that the predictions from the set ˆ Y (xi) are combined with the original class label yi to form the set Smeta = {(ˆyC1(xi),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=', ˆyCL(xi)),yi} of meta level data instances subsequently used in another round of algorithm training to produce a new classifier, which is used to obtain the final predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' The novel method we propose in this paper is inspired by the cost-sensitive weights for model votes paradigm described in [25], and consequently called MEC-weighted stacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' To each classifier C, we can assign a weight wC based on that classifier’s cost-performance on the validation set: wC = f(ε), where ε is the sum of the misclassification costs of all data 3 Table 2: Characteristics of the datasets used in our experiments Application Dataset alias # instances # Attr % positives Instance-dependent domain costs source 1 Bankruptcy bankruptcy (private) 404999 221 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='31 this publication 2 Churn churn kgl (Kaggle*) 7043 21 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='54 [25] 3 Churn churn AB [3] 9410 45 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='83 [3] 4 Credit risk credit kgl (Kaggle*) 112915 15 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='70 [2] 5 Credit risk credit de uci [12] 1000 20 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='00 this publication 6 Credit risk credit kdd09 [28] 38938 39 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='89 [2] 7 Credit risk credit ro vub [24] 18918 24 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='95 [24] 8 Direct marketing dm pt uci [12, 22] 45211 17 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='27 [4] 9 Direct marketing dm kdd98 [12] 95412 (train) 479 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='08 [25] 96367 (test) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='06 10 Fraud detection fraud ulb kgl [21] 284807 31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='17 [25] 11 Fraud detection fraud ieee kgl (Kaggle*) 590540 432 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='50 [25] 12 HR analytics absenteeism be (private) 36853 (train) 71 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='50 [20] 35884 (test) 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='76 Kaggle: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='kaggle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='com/ instances incorrectly classified by C on a validation set and f(ε) is a transformation function, which for example can take one of the following forms: f(ε) = ln((1−ε)/ε), f(ε) = 1−ε, f(ε) = exp((1−ε)/ε), and f(ε) = ((1−ε)/ε)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' The general stacking procedure is thus modified with the additional step of collecting the MEC-weights for each of the predictions from the set ˆ Y (xi), yielding the weighted set of predictions ˆ YMEC(xi) = {(wC1 ˆyC1(xi),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=',wCL ˆyCL(xi)),yi}, which is used in meta classifier training instead of ˆ Y (xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' 4 Experimental setup 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='1 Data In this study we use a collection of 10 publicly available datasets and 2 private datasets, for which misclassification costs have either already been defined or will be defined here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' This collection of datasets represents a number of application domains: credit scoring, customer churn prediction, direct marketing, credit card fraud detection, and HR analytics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='2 Misclassification costs Table 2 presents the references both to the datasets and to relevant publications where the instance-dependent misclassifi- cation costs for a given domain were introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' Most of the datasets are large, the number of instances ranging between 1000 and almost 600000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' The number of input features ranges from 15 to 479.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' All of these datasets demonstrate a large degree of class imbalance, where the percentage of positives reaches at most 30%, and in dataset fraud ulb kgl less than 1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' In this work we propose instance-dependent costs for these two datasets, for which no costs were previously defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' The German credit dataset is well known and is referred to as credit de uci in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' Only class-dependent costs were available for this data set, where the prediction task is to identify customers that will default on their loan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' We define instance-dependent costs using the conceptual framework proposed by [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' For any data instance i, the cost of a false negative Ci FN is defined as loss given default and constitutes 75% of the credit line, while the cost of a false positive Ci FP is the loss of the potential profit from rejecting a good customer, plus the sum of the average expected loss and the average expected profit estimated on the training sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' We define profits as simply the interests earned on the credit line in the current year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' The profits are calculated using historic interest rates for the year 2000 in Germany, which we apply randomly and uniformly to the whole sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' The bankruptcy dataset was provided by the credit risk department of a European utilities-provider, who was interested 4 in predicting the risk of corporate bankruptcy for new customers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' With minor modifications, it readily transfers to the same credit risk model described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' Here the credit line is equivalent 90 days of utilities usage by the customer, which, in case of default, the provider loses in full, so Ci FP equals the credit amount.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' The profit margins were provided to us and are calculated per customer based on the assumption of a 12-month contract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' Thus, the Ci FP then equals the annual profit margins for the potentially good customer plus the expected average loss and expected average profit calculated on the given sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='1 Data preprocessing We take care to employ the same preprocessing steps for each of the datasets in the sample, as recommended by the works that first published them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' In addition to that, we apply the following preprocessing steps to all datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' All numeric variables are rescaled using the quantile statistics, which are robust to outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' Missing values of numeric variables are imputed with sample median, and of categorical variables are encoded as a separate category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' All categorical variables are transformed using weight-of-evidence coding [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='2 Data partitioning The classifier performance estimates are obtained by means of repeated stratified k-fold cross-validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' The 5×2 cross- validation suggested by [10] is used to train and evaluate stacking ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' This resampling is repeated 5 times using different random seeds, and the results are averaged across folds and across iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' Large datasets with more than 100000 observations, to keep training times manageable, were split into five disjoint subsets, uniformly at random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' We note that two datasets in our sample are provided with a separate test set, used to evaluate model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' In this case, for fairness of comparison, we perform the split into folds on each of the training and test datasets using the same seed, we then proceed using the training partition of the training set and the test partition of the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' The training partition of the test set remains unused in evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' When training and test data sets contain the same observations at different time periods (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' in bankruptcy prediction) we ensure that training and test datasets are disjoint and do not contain overlapping data instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='3 Learning algorithms The choice of the algorithms for the base- and meta-level of stacking remains one of the challenges of stacked generaliza- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' To the best of our knowledge, no study exists that demonstrates the necessity to use a specific algorithm combination in either base- or meta-level of stacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' The main requirement for the base classifiers of any ensemble is that they are sufficiently accurate (meaning they predict better than a random guess) and sufficiently diverse (meaning their errors are uncorrelated) [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' In a heterogeneous ensemble, where the decisions of different learning algorithms are combined, the number of base-learners need not be large [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' All algorithms below have previously been described and discussed in detail in a number of machine learning textbooks, for example [16], so we refrain from repeating these descriptions here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='1 Base-learners The base learners in our experiments are four well known classification algorithms, which are: CART Decision Tree (DT), K-Nearest Neighbors (KNN), Support Vector Machines (SVM) and Logistic Regression (LR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' Unlike [7] and [33] before us, we choose not to use ensembles such as Random Forest or Extremely Randomised Trees in the base level of stacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' The reasons for this are two-fold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' Firstly, ensembles in general, and stacking in particular are typically built on weak base-learners, which these very powerful models, which are themselves ensembles, certainly are not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' Secondly these methods are based on decision trees and their errors will be correlated with DT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' In our choice we also considered the recommendations of [8], one of the largest empirical studies known to date comparing algorithm performance on 121 datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' Their results on binary problems (55 UCI datasets) demonstrate that Random Forest, SVM, Bagging and Decision Trees have the highest probability of obtaining more than 95% of accuracy, while classifiers of the Naive Bayes (NB) family are not competitive in comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' We therefore do not include NB in our experiments, unlike some previous studies in cost-sensitive stacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' 5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='2 Meta-learners The choice of the meta-learner constitutes a challenge as well, as was called ’the black art’ by the original author of stacked generalization [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' To keep the scale of our experiments manageable and to allow for statistical comparison between stacking and base classifiers, we use the same four algorithms that were used in the level-0 of stacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' In addition to that we also use two homogeneous ensemble methods that, according to [8], perform well on most problems, namely Adaboost (Ada) and Random Forest (RF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='3 Cost-sensitive learners While many variants of cost-sensitive learning algorithms exist that can incorporate the misclassification costs during classifier training [25], in this study we are not interested in comparing cost-sensitive learning algorithms, but in ways of combining cost-sensitive and cost-insensitive learners in a single ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' For our purposes it is important that the two classifiers we compare are different in all but one thing, that is the composition of the ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' We therefore choose to turn known cost-insensitive classifiers cost sensitive by applying a cost-sensitive threshold adjustment method called DMECC [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' In this method, each data instance is classified according to its individual cost-sensitive decision threshold, which is based on the ratio of misclassification costs of that particular data instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' The threshold is calculated as follows: T i cs = Ci FP−Ci TN Ci FP−Ci TN+Ci FN−Ci TP , where Ci TN and Ci TP refer to the costs of correct classification, and Ci FN and Ci FP refer to the misclassification costs of the positive and negative data instances respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' A given record is classified as positive when its estimated probability of being positive exceeds its individual cost-sensitive threshold T i cs [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' Since some learning algorithms (such as DT or SVM) are known not to produce reliable probability estimates, we applied isotonic calibration [36] to all base-learners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='4 Cost-sensitive stacking To the best of our knowledge no definition exists of what constitutes cost-sensitive stacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' Based on the insights from the literature earlier discussed in Section 2, we see three main possibilities of introducing cost-sensitivity into the ensemble structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' Level-0 classifiers are cost-sensitive, level-1 classifiers are cost-insensitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' Level-0 classifiers are cost-insensitive, level-1 classifiers are cost-sensitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' Both level-0 and level-1 classifiers are cost-sensitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' We consider 4 functional forms for the MEC-weights as introduced in Section 3, which resulted in a total of 15 stacking setups to be compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' The complete list of ensemble compositions is presented in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' MEC-weighted stacking renders the level-1 classifier cost sensitive through manipulation of the training data in a cost-sensitive way by applying MEC-weights to the training data of the level-1 classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' We consider it an alternative to obtaining an ensemble where both training levels are cost-sensitive, which is the third stacking setup stated above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='5 Software used All of our experiments were performed using the Python programming language (version 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' Cost-insensitive algorithm implementations came from the scikit-learn (version 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='1) Python library [23], while the cost-sensitive implementations are our own.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='4 Evaluation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='1 Evaluation metrics Contrary to previous studies in cost-sensitive stacking, we would like to emphasise the importance of using appropriate evaluation metrics for cost-sensitive classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' Most authors use traditional evaluation metrics such as ROC AUC, Preci- sion or F1 metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' ROC AUC is known to be cost-invariant, since it is a measure that aggregates classifier performance over all possible class-dependent thresholds, thus implicitly averaging performance over multiple class-dependent costs, which 6 Table 3: The complete list of stacking setups compared in our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='Stacking setup ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='Level-0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='Level-1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='Level-1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='alias ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='algorithm type ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='input weights f(ε) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='algorithm type ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='1 ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='type-3 acc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='CS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='1−ε ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='CS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='is not appropriate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' Other error-based metrics typically assume equal class-dependent costs, which, again, is not appropri- ate, when instance-dependent costs are known at estimation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' Cost-sensitive learning aims to adapt the classification decision of a learning algorithm to the differences between misclassification costs assigned to each of the classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' It is therefore important that the evaluation metrics used to assess the performance of cost-sensitive classifiers is also adapted to account for the difference in misclassification costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' The typical evaluation metric used in cost-sensitive literature is the total misclassification cost [14], that simply adds up the errors weighted with their individual misclassification costs, as defined on the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' Another option is to normalise the total misclassification cost over some budget constraint, which will depend on the application domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' A more general way to do this is to use the savings score proposed in [2], where the total misclassification costs are normalised with the cost of either misclassifying all positives as negatives, or misclas- sifying all negatives as positives, which ever is smallest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' This gives a metric on the interval between 0 and 1, facilitating comparison across different datasets, when necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' Since the majority of comparisons in our study is performed based on average ranks, it requires no commensurability of the evaluation metrics, so the models are ranked according to their total misclassification costs, which allows for more precise outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='2 Multiple classifier comparison In order to compare multiple classifiers on multiple datasets, we use the standard approach of the combination of the Friedman omnibus test and post-hoc Nemenyi test [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' The Friedman test is conducted under the null-hypothesis that all algorithms in comparison are equivalent in performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' If this null-hypothesis is rejected, the post-hoc test can be performed to identify pairs of classifiers whose performance is significantly different, which is measured using the critical difference statistic, and can be visualised using the critical differences diagrams [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' The non-parametric tests, such as the Friedman test, are preferred in case where the number of datasets in comparison is less than 30, which is the number of datasets necessary to satisfy the normality assumptions of parametric statistical tests, such as ANOVA [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' The post-hoc test is known to be of low power, not rejecting the null even if the null was rejected for the Friedman test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' In this case, we additionally apply Wilcoxon signed-ranks test, as appropriate, which is used for pairwise comparisons of classifiers on multiple datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' This test ranks differences in performances of a given pair of classifiers, under the null hypothesis that the median difference in ranks is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' It therefore allows establishing whether the observed differences in performance between two classifiers are significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' It is considered more powerful than its parametric equivalent, the paired t-test when the assumptions of the latter cannot be guaranteed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' It is also considered more powerful than the Sign test, which counts the number of wins, losses and ties [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' 7 Ada DT KNN LR RF SVM Level-1 algorithm type-1 type-1_acc type-1_exp type-1_ln type-1_sq type-2 type-2_acc type-2_exp type-2_ln type-2_sq type-3 type-3_acc type-3_exp type-3_ln type-3_sq Stacking setup 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='10 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='30 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='70 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='60 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='30 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='10 72.' metadata={'source': 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classifiers by average rank across 10 datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' Lower numbers correspond to better rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' 5 Experimental results The purpose of our experiments is twofold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' Firstly, we would like to compare the performance of the different cost- sensitive stacking setups in order to determine which of them results in the lowest cost-loss and can be recommended to practitioners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' Secondly, we aim to empirically evaluate MEC-weighted stacking, which is a new cost-sensitive stacking method we earlier described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' Despite our best efforts, not all classifiers trained successfully on all 12 datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' In particular, we were unable to collect results for the MEC-weighted stacking where the weights were defined by the logarithmic function on the credit scoring problem credit ro vub, and MEC-weighted stacking with exponential weights were missing on the fraud detection dataset fraud ulb kgl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' The full results for all 15 stacking setups are thus available on 10 datasets, instead of 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' Unweighted stacking results, however, are available on all 12 datasets, which we briefly discuss, for completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='1 Finding the best cost-senstive stacking setup 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='1 Overall comparison We begin with an overall comparison, where all classifiers are evaluated and ranked on each of the 10 datasets, and for each of them an average rank is computed across all datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' Figure 1 presents the average ranks for all stacking classifiers, where the vertical axis shows the stacking setup and the horizontal axis shows the corresponding level-1 algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' The comparison consists of a total of 90 classifiers (6 algorithms and 15 stacking setups).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' For brevity, we adopt the aliases for each of the stacking setups earlier presented in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' We notice immediately that the ranking demonstrates clusters with stacking ensembles of type-3 ranking the best, while type-1 ensembles rank the worst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' We note that models built with KNN and SVM algorithms tend to rank lower than decision tree based models or logistic regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' However, the general picture of type-3 stacking ranking the best and type-1 ranking the worst remains unchanged for KNN and SVM, although the differences in ranks between the three groups are smaller than for other algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' Whether these differences in ranks are statistically significant will be discussed in the following subsection, where we demonstrate the outcomes of statistical tests that compare the performance of various stacking classifiers across multiple domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='2 Comparing unweighted stacking setups on 12 datasets We begin by testing the null hypothesis that the three unweighted stacking setups show no difference in performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' The comparison is performed for each of the six classification algorithms used as level-1 learners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' The null hypothesis of the 8 Table 4: The outcome of the Friedman multiple hypothesis testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' Test statistic (χ(k−1)) Ada DT KNN LR RF SVM Unweighted (k = 3,n = 12) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='5 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='69** 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='59** 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='79** 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='69** 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='59** 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='59** All (k = 15,n = 10) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='94 132.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='44** 132.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='09** 132.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='35** 132.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='29** 132.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='09** 132.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='09** ** significant at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='01 level k: number of stacking setups in comparison, n: number of datasets in comparison (a) Adaboost (b) Decision Tree (c) K-Nearest Neighbors (d) Logistic Regression (e) Random Forest (f) Support-Vector Machine Figure 2: Pairwise comparison of the three unweighted stacking setups on 12 datasets using Nemenyi test at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='05 significance level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' Friedman test was rejected for all 6 comparisons, and the test statistics are presented in row 1 of Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' We proceed with the post-hoc Nemenyi test to evaluate the alternative hypothesis that the performance of three stack- ings setups is not equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' Figure 2 presents the results of the post-hoc tests at the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='05 significance level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' We find that type-3 stacking ranks best and is significantly different from both type-2 and type-1 for all algorithms except SVM, where the difference is only significant for the comparison between type-3 and type-1, but no conclusions can be made regarding the differences between ensembles of type-3 and type-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' Similarly, no conclusions can be made regarding the differences in rank between type-2 and type-1 stacking ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' Since the outcome of the post-hoc tests are ambiguous in the case of SVM, we also perform the Wilcoxon rank sum test under the null hypothesis that the median of the paired differences is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' For the comparison between type-3 and type-2 unweighted stacking the null is rejected at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='05 level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' We conclude from these tests that type-3 stacking performs significantly better than the other two stacking setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='3 Comparing all cost-sensitive stacking setups on 10 datasets We proceed to compare all 15 stacking classifiers on 10 datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' The outcome of the Friedman rank sum test can be found in row 2 of Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' The null hypothesis of the Friedman test is rejected for every meta-learner at the 1% significance level, so we conclude that the performance of all 15 models is not equal and proceed with the post-hoc test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' Figure 3 shows the outcome of the Nemenyi test at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='05 significance level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' These are for the most part consistent with what we observed in Figure 1, where the classifiers tend to cluster by stacking setup, type-3 being the leader, type-2 the second-best and type-1 ranking worst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' Similar to what we observed above with unweighted stacking, we can reject the null that type-3 stacking and its MEC-weighted variants are equal in performance to type-1 stacking and variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' This holds for all algorithms except KNN and SVM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' For stacking ensembles with KNN in level-1 type-3 and type-3 acc classifiers are not significantly different from type-1 exp and type-1 sq, while for SVM no significant differences were detected between type-3 exp and type-3 sq and other type-1 ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' Since Nemenyi post hoc test is not powerful enough to establish whether the differences between the three stacking setups are statistically significant, additional testing is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' From the outcomes of the post hoc test we observed that type-3 stacking generally tends to rank highest, and is therefore of most interest to us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' We therefore perform the Wilcoxon rank sum test for all combinations of pairwise comparisons of stacking algorithms of type-3 vs type-1 and of type-3 vs type-2 under the null hypothesis that the median of the rank differences between the two groups is equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' The complete tables with the obtained test statistics and p-values can be found in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' We find that the null could be confidently rejected for all comparisons between type-3 and type-1 stacking ensembles,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' we refer the reader to the Table 7 9 CD 1 2 3 type-3 type-1 type-2CD 1 2 3 type-3 type-1 type-2CD H 1 2 3 type-3 type-1 type-2CD H Y 1 2 3 type-3 type-1 type-2CD 1 2 3 type-3 type-1 type-2CD H 2 type-3 type-1 type-2(a) Adaboost (b) Decision Tree (c) K-Nearest Neighbors (d) Logistic Regression (e) Random Forest (f) Support-Vector Machine Figure 3: Comparing all stacking setups on 10 datasets using Nemenyi post-hoc test at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='05 significance level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' in the Appendix for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' As for the comparison of stacking type-3 with type-2, the only algorithm where the null could not be rejected was SVM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' We found that the differences between all type-3 MEC-weighted stacking variants and type-2 stacking ensembles were not significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' However, type-3 unweighted stacking was significantly different from all type-2 stacking variants, see Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='2 for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' We can therefore recommend type-3 stacking, where both levels of stacking are cost-sensitive, as the winner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='2 Evaluating MEC-weighted stacking Our next research question is whether within the same setup, MEC-weighted stacking offers any improvement over the unweighted stacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' To determine whether there is a statistically significant difference in performance between the MEC- weighted stacking models and their unweighted counterparts we perform pairwise comparison using Wilcoxon rank sum test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' The test statistics and corresponding p-values from the 72 comparisons are reported in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' Values that are significant at 5% level were highlighted with boldface text, weakly significant values at 10% level were highlighted with italics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' As previously, the results are reported per learning algorithm used as the level-1 classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' We find almost no significant differences in performance between unweighted and weighted stacking for setups of type-1 and type-2 with rare exceptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' Surprisingly, only one comparison of stacking type-1, where the level-1 classi- fier is cost-insensitive shows a significant difference in performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' Namely, the stacking setup type-1 sq learned with Logistic Regression in level-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' Referring back to the average rankings reported in Figure 1 it happens to be the best performing Logistic Regression stacking of type-1, so in this instance MEC-weighted stacking is significantly better than its counterpart with equally weighted meta-inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' It is, however, an exception, and we must conclude that introducing cost-sensitivity through MEC-weights into level-1 of stacking has no positive impact on type-1 stacking performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' Similar conclusions can be drawn for stacking of type-2, where base classifiers are cost-insensitive but the meta- classifier is made cost-sensitive using DMECC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' Here we observe only two statistically significant test outcomes, both of which have lower average ranks than the unweighted stacking of type-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' For the setup of type-3 where both level-0 and level-1 of stacking are made cost-sensitive using DMECC, the null could not be rejected for AdaBoost, Logistic Regression and KNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' Most of the MEC-weighted ensembles built with Decision Tree, Random Forest and SVM were significantly different from unweighted stacking of type-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' Looking at the differences in the average ranks, however we note that unweighted stacking of type-3 ranks noticeably better than any of the MEC-weighted models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' We must therefore conclude that MEC-weighted stacking does not offer a statistically significant improvement over conventional stacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' ' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='type-1_acc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='type-2_sq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='type-2_in ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='type-2_acc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='type-2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='type-2_expTable 5: Pairwise comparison of unweighted and MEC-weighted stacking using Wilcoxon rank sum test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' Statistically significant values are marked with boldface (significance 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='05) and italics (significance 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' Level-1 Unweighted stacking type-1 vs algorithm type-1 acc type-1 exp type-1 ln type-1 sq Ada 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='3) 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='3) 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='3) 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='3) DT 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='63) 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='63) 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='63) 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='63) KNN 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='5 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='0) 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='56) 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='32) 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='5 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='92) LR 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='96) 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='15) 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='56) 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='5 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='08) RF 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='63) 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='63) 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='63) 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='63) SVM 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='3) 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='96) 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='56) 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='96) Unweighted stacking type-2 vs type-2 acc type-2 exp type-2 ln type-2 sq Ada 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='56) 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='63) 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='63) 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='63) DT 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='5 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='0) 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='5 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='0) 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='5 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='92) 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='5 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='35) KNN 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='8) 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='5) 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='8) 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='5) LR 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='5 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='08) 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='5 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='47) 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='5 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='76) 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='5 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='26) RF 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='5 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='0) 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='5 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='61) 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='5 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='3) 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='5 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='92) SVM 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='5 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='68) 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='5 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='41) 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='5 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='36) 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='5 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='08) Unweighted stacking type-3 vs type-3 acc type-3 exp type-3 ln type-3 sq Ada 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='16) 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='43) 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='43) 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='43) DT 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='01) 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='11) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='0) 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='08) KNN 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='77) 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='32) 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='77) 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='28) LR 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='16) 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='43) 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='19) 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='77) RF 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='01) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='05) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='01) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='06) SVM 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='08) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='0) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='06) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='04) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='3 Comparing cost-sensitive stacking with single cost-sensitive models Finally, one might ask whether the effort involved in training the level-1 classifier is worth it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' To answer this we compare the best stacking classifier with the corresponding single classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' Having previously determined the best stacking setup, where DMECC was applied in both levels, and no MEC-weights are applied, we will omit other classifiers from this analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' We average classifier performance across cross-validation folds using the savings metric (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='4) for commensurability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' We also rank the resulting selection of classifiers by savings and average ranks across 12 datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' The results are reported in Table 6, where the winning classifier (per algorithm) is marked with boldface font, the best performer per dataset is marked with italics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' We note that cost-sensitive stacking always achieves positive savings, meaning its total misclassification costs are lower than the predetermined budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' Stacking has higher average savings on all algorithms except KNN and Random Forest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' In terms of average ranks, stacking wins for all level-1 algorithms except KNN, which, we note, is one of the worst ranking algorithms in our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' 6 Discussion Outcome 1: using cost-sensitive models in both levels of stacking is recommended The results presented in this paper, have demonstrated that there is a statistically significant difference in performance between the three different stacking setups considered in our experiments, namely CiS-CS, CS-CiS, and CS-CS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' Contrary to the majority of cost-sensitive stacking papers that assumed that one level of cost-sensitive decision-making is sufficient, our experiments demonstrate that stacking models where the DMECC was applied in both levels of stacking achieved the highest ranking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' While these conclusions hold for this particular post-training method, cost-sensitivity can be introduced either before or during training of the learning algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' Further experiments are required to investigate how different cost-sensitive methods affect our conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' Now that we have established how cost-sensitive stacking should be built, future work can focus on combining various kinds of cost-sensitive algorithms, including pre-, during- and post-training cost-sensitive methods [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' Another interesting avenue for future research would be investigating homogeneous cost-sensitive stacking, an example of which was proposed in [5] using cost-sensitive decision trees as base classifiers and cost-sensitive logistic 11 Table 6: Comparing single classifiers with type-3 unweighted stacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' Savings score is reported for each classifier, higher is better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' Best model per dataset is marked with italics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' The winning classifier is marked with boldface letters.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='25 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='08 regression as level-1 classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' Outcome 2: cost-insensitive classifiers do not perform well when costs are known, even in stacking As was pre- viously shown in [20] cost-insensitive classifiers, having no way to account for differences in misclassification costs, typically perform worse than cost-sensitive models when evaluated using cost-based performance metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' In our study, we observed yet another confirmation to this in the context of heterogeneous ensembles, where base-learners were cost- sensitive and meta-learners were cost-insensitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' It is, however, surprising that applying MEC-weights has no positive impact on the performance of these cost-insensitive meta-learners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' So we conclude that the transfer of cost information via cost-sensitive decision-making of the base-classifiers, and via MEC-weights was not sufficient to influence the final decision of the meta-learner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' And even though application of MEC-weights to the meta-inputs makes the meta-level cost- sensitive, the performance of this method is inferior to unweighted stacking models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' We can hypothesise that it may be different if the meta-learner used misclassification costs internally, but this questions is left for future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' Limitations Our current work is not without limitations, which we address below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' The choice of the algorithms to be used in stacking is likely to impact its performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' In order to keep our experiments manageable, we limited ourselves to algorithms used previously in the cost-sensitive stacking literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' No parameter tuning was performed to preserve the same base-classifier composition across domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' Those familiar with SVM classifiers could remark that not tuning this algorithm is a mistake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' We are aware of this limitation, which resulted in possibly poor comparative performance of SVM-based ensembles, however the purpose of the work was to ensure that the ensembles compared differ in only one thing, which is the inclusion of cost-sensitive decision-making into different levels of the ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' We are interested in relative performance of stacking setups, not in optimal performance of every learning algorithm on every domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' In order to perform statistical tests, we had to ensure that the classifiers were the same in every ensemble for every dataset, while parameter tuning will have resulted in different parameter settings on different datasets, which would have prevented us from performing statistical comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' In future work we may experiment with homogeneous stacking, where the diversity of the ensemble will be created by hyperparameter tuning of the base classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' 7 Conclusions Stacking is a well established state-of-the-art ensemble method, that has been widely applied to many application domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' In this work we provide insights into ways to make stacking cost-sensitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' We compare 90 stacking models built with 15 different compositions of the stacking ensemble using 6 well known classification algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' We evaluate on 12 real- world cost-sensitive problems with clearly defined, non-synthetic, instance-dependent misclassification costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' In contrast 12 to the absolute majority of cost-sensitive literature, our experimental results demonstarate that for the best results, not one, but two layers of cost-sensitive decision-making are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' We also found that applying MEC-weights to the training inputs of the level-1 classifier in stacking did not significantly change the performance of stacking models where the level-1 algorithm applied the default decision threshold to classify.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' Moreover, MEC-weighted stacking models where both levels were cost-sensitive performed worse than the unweighted stacking of the same type, 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='05) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='0) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='03) 15 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='2 Pairwise comparison of stacking setups of type-3 and type-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' Table 8: Wilcoxon test statistics (p-values).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content=' Level-1 algorithm type-3 type-3 acc type-3 exp type-3 ln type-3 sq Ada type-2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='0) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfx_7e/content/2301.01748v1.pdf'} +page_content='0 (0.' metadata={'source': 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-0,0 +1,2404 @@ +Gravastar-like black hole solutions in q-theory +M. Selch, J. Miller, and M.A.Zubkov +Ariel University, Ariel, 40700, Israel +(Dated: January 10, 2023) +We present a stationary spherically symmetric solution of the Einstein equations, with a source +generated by a scalar field of q-theory. +In this theory Riemannian gravity, as described by the +Einstein - Hilbert action, is coupled to a three - form field that describes the dynamical vacuum. +Formally it behaves like a matter field with its own stress - energy tensor, equivalent to a scalar +field minimally coupled to gravity. The asymptotically flat solutions obtained to the field equations +represent black holes. For a sufficiently large horizon radius the energy density is localized within +a thin spherical shell situated just outside of the horizon, analogous to a gravastar. The resulting +solutions to the field equations, which admit this class of configurations, satisfy existence conditions +that stem from the Black Hole no - hair theorem, thanks to the presence of a region in space in +which the energy density is negative. +I. +Introduction +General Relativity (GR) is based on the axiom that the gravitational field is encoded by the geometry of spacetime: +in a perfect vacuum spacetime is flat, while massive objects distort the surrounding spacetime from being otherwise +flat to having a curved geometry. Test particles move along geodesics of the background spacetime in a way such +that the trajectory depends on the geometry of the spacetime. In flat space the trajectory is a straight line, while in +curved space it gets accelerated away from straight line motion. This is the equivalence principle: the gravitational +field is equivalent to acceleration in the background spacetime. In this manner Newton’s ‘action at a distance’ theory +of gravity is replaced by Einstein’s field theory approach, in which the field is the very geometry of the spacetime. +The vacuum Einstein field equations contain a cosmological constant [1], which relates the large-scale expansion of +the universe in cosmological models. In turn the cosmological constant is related to the vacuum energy density [2–6]. +Its value according to astronomical observations lies at a typical energy scale of the order of 10−3 eV [7–9], while its +range of values as inferred from theoretical models is much larger [10–14]. +Volovik and Klinkhamer have suggested [15, 16] that the smallness of the observed vacuum energy density can +be explained on the basis of a thermodynamic argument by which the vacuum energy density is exactly canceled in +equilibrium (perfect quantum vacuum). q-theory appears as a low-energy effective theory [17–20] as opposed to a +purely fundamental theory [10–14], and as a result contributions to the vacuum energy density become suppressed at +macroscopic scales and reside in small perturbations of the equilibrium state. In the Klinkhamer Volovik model, the +effective vacuum energy density that enters the low-energy field equations is described by a vacuum field variable. At +high energies it may be described by the 3-form field Aβγδ, which is antisymmetric with respect to permutation of +indices. From this tensor the scalar q-field, the low energy vacuum variable, is composed as q2 = − 1 +24FαβγδF αβγδ, +where Fαβγδ = ∇[αAβγδ] is the field strength. The equilibrium value of q alters if the vacuum is perturbed towards a +new equilibrium state. More details can be found in §II. +Black holes (BHs) are unstable due to several effects. They may evaporate gradually resulting in Hawking radiation +[21]. Alternatively a BH may undergo a transition to a white hole [22], through quantum mechanical tunneling from +inside a trapped region to an anti-trapped region. Another possible outcome is the formation of a vacuum star. In this +model the event horizon takes the form of a boundary between different phases of the quantum vacuum [23]. There +are a number of similar phenomena between semi-metals and BHs, for example the event horizon emerging on the +boundary between type I and type II Weyl semimetals. Volovik [24] has discussed an analogous process that occurs +in Dirac and Weyl semimetals that suggests the viability of the formation of a gravastar or vacuum star after vacuum +reconstruction, once Hawking radiation has been ended. According to [24] (and unlike conventional gravastars [25]) +the gravastar admits three distinct regions: the vacuum inside the Cauchy horizon with the de Sitter metric, the +vacuum inside the thin shell between the Cauchy horizon and the event horizon, and the vacuum outside the event +horizon with the ordinary Schwarzschild metric. Even classically, BHs may be unstable. Regarding it as an ‘excited +state’, decay due to exponential growth of the metric (plus soliton) perturbations becomes possible [26]. +The geometry of the spacetime in a neighbourhood of a gravastar can be described using the action of a q-field +coupled to gravity, where the q-field, as mentioned above, is related to the energy density of the quantum vacuum +[27, 28]. The goal of this work is to show (by numerical means) that (static and spherically symmetric) “scalar-haired” +BHs exist within q-theory induced by the scalar q-field minimally coupled to gravity. The solutions are discussed +thoroughly and are interpreted within the context of gravastars. A similar (numerical) calculation by Nucamendi and +Salgado can be found in refs. [29], in which solutions to the field equations are derived for the case of a scalar field +arXiv:2301.02914v1 [gr-qc] 7 Jan 2023 + +2 +coupled minimally as well, in a generic static, spherically symmetric and asymptotically flat spacetime very similar +but not identical to our considerations within q-theory. There, “scalar-hair” BH solutions were shown to exist. +As mentioned above, such BH solutions admit non-trivial “hair” associated with the scalar field. A general BH +“no-hair” conjecture was originally proposed by Ruffini and Wheeler [30] (see also Hawking 1975 [21] for a thorough +pedagogical overview). A set of conditions arise from no-hair theorems [31] for the existence of a solution to the +Einstein equations with a scalar field source, one of which is the the no-hair integral, defined in (60), of the solution. +It is shown in §IV that these criteria are satisfied for the solutions obtained in this work. +For a given spacetime the presence of an event horizon can be inferred from analyzing ingoing null trajectories, +as explained in §III. A set of coordinates convenient for both tracking null trajectories and describing the whole +neighborhood of an event horizon are Painlevé-Gullstrand (PG) coordinates, originally suggested in [32, 33]. Below +can be found a brief description of how they are derived and the manner in which PG coordinates describe null +trajectories. More complex spacetime tensors including the stress-energy and Einstein tensors in PG coordinates are +given in §III A. +PG coordinates were originally suggested as an alternative to Schwarzschild coordinates for describing radial null +trajectories in Schwarzschild spacetime. The unique solution of the Einstein equation that is spherically-symmetric, +stationary, non-spinning with no net charge is the Schwarzschild metric with the form +ds2 = −fdt2 + 1 +f dr2 + r2dΩ2, +(1) +where dΩ2 = dθ2 +sin2 θdϕ2 is the line-element of a unit two-sphere and f = 1−2M/r is the Schwarzschild term with +M being the mass of the background. The four-velocity U µ = dxµ/dτ ≡ ˙xµ (τ being proper time along the worldline) +satisfies the normalization condition −1 = gµνU µU ν = −f ˙t2 + f −1 ˙r2 = U 2 � +−f + f −1(dr/dt)2� +where U ≡ dt/dτ. +The quantity ε = −gµνξµU ν = fU is a constant of motion, since ξµ = (∂/∂t)µ is a timelike Killing field. Accordingly, +the four velocity of a radially outgoing or ingoing spherically-symmetric worldline is +U µ = +� ε +f , − +� +ε2 − f, 0, 0 +� +. +(2) +In PG coordinates, the time coordinate, denoted tp to distinguish it from the t in Schwarzschild coordinates, is the +proper time along the worldline of the geodesic. As such the four velocity now has the more natural form +U µ +p = +� ˙tp, ˙r, ˙θ, ˙φ +� += +� +1 , − +� +ε2 − f, 0, 0 +� +≡ (1, −v, 0, 0) +(3) +where a ‘dot’ refers to a derivative with respect to tp, and +v = +� +ε2 − f +(4) +is the radial component of the velocity on the free-falling trajectory. The PG time coordinate tp is related to the +Schwarzschild time coordinate as +dtp = εdts + +� +ε2 − f +f +dr. +(5) +After writing the Schwarzschild metric (1) in PG coordinates the Painlevé-Gullstrand metric is obtained with the +form +ds2 = −dtp +2 + 1 +ε2 (dr + vdtp)2 + r2dΩ2. +(6) +Note that as pointed out in [34], the form of the metric in (6) is somewhat analogous to the conserved Newtonian +energy +E = 1 +2 +� dr +dtp +�2 ++ Φ(r) +(7) +where Φ(r) = −M/r is a Newtonian type potential and E = (ε2 − 1)/2 is constant. If the particle falls from rest at +infinity, ε = 1, E = 0, such that (6) reduces to the standard Painlevé Gullstrand metric +ds2 = −dtp +2 + +� +dr + +� +2M +r dtp +�2 ++ r2dΩ2. +(8) + +3 +Both forms of the metric in (6) and (8) are regular at the horizon r = 2M, ergo the spacetime geometry inside and +outside the horizon of a black hole can be related without the emergence of any singularities. As explained in [34], the +Newtonian type energy motivates the following ansatz for the metric in the generalized Painlevé-Gullstrand form: +ds2 = −dtp +2 + +1 +1 + 2E(tp, r) +� +dr + v(tp, r)dtp +�2 ++ r2dΩ2, +(9) +where +v(tp, r) = +� +2E(tp, r) + 2m(tp, r) +r +. +(10) +Here E and M are not constant values, rather they are functions of tp and r. The metric in (10) is the one used in +this paper, but without an explicit dependence on tp, namely only stationary solutions are considered. +The metric signature is taken to be (−1, 1, 1, 1), and we use natural units, namely c = ℏ = 1 is assumed. In the +opening section we defer setting the Newtonian constant G = 1 for the purpose of offering clarity in our calculations, +but later it will be set to unity. +This paper is structured in the following way. In section II we introduce our model for q-theory, which is effectively +a scalar field theory with a double-well potential interaction, minimally coupled to Einstein gravity. +In section +III the Einstein equations are given for the case of a static spherically symmetric spacetime, in two different sets +of coordinates: generalized Painlevé-Gullstrand, and those that shall be referred to as generalized Schwarzschild +coordinates. In section IV the restrictions on solutions to the field equations are explain, which arise due to the no- +hair theorems that hold for scalar field theories minimally coupled to gravity. Section V contains a detailed discussion +of one specific static and spherically symmetric q-theory solution to the field equations. Section VI builds on section V, +containing a local scan of the space of solutions around the solution considered in section V. In section VII instabilities +of the obtained solutions are addressed, both due to classical perturbations as well as due to Hawking radiation. We +end our work in VIII with a conclusion of our findings. +II. +The model under consideration +In this paper we consider a gravitating dynamical vacuum of the type introduced in Refs. [15, 16]. One way to +describe such a system is through a 3-form field Aβγδ, antisymmetric with respect to permutation of indices. From a +stand point, this system can be considered to be matter described by a field Aβγδ, interacting with the gravitational +field. The secondary scalar field q, which is the effective degree of freedom at low energies, is composed of a three +form field Aβγδ as +q2 = − 1 +24FαβγδF αβγδ, +Fαβγδ = ∇[αAβγδ], +(11) +Fαβγδ = ±q√−gεαβγδ, +F αβγδ = ±q +1 +√−g εαβγδ , +(12) +where εαβγδ and εαβγδ are completely antisymmetric, namely ε0123 = 1 and ε0123 = −1. The square brackets denote +antisymmetrization of indices. From these relations it follows that +δq +δgαβ = 1 +2qgαβ . +(13) +The action of the model has the form +S = +� +d4x√−g +� +R +16πG − ϵ(q) − 1 +2gαβ∇αq∇βq +� +(14) +where G is Newton’s constant. ϵ is a polynomial function in q that has the form +ϵ(q) = λ +4 +� +q4 − 1 +aGq2 +� +. +(15) +λ and a are real numbers assumed to be O(1)-parameters with λ, a > 0. The scale associated with the potential +function is due to G and, therefore, it is the Planck scale. + +4 +Variation of the action with respect to the metric results in the Einstein equations: +Rαβ − 1 +2gαβR = −8πG +� +gαβ +� +ρ(q) + 1 +2∇αq∇αq +� +− ∇αq∇βq + 2□q δq +δgαβ +� +(16) +where +ρ(q) = ϵ(q) − dϵ +dq +�1 +2gµν δq +δgµν +� += ϵ(q) − dϵ +dq q +(17) +follows directly from (13). The function ρ enters the Einstein equations in the same way as the cosmological constant. +The shift from ϵ to ρ, as well as the final term on the right hand side of the Einstein equations, follow from the relation +q = q(gµν). A self-sustained quantum vacuum fulfills +0 = P = −ρ +(18) +in thermodynamic equilibrium, where P refers to pressure and ρ is the energy density. This yields, in our example, +the equilibrium values qeq = 0, ± +1 +√ +3aG, which satisfy ρ(qeq) = 0. By further analogy with thermodynamics, a chemical +potential µ may be defined (up to a constant) as +µ = dϵ +dq +���� +q=qeq +. +(19) +Given that the original potential function is even with respect to q, and assuming that the vacuum has a non-trivial +equilibrium configuration, it shall be assumed that qeq = +1 +√ +3aG from now on. The energy density function is then +given by +ρ(q) = ϵ(q) − µq = λ +4 +� +q4 − 1 +aGq2 + +2 +(3aG) +3 +2 q +� +. +(20) +It has a local minimum at q = qmin = −( 1 +√ +3 + 1) +1 +√ +4aG, a local maximum at q = qmax = (− 1 +√ +3 + 1) +1 +√ +4aG and another +local minimum at the equilibrium value q = qeq = +1 +√ +3aG with ρ(q = qeq) = 0. Consequently, the equilibrium value is +also a double root of ρ. The other roots are located at q = 0 and q0 = − +2 +√ +3aG. +Vacuum stability requires the vacuum compressibility χvac to be positive : +χ−1 +vac = +� +q2 d2ϵ +dq2 +� ���� +q=qeq += +λ +6a2G2 ≥ 0 , +(21) +fulfilled for λ, a > 0. +The energy density function for λ = 1 and different values of a is plotted in Fig. (1) with the black line marking +the level of vanishing energy density. The potential has the same basic characteristic of two wells: the well containing +the equilibrium value of the q-field as a minimum with zero energy density, and the well that is deeper and hence +allows for negative energy densities. The region of negative energy densities starts at q = q0 and ends at q = 0. +The action in (14) lacks higher derivative terms of the q-field, which usually appear in the effective field theory +description without fine tuning. +These terms are (in the absence of an intermediate high-energy physics scale) +suppressed by the Planck energy scale. As long as the q-field varies slowly (q′ ≪ 1 in units with G = 1), these +contributions are negligible. This is the approach adopted in the following part of the discussion. +Variation of the action with respect to the three - form field Aαβγ yields the generalized Maxwell equation +∇α(√−g +� +−dϵ(q) +dq +δq +δFαβγδ ++ □q +δq +δFαβγδ +) +� += 0 +(22) +⇔ ϵαβγδ∇α +� +−dϵ(q) +dq ++ □q +� += 0 +(23) +⇔ dϵ(q) +dq +− □q = µ +(24) +Here µ is the integration constant. Inserting this into the Einstein equations (16) yields +Rαβ − 1 +2gαβR = −8πG +� +gαβ(ϵ(q) − µq + 1 +2∇αq∇αq) − ∇αq∇βq +� +(25) + +5 +Figure 1. The energy density of the q-field for λ = 1 and different values of a is depicted. While λ sets the absolute scale, a +determines the depth and separation width of the potential wells. +which comprises both the gravitational field and the matter (generalized Maxwell) equations. The fact that the q-field +is not fundamental, but rather only an effective degree of freedom leads to the effective replacement of ϵ with ρ. In +equilibrium, the scalar-field value corresponds to a minimum of the energy density ρ, which is located at the value +zero. +This shows that the problem reduces to that of solving the (modified) Einstein equations given by (25). +It is +equivalent to a scalar field theory minimally coupled to gravity with a scalar field potential ρ. +III. +Static and spherically symmetric solutions of the Einstein equations +The discussion is now about solving the static, spherically symmetric Einstein equations in order to find asymptotically +flat solutions that describe a BH with a non-trivial q-field behavior. The q-field, as well as all the other functions, +depend on a single coordinate only, which we choose to be the standard radial coordinate. The q-field is expected +to relax to the equilibrium value at large values of the radial coordinate, and to deviate from the equilibrium value +approaching smaller and smaller values for the radial coordinate. +In the following, two different ansätze are used for the metric in order to solve the Einstein equations: +Gα +β = 8πG(Tq)α +β, Gα +β = Rα +β − 1 +2gα +βR +(26) +where the Einstein tensor is Gα +β and the q-field energy-momentum tensor is (Tq)α +β. Primes above symbols label +derivatives with respect to the radial variable, throughout this work. +A. +Generalized Painlevé coordinates +The first ansatz is a generalized Painlevé-Gullstrand metric with the form +ds2 = −dt2 + +1 +1 + 2E(r)(dr + v(r)dt)2 + r2dΩ2 +(27) +and +v(r) = +� +2E(r) + 2Gm(r) +r +(28) +while +dΩ2 = dθ2 + sin2(θ)dφ2 . +(29) + +energy density p(q) +energy density pgi,zoomed +14.0 +a = 0.1 +a = 0.1 +a = 0.2 +7.5 +a = 0.2 +a=0.5 +a=0.5 +50 +a=l +a=l +a=2 +5.D +a=2 +40 +a= 5 +a= 5 +25 +(b)d +30 +(b)d +0.D +24 +2.5 +5.0 +0 +-7.5 +-10 +-2 +L- +2 +10.0 +4 +E- +0 +1 +3 +4 +1.0 +0.5 +0.D +0.5 +1D +1'5 +2D +a +q6 +As explained in the introductory remarks, the terms E and v in the metric are related to the kinematical quantities +of a particle in motion in the background. As elucidated in [34], v(r) may be interpreted as the velocity of a freely +falling test particle as it falls in towards a (spherically symmetric) gravitating object from infinity, while E(r), at least +asymptotically, can be related to the normalized total energy of a test particle (E(∞) = (e2−1) +2 +, where e represents +the total energy per unit rest mass of a test particle at infinity). This motivates labeling v a velocity function, and E +an energy function. +For the first metric ansatz, the non-vanishing Einstein tensor components are +Gt +t = −2Gm′ +r2 +, Gt +r = +−2E′ +r(1 + 2E) +� +2E + 2Gm +r +, +(30) +Gr +r = −2rE′ + 4E′Gm − 4EGm′ − 2Gm′ +(1 + 2E)r2 +, +(31) +Gθ +θ = Gφ +φ = (3r2(1 − 2Gm +r +)(E′)2 + 3r(1 + 2E)E′Gm′ − (r + m)(1 + 2E)E′ +(32) +− r2(1 − 2Gm +r +)(1 + 2E)E′′ − r(1 + 2E)2Gm′′)/((1 + 2E)2r2) +The energy momentum tensor (Tq)α +β for the q-field takes the form +(Tq)α +β = −(gα +β(ρ(q) + 1 +2gαβ∇αq∇βq) − gαγ∇γq∇βq) +(33) +with non-vanishing components +(Tq)t +t = (Tq)θ +θ = (Tq)φ +φ = −(ρ(q) + 1 +2(1 − 2Gm +r +)(q′)2), +(34) +(Tq)t +r = +� +2E + 2Gm +r +(q′)2, +(35) +(Tq)r +r = −(ρ(q) − 1 +2(1 − 2Gm +r +)(q′)2). +(36) +The Einstein equations can thus be brought into the following form +Gm′ = 4πGr2(ρ(q) + 1 +2(1 − 2Gm +r +)(q′)2)) +(37) +2E′ +1 + 2E = −8πGr(q′)2 +(38) +q′′ = (1 − 2Gm +r +)−1(−2q′ +r + 2Gmq′ +r2 ++ 8πGrρq′ + dρ +dq ). +(39) +The first two equations are obtained from the radial and time components of the Einstein equations. Using these to +simplify the equation due to the angular components leads to the third equation. +The second equation can be solved for E. With the definition F = ln(1 + 2E) and E(∞) = lim +r→∞ E(r) the result is +F(r) = ln(1 + 2E(∞)) + 8πG +� ∞ +r +s(q′(s))2 ds . +(40) +This leaves the first and third equations, which are solved numerically. +B. +Generalized Schwarzschild coordinates +An analogous procedure for the second ansatz yields the metric in the form +ds2 = −f(r)dt2 + +1 +h(r)dr2 + r2dΩ2, +(41) + +7 +which shall be referred to as generalized Schwarzschild coordinates. +From this form of the metric the following +non-vanishing components of the Einstein tensor are derived: +Gt +t = rh′ + h − 1 +r2 +, Gr +r = rhf ′ + hf − f +r2f +, +(42) +Gθ +θ = Gφ +φ = −1 +4(rh(f ′)2 − 2rfhf ′′ − 2fhf ′ − (rff ′ + 2f 2)h′)/(rf 2) +(43) +The non-vanishing components of the corresponding energy momentum tensor are +(Tq)t +t = (Tq)θ +θ = (Tq)φ +φ = −(ρ(q) + 1 +2h(q′)2), (Tq)r +r = −(ρ(q) − 1 +2h(q′)2). +(44) +The Einstein equations can finally be brought into the following form +1 − h − rh′ = 8πGr2(ρ + 1 +2h(q′)2) +(45) +f ′ +f − h′ +h = 8πGr(q′)2 +(46) +q′′ = 1 +h +dρ +dq − h′ +h q′ − 2 +r q′ − 4πGr(q′)3. +(47) +The first two equations are obtained from the radial and time components of the Einstein equations, and subsequently +used to simplify the relation for the angular components. This results in the third equation. The second equation can +be solved and reads, with k = ln(f) and the assumption k(r = ∞) = 0, as +k(r) = − +� ∞ +r +�h′(s) +h(s) + 8πGs(q′(s))2) +� +ds . +(48) +The connection between the two parameterizations is provided by the relation +h(r) = 1 − 2Gm(r) +r +(49) +according to which the first and third equations of the two ansätze are identical. For convenience and for the purpose +of being consistent with other literature on this topic, we introduce the notation +f(r) = h(r)e2δ(r) +(50) +and +˜δ(r1, r2) = +� r2 +r1 +4πGs q′(s)2 ds . +(51) +C. +Black hole spacetime characteristics +In §V we will start from the assumptions that there exists an event horizon at a certain radial coordinate value r = rh, +and that spacetime is asymptotically flat. It then follows that +δ(r) = −˜δ(rh, ∞) + ˜δ(rh, r), +F(r) = ln(1 + 2E(∞)) + 2˜δ(r, ∞). +(52) +In order to check for the existence of an event horizon, radial null geodesics in generalized Painlevé-Gullstrand +coordinates or generalized Schwarzschild coordinates need to be discussed. The requirement ds2|θ=θ0,φ=φ0 = 0 leads +to +−dt2 + (dr + vdt)2 +1 + 2E += 0 ⇔ dr +dt = ± +√ +1 + 2E − v +(53) +in generalized Painlevé-Gullstrand coordinates and +−fdt2 + 1 +hdr2 = 0 ⇔ dr +dt = ± +� +fh +(54) + +8 +in generalized Schwarzschild coordinates. The plus sign is associated with outward motion, while the minus sign +corresponds to inward motion. If dr +dt < 0 for both signs and every r < r0, then r0 marks an event horizon. More +precisely, it marks the locus of an apparent horizon. In a static (or more generally in a stationary) spacetime, the +apparent and the event horizon coincide. Generalized Schwarzschild coordinates are only valid on one side of the +event horizon, since they become singular at an event horizon (with at least either f = 0 or h = 0). In the vicinity of +an event horizon, h(r) ≪ 1, and the outward velocity of a massless particle may be approximated by +(dr +dt )out = +� +1 + 2E(r) − v(r) = +� +1 + 2E(r) − +� +2E(r) + 2Gm +r +(55) += +� +1 + 2E(r) − +� +2E(r) + 1 − h(r) = +h(r) +2 +� +1 + 2E(r) ++ O(h(r)2). +As a result, r = rh marks an event horizon if h(rh) = 0. Taking a look back at the Einstein equations in the generalized +Painlevé-Gullstrand ansatz (39), it can be observed that, although the coordinates are regular on the horizon, the +inclusion of a scalar field in Einstein gravity leads to singular behavior as the horizon is approached. This is manifest +in the third Einstein equation for both ansätze. +Of particular interest about solutions in q-theory is the distribution of energy density inside and outside the event +horizon, as well as the question of whether or not out solutions are singular at the origin. +To answer these questions it is convenient to define +−(Tq)t +t = V (r) + T(r), +V (r) = ρ(q(r)), +T(r) = 1 +2h(r)(q′(r))2 +(56) +such that the energy density into a potential part V and a kinetic part T. +The latter quantity may be deduced from the Kretschmann invariant +K(r) = RµνρσRµνρσ +(57) +as r → 0, where Rµνρσ is the Riemann curvature tensor. For generalized Schwarzschild coordinates it takes the +explicit form +Kq−theory(r) = 1 +r4 [4r4h2(r)(δ′(r))4 + 8r4h2(r)(δ′(r))2δ′′(r) + 4r4h2(r)(δ′′(r))2 ++ 8r2h2(r)(δ′(r))2 + r4(h′′(r))2 + (9r4(δ′(r))2 + 4r2)(h′(r))2 ++ 4(3r4h(r)(δ′(r))3 + 3r4h(r)δ′(r)δ′′(r) + 2r2h(r)δ′(r))h′(r) ++ 2(2r4h(r)(δ′(r))2 + 2r4h(r)δ′′(r) + 3r4δ′(r)h′(r))h′′(r) + 4(h(r) − 1)2]. +(58) +For Schwarzschild spacetime with h(r) = f(r) = 1 − 2GM +r +and Schwarzschild mass parameter M the Kretschmann +scalar reads +KSchwarz(r) = 48G2M 2 +r6 +. +(59) +Black hole spacetimes with minimally coupled scalar fields underlie a series of criteria to be fulfilled as dictated by +the black hole no-hair theorems. These criteria are the topic of the next chapter. From now on we will work in units +with G = 1. +IV. +Black hole no-hair theorems +In this section the BH no-hair theorems are explained, that restrict the set of allowed solutions of scalar hair black +holes (SHBH’s) in curved spacetime, which is the same class of solutions considered in this work. +The BH no-hair theorems, as discussed in [31, 35], assert the following: +1. In the absence of event horizons there exist no non-trivial, regular scalar soliton solutions, that satisfy the dom- +inant energy condition but violate the strong energy condition, at every point in asymptotically flat spacetimes. +2. In the presence of an event horizon in a static, spherically symmetric and asymptotically flat spacetime, no +non-trivial regular scalar-field solution exists outside the event horizon, if the dominant energy condition holds +but the strong energy condition is violated. + +9 +3. Any SHBH solution must necessarily have V (rh) < 0 where rh denotes the radial location of the event horizon, +and V is the potential energy density of the scalar field. The vicinity of the event horizon is enveloped by a +region of negative scalar-field energy density. +The strong energy condition is defined by the requirement that Tαβkαkβ ≥ 0, where Tαβ is the covariant energy +momentum tensor and kα is an arbitrary null vector field, and the requirement that −T α +βpβ must be a future +pointing causal vector field whenever pα is. The strong energy condition stipulates that for every timelike vector field +uα, the condition (Tαβ − 1 +2Tµνgµνgαβ)uαuβ ≥ 0 holds. +If in the first case, spherical symmetry is assumed with a positive scalar field potential, hence the dominant energy +condition does not need to be imposed in order to infer the absence of non-trivial scalar field solutions. From the +first two criteria, it is immediate that within the domain of outer communications (the region from the event horizon +to asymptotically flat infinity), the scalar field energy density must have negative values. The third criterion then +specifies where this region of negative energy density must be located. +This leaves as the only possibility for a non-trivial SHBH solution in the considered setup a q-field which asymptot- +ically relaxes to its equilibrium value but sweeps over field values corresponding to negative potential energy density +as the horizon is approached, for sure in the horizon proximity. +Both SHBH’s, and scalar solitons, have to fulfill an integral equation as a necessary condition for existence, as +derived from a scaling argument [31]. These conditions are written below in our notation convention. A necessary +condition for the existence of a scalar soliton (scalaron) in curved spacetime, in a non BH geometry is +� ∞ +0 +4πr2 exp (2δ(r)) +� +Eflat +kin (r) + 3V (r) +� +dr = 0 +(60) +It comprises the flat space kinetic energy density Eflat +kin (r) = 1 +2(q′(r))2, as well as the potential energy density V (r) = +ρ(q(r)). Analogously, a necessary condition for the existence of a SHBH solution is +� ∞ +rh +4πr2 exp(2δ(r)) +�2rh +r +� +1 − m(r) +r +� +− 1 +� +Eflat +kin (r) + +�2rh +r +− 3 +� +Vpot(r))dr = 0. +(61) +where rh is the radial coordinate of the event horizon. The fulfillment of this latter condition is taken as a tool to +fine-tune the shooting parameter q(rh) (introduced and discussed in section V), in finding SHBH solutions. In the +limit rh → 0 (60) is recovered. For future convenience we introduce the function +nhf(r) = +� r +rh +s2 exp(2δ(s)) +�2rh +s (1 − m(s) +s +) − 1 +� +Eflat +kin (s) + +�2rh +s +− 3 +� +V (s)) ds +(62) +which is then supposed to fulfill lim +r→∞ nhf(r) = 0. +V. +A representative SHBH solution for q-theory +The existence of SHBH solutions has been known for some time and was first considered numerically in [29] for a scalar +field minimally coupled to gravity in a double well scalar field interaction potential. Subsequently, these solutions +have been discussed in the framework of the of isolated horizons [36]. The latter formalism has been treated in detail +in [26] and references cited therein. The difference between this work and [29] is in the more restrictive potential +of q-theory. There are only two free parameters, the absolute scale provided by λ and the depth or separation of +the wells as parameterized by a. A shift in the q-field, accompanied by a corresponding change in the scalar energy +density function, does not lead to a further quantitative change in the solution as induced by the shift in the q-field +itself. It can be seen as fixed by the condition ρ(q = 0) = 0. The scalar potential in [29] has three free parameters +and allows for adjusting the positions of the wells, independently. Consequently, the solutions found and presented +within that work cannot be used for q-theory, since they lie outside the parameter space spanned by λ and a. In the +following, we replace the parameter a by the location of the minimum qeq = +1 +√ +3a of the shallow potential well. +We use Python for plotting and numerically solving the Einstein equations for the minimally coupled q-field. The +solver is non-adaptive and makes use of a refined (fourth order) Runge-Kutta method. Refined means that the grid +size close to the horizon is smaller than further away. This is due to the observation that the equations become +singular at the horizon. We will nevertheless employ a prescription of how to start “close” to the horizon. +This comes about since only one of the three boundary conditions of the differential equations (q(r = rbound), +q′(r = rbound) and m(r = rbound) at radial coordinate boundary position r = rbound) is free for an asymptotically +flat black hole spacetime regular at the horizon which we are searching for. +This freedom resides in the radial + +10 +Figure 2. A representative SHBH solution for q-theory outside the event horizon is shown for a horizon radius of rh = 1, q-field +potential parameters λ = 1 and qeq = 0.158, a grid size of 106 and shift parameter ϵ = 10−6. +coordinate location of the event horizon for a fixed scalar field potential function, as is the case for Schwarzschild +spacetime. The dependencies of the boundary values on the horizon radius are known precisely only at the horizon +except for one parameter, the “shooting” parameter. How the singular behavior at the horizon is circumvented is +explained further below, together with an account of numerical errors. The results have been confirmed by an adaptive +routine of the NDsolve-method in Mathematica, which is the most accurate. We will still stick with Python in the +following discussion, since the difference between the solving routines is negligibly small for tiny grid sizes (of the +order ∼ 105 − 106), as used in Python. +Both a discussion of how to numerically avoid the singularity in the third Einstein equation at the horizon, as well +as a discussion of numerical errors are lacking in [29], but will be provided in the following. We shall start with a +qualitative discussion of a representative solution in this section before focusing on the space of solutions in the next +section. +A. +Outside the event horizon +The method of finding solutions for q-theory, makes use of previously discovered solutions, insofar as they are used +as a starting point in an interpolation of solutions between the respective parameter spaces. For a horizon radius of +rh = 1, q-field potential parameters λ = 1 and qeq = 0.158, a grid size of 106 and shift parameter ϵ = 10−6 (to be +introduced further below) the solution is shown in Fig. (2) for the region outside the event horizon (also termed the +domain of outer communications). It is in qualitative agreement with that of [29]. +The q-field starts with a negative value such that q0 < q(r = rh) < 0, which is equivalent to V (r) = ρ(q(r)) < 0, as +is required by the no-hair theorems. More specifically we have qmin < q(r = rh) < 0. The q-field starts in the right +half of the deep potential well shown in Fig. (1) with the characteristic points of the double well potential named +as discussed in section I. It increases into the positive energy density domain, passes the local maximum and relaxes +asymptotically to the equilibrium value qeq. +The metric function m(r), termed mass function, starts from the horizon with m(rh) = 1 +2rh, initially decreases as + +q-field +potential and kinetic energy densities versus q' +0.02 +q (r) +0.1 - +Vr vs. T(r) vs. q'(r) +0.D1 +T(r) +.. +50V(r) +0.D +q(r) +0.00 +0.02 +0.2 +0.03 +0.3 +15 +25 +0.04 +0.0 +0.5 +1b +20 +3.D +0.0 +S0 +15 +2D +25 +3.D +logia(r/rh? +logia(r/rh? +evolution of the mass function +metric functions +3.D +h(r) +25 +f(r) +5 - +(μ)g +20 +m(r) +0 +5- +LD + OT- +0.5 +15 + +0.D +0.0 +0.5 +1D +15 +2D +25 +3.D +0.D +15 +20 +25 +3.D +logia(r/rh? +logia(rfrh? +relabive ermor of the q-field solution +no-hair function +2.00 +0 +f.n.d.,5-p.s.m for g +f.n.d.,5-p.s.m for q +175 +-2 +Jogia of the relative +-4 +f.n.d.,5-p.s.m for m +150 +6 +125 +1LDo +oT- +0.75 +-12 +0.50 - +-14 +0.25 +-16 +00 +0.2 +0.4 +0.6 +0.B +12 +16 +0.DO +14 +0.D +0.5 +15 +20 +25 +3.D +logia(r/rh? +logia(r/rh?11 +ρ(q) < 0 until reaching a global minimum. It is located at a radial coordinate value shortly advancing that of q = 0, +since the kinetic energy density T(r) = 1 +2h(r)(q′(r))2 is positive everywhere outside the horizon. The mass function +increases until finally approaching its asymptotic value, the ADM mass of the spacetime, from below. The potential +and kinetic energy densities, as introduced in equation (56), are shown together with the derivative of the q-field on +the upper right. The potential energy density is seen to be of minor size as compared to the kinetic part. +Further shown are the metric functions in generalized Schwarzschild coordinates in the central plot on the right +hand side. As is required by asymptotic flatness we have the limits +lim +r→∞ f(r) = 1, +lim +r→∞ h(r) = 1, +lim +r→∞ δ(r) = 0, +(63) +while limr→∞ E(r) remains a free parameter for the generalized Painlevé-Gullstrand coordinates. This free parameter +is mostly irrelevant for the discussion of SHBH solutions and only of interest for test particle motion. Its variation +will be discussed in Appendix A but is of no crucial importance elsewhere in the discussion. +The lowermost plot on the left shows the error sizes. We calculate the relative error of the solution by taking the first +numerical derivative (f.n.d.) of q, q′ and m and comparing it with the solution as given by q′, q′′ and m′ and obtained +from the non-adaptive, refined (fourth order) Runge-Kutta algorithm. The first numerical derivative is calculated +using a central 5-point stencil method (5-p.s.m.). On a formal level this method approximates the derivative of a +five times differentiable function accurately up to O((∆r)4)-corrections where ∆r represents the grid spacing. The +grid size is 106. The first four peaks mark the radial coordinate values where the grid size changes abruptly and are +artificial, since the 5-p.s.m. formula we employed is based on equal grid size spacing. The final peaks for the relative +errors are due to the appearance of the local extrema of q′ as well as m. Apart from these points the relative error +can be seen to be smaller than 10−7 throughout. +The lowermost plot on the right shows the function nhf(r) introduced in the last section in equation (62) which +indeed fulfills limr→∞ nhf(r) = 0. This latter property was taken to fine-tune the shooting parameter q(rh), though, +which will be discussed shortly. +It can be seen that the plot of the relative errors do not reach as far out as the other plots. An asymptotic analysis +of the third Einstein equation with δq(r) = q(r) − qeq yields the linear approximation +(δq)′′ = −2(δq)′ +r ++ ρ′ = −2(δq)′ +r ++ λ +2aδq , +(64) +which has the solution +δq(r) = c1 +1 +r exp +� +− +� +λ +2ar +� ++ c2 +1 +r exp +� ++ +� +λ +2ar +� +. +(65) +The asymptotic behavior of the relevant functions can be deduced from the first Einstein equation and the condition +limr→∞ m(r) < ∞. A first order Taylor expansion of the potential energy density and its derivatives around q = qeq is +well justified for δq ≪ +1 +√a. The approximation of the third Einstein equation is well justified for 2m(r) +r +, 4πr2ρ(q(r)) ≪ +1. Taken together we then obtain (64). The asymptotic solution has an exponentially growing and an exponentially +depleting part. +In the search of a solution it is found that when the q-field and the mass function approach their asymptotic +values they leave them again after some critical value. This can not be avoided and is an artifact of the numerical +approximation. It induces a non-vanishing coefficient c2 and therefore exponential growth due to numerical uncer- +tainty. Therefore the exponentially depleting part is fitted onto the functions q, q′ and m after they tend to change +very slowly by approaching their asymptotic values. This fitting of the free parameter c1 takes place at those radial +coordinate values where the relative error plots end. At this point δq(r) +qeq , m(∞)−m(r) +m(∞) +, nhf(r) ≪ 1 hold. +We now turn to the near horizon region. We search for solutions regular at the horizon by imposing limr→rh q′′(r) < +∞. The third Einstein equation as well as the assumptions of the existence of an event horizon and of asymptotic +flatness then reduce the freedom of the boundary conditions of q, q′ and m to one parameter. This parameter may +be declared as the horizon radius. +After fixing the scalar field potential parameters and thereby the theory, the +space of solutions is one dimensional and parameterized by rh as is Schwarzschild spacetime. Up to one degree of +freedom, the boundary values are known at the horizon. The event horizon condition h(r) = 0 on the one hand and +limr→rh q′′(r) < ∞ on the other hand imply +m(rh) = rh +2 , +q′(rh) = rh +dρ +dq +(q(rh)) +(1 − 8πr2 +hρ(q(rh))) . +(66) + +12 +Figure 3. A representative SHBH solution for q-theory inside the event horizon is shown for a horizon radius of rh = 1, q-field +potential parameters λ = 1 and qeq = 0.158, a grid size of 106 and shift parameter ϵ = 10−6. +The remaining freedom then resides in the value of q(rh). It is adjusted so as to yield ("shoot" towards) an asymptot- +ically flat solution and therefore termed shooting parameter. By choosing it more and more accurately the approach +of q, q′ and m to their asymptotic values may be improved. This suggests that there exists exactly one shooting pa- +rameter which is appropriate for ensuring asymptotic flatness. We can only approach it to within a certain numerical +accuracy. As soon as the solutions to the first and third Einstein equations are obtained, the horizon values of E(r) +and δ(r) may be deduced by integration of the second Einstein equation. They are given by +E(rh) = ((1 + 2E(∞)) exp( +� ∞ +rh +r(q′(r))2dr) − 1)/2, +δ(rh) = −˜δ(rh, ∞). +(67) +The singular behaviour of the third Einstein equation at r = rh is avoided by the prescription r → r(1 + iϵ). We call +ϵ the shift parameter and choose it such that 0 < ϵ ≪ 1. The quantities plotted in Fig. (2) are then understood as +the real parts of the functions of the (total, complex valued) solution. The accuracy of the numerical calculations due +to finiteness of grid size as well as shift parameter is analyzed in Appendix B. +B. +Inside the event horizon +In extension of the plots of Fig. 2, the solution for a horizon radius of rh = 1, q-field potential parameters λ = 1 +and qeq = 0.158, a grid size of 106 and shift parameter ϵ = 10−6 is shown in Fig. 3 for the region inside the event +horizon. In distinction to the corresponding figure for the outside region the lowermost plot on the right hand side +highlights the energy function. The relative error size remains below 10−5 for all of the functions q′, q′′ and m′. The +peaks mark again those radial coordinate values where the grid size changes abruptly. Close to the event horizon it +has been chosen smaller as in the case of the region outside the event horizon. Of interest is the behaviour in the limit +where the radial coordinate tends to zero. The solution is found to be singular in the q-field in this limit. The mass +function, in contrary, tends to a constant value of about limr→0 m(r) = 0.503. An expansion of the third Einstein + +q-field +potential and kinetic energy densities versus q +0.19B +40 +0.200 +V(r) vs. T(r vs. q'(r) +log 1o(T(r) +log 1o(V(r) +0.202 +因 +0.204 +0.206 +0.20B +10 +.210 +5 +4 +-2 +L- +4 +-2 +0 +logia(r/rh? +Iogia(rfrh) +evolution of the mass function +metric-functions +0.504 +log 1o(h(r) +EOS'0 +4 +2 +0.502 +(sjuu +0 +0.501 +azis +0.50 +4 +0.499 +6 ++ 60 +5 +-4 +E- +-2 +-1 +4 +E- +-2 +-1 +logia(r/rh? +logia(rfrn) +relabive ermor of the q-field solution +energy function +0 +fogia of the relative error +f.n.d.,5-p.s.m. for q +2 +f.n.d.,5-p.s.m. for q +f.n.d.,5-p.s.m. for m +Jogia(E(r) - E(rh)) +6 +9- +ot- +-12 +714 +-16 +-3 +4 +E- +-2 +-1 +- +logia(r/rh? +Iogia(rfrh?13 +Figure 4. The Kretschmann invariant for a representative SHBH solution for q-theory inside and outside the event horizon is +shown for a horizon radius of rh = 1, q-field potential parameters λ = 1 and qeq = 0.158, a grid size of 106 and shift parameter +ϵ = 10−6. +equation for r ≪ 1 yields the approximate equation +q′′ = −q′ +r +(68) +with solution +q(r) = d1 log10(r) + d2. +(69) +It is found that d1 = 0.0010±0.0001 and d2 = −0.2010±0.0001 by a straight line fit. This implies the approximations +F(r) ≈ ln(1 + 2E(rh)) + 8π +� rh +r0 +r(q′(r))2dr + 8πd2 +1(ln(r0) − ln(r)) +(70) += C + 8πd2 +1ln(1 +r ) +⇔ E(r) ≈ exp(C) +2 +r−8πd2 +1 − 1 +2 +δ(r) ≈ δ(rh) − +� rh +r0 +4πr(q′(r))2dr − 4πd2 +1(ln(r0) − ln(r)) +(71) += D − 4πd2 +1ln(1 +r ) ⇔ exp(2δ(r)) ≈ exp(2D)r8πd2 +1, +h(r) ≈ 1 − 2m(0) +r +(72) +for r < r0 ≪ 1 with for the further discussion irrelevant constants C and D. Consequently the limits +lim +r→0 q(r) = −∞, +lim +r→0 q′(r) = ∞ +(73) +for the q-field and its derivative and +lim +r→0 E(r) = ∞, +lim +r→0 h(r) = −∞, +lim +r→0 δ(r) = −∞ +(74) +for the metric functions follow. Since d1 ≪ +1 +√ +8π, E(r) and exp(2δ(r)) vary very slowly as compared to h(r). Therefore +the behavior of the metric as r → 0 will asymptote that of Schwarzschild spacetime. Especially, very close to the +center the energy density will change sign again and becomes positive. The metric can not be continued to the radial +coordinate origin. There is a curvature singularity in the limit r → 0 as is well known for Schwarzschild spacetime. +The Kretschmann invariants for q-theory and Schwarzschild spacetime are shown in Fig. (4). While they differ visibly +asymptotically far from the horizon where both tend to zero, they show the same +1 +r6 -divergence as r → 0. +After discussing one solution in detail we proceed with a local scan of the space of solutions around that just +presented. Both grid size and shift parameter will no longer be mentioned from now on and chosen to be very large +in the former case and negligibly small in the latter as has been done within this section. + +Kretschmann invariant +30 +K5chaz(r) +25 +21 +((u)xjDTbo) +15 +1 +5 - + 5- +-10 +E- +-2 +-1 +1 +fogia(rfrh?14 +VI. +The parameter space of SHBH solutions for q-theory +The qualitative features of the representative solution presented in the previous section are common to all SHBH +solutions in q-theory. The space of solutions is parametrized by the horizon radius rh as well as the q-field potential +parameters λ and qeq (or as well a). In order to understand the differences between solutions we perform a local +scan around the representative solution in the space of solutions and extract different quantities for each individual +solution, in part in analogy with [29, 36]. +The ADM mass of a (static and spherically symmetric) solution is given by +MADM(rh, λ, qeq) = lim +r→∞ mrh,λ,qeq(r). +(75) +We decompose it into two contributions +MADM(rh, λ, qeq) = M Schwarz +ADM +(rh) + Mhair(rh, λ, qeq). +(76) +The first contribution is the mass parameter of a Schwarzschild black hole of horizon radius rh, M Schwarz +ADM +(rh) = rh +2 . +It is independent of the scalar field potential parameters, as it describes a (static and spherically symmetric) black +hole in vacuum. The dressing by the non-constant q-field outside the event horizon yields the non-vanishing additional +contribution +Mhair(rh, λ, qeq) = − +� ∞ +rh +4πr2T t +tdr +(77) += +� ∞ +rh +4πr2(ρ(q(r)) + 1 +2h(r)(q′(r))2)dr. +(78) +Further quantities of interest are the shooting parameter as well as the radial coordinate location relative to rh and +absolute depth of the global minimum of the mass function as functions of the parameters of the space of solutions. +The relative radial coordinate location of the global minimum of the mass function is important insofar as it marks +the region where both the q-field and the mass function vary significantly. +A. +Dependence on the horizon radius +Fig. 5 illustrates the characteristic quantities introduced above for a horizon radius range 10−3 < rh < 103 and scalar +field potential parameters coincident with those of the representative solution presented in section V. The lower limit +has been chosen such that the asymptotic dependence on rh is visible. The upper bound is due to lack of precision of +fitting the asymptotic part of the q-field and mass function. The asymptotic plateau becomes difficult to identify, since +both q-field and mass function behave less and less smooth in the fitting region. The asymptotics for large horizon +radii seem to be deducible from the plots as well, though. We will assume that the plots show the true asymptotics +in both extreme situations rh ≪ 1 and rh ≫ 1. +We may then draw the following conclusions: +1) The shooting parameter qshoot(rh) = q(rh) varies significantly. Its asymptotic values are limrh→0 qshoot(rh) = +0.913qmin and limrh→∞ qshoot(rh) = 0.500qmin, respectively. +2) For rh ≪ 1 the ADM mass and scalar hair mass converge towards the value +limrh→0 log10(MADM(rh)), log10(Mscalar hair(rh)) = 0.817. For rh ≫ 1 both ADM mass and scalar hair mass +increase linearly with the horizon radius and may be parameterized by +log10(MADM(rh)) = c1 log10(rh) + c2 +(79) +log10(Mscalar hair(rh)) = c3 log10(rh) + c4 +(80) +with c1 = 1.0031 ± 0.0001, c2 = −0.1985 ± 0.0001, c3 = 1.0136 ± 0.0001 and c4 = −0.8747 ± 0.0001 obtained +by a straight line fit. The q-theory ADM mass is a monotonically increasing function of the horizon radius and +everywhere larger than the corresponding Schwarzschild spacetime ADM mass (which coincides with the mass +parameter). +3) The region outside the event horizon where the q-field and mass function vary significantly approaches the event +horizon relative to its size for rh ≫ 1. So it seems that in this regime the q-field behaves non-trivially only just + +15 +Figure 5. The variation of several characteristic quantities of SHBH solutions for q-theory with respect to the horizon radius +rh is shown for the q-field potential parameters λ = 1 and qeq = 0.158. These quantities include shooting parameter, ADM +mass, scalar-hair mass, relative location of the global minimum of the mass function as well as the absolute value of the global +minimum of the mass function. +outside the horizon and relaxes to its equilibrium value very quickly in the near horizon regime. This region of +significant change does not approach the horizon infinitely close but relaxes to the value +limrh→∞ rmin(rh)/rh = 1 + e−0.916 = 1.400. +In the opposite limit rh ≪ 1 the region of significant change of both q-field and mass function gets pushed +further and further away from the horizon region. This implies that for the very limit rh → 0 no scalar soliton +(scalaron) exists (contrary to the conclusions drawn in [29]). Rather the q-field remains constant outside the +event horizon with a value of qshoot(0) = limrh→0 qshooot(rh) = 0.913qmin. The corresponding energy density is +negative resulting in a Schwarzschild-anti de Sitter spacetime with cosmological constant Λ = 8πGρ(qshoot(0)). +4) The size of the global minimum of the mass function converges to a constant for rh ≪ 1 with value +limrh→0 log10(−min(m)(rh)) = 1.125. +For rh ≫ 1 it decreases linearly and faster than the negative ADM +mass. It may be parametrized by +log10((−min(m)(rh)) = d1 log10(rh) + d2 +(81) +with d1 = 3.0268 ± 0.0001 and d2 = −3.5790 ± 0.0002. +B. +Dependence on the parameters of the scalar field potential +We now consider changes in the scalar field potential parameters. The effect of a variation of the scalar field potential +parameter λ for a horizon radius range +1 +10 ≤ rh ≤ 10 and fixed scalar field potential parameter qeq = 0.158 for the +characteristic functions presented previously in Fig. (5) is visualized in Fig. (6). The radial parameters and masses +have been rescaled in a particular way following [29]. It can be seen that the rescaled quantities seem to depend only + +value of the shooting parameter +ADM-massoftheblackholesolution +0.9 +2 +MSchwa2 +0.B +1 +0 +0.7 +-1 +0.6 +-2 +-3 +0.5 +-2 +-1 +0 +i +2 +3 +E- +-2 +-1 +0 +i +2 +3 +Iogia(rh) +fogia(rh) +relative location of the global minmum of m +global minimum of m(rh) +log 1o(rmin/rn) +log 1o(rmin rn)/rn) +Jogia(Fmin/rh) vs. Jogia(Fmin - Fh)/rh) +5 - +3 +logia(-min(m(rh))) +3 +1 +2 +0 +1 +-1 +0 1 +E- +-2 +-1 +0 +1 +2 +3 +E- +-2 +-1 +0 +1 +2 +3 +logia(rh? +Iogia(rh?16 +Figure 6. The variation of several characteristic quantities of SHBH solutions for q-theory with respect to the scalar field +potential parameter λ is shown for a horizon radius of rh = 1 and remaining q-field potential parameter qeq = 0.158. These +quantities include shooting parameter, ADM mass, scalar-hair mass, relative location of the global minimum of the mass +function as well as absolute value of the global minimum of the mass function. +on two instead of three parameters. To be more precise about the parameter dependencies define +�rh = +√ +λrh, +� +rmin = +√ +λrmin, +ˆ +MADM = +√ +λMADM, +� +min(m) = +√ +λmin(m). +(82) +It then follows that +� +rmin = 1, +ˆ +MADM = ˆ +MADM( �rh, qeq), +� +min(m) = +� +min(m)( �rh, qeq). +(83) +The effect of a variation of the scalar field potential parameter qeq for a horizon radius range +1 +10 ≤ rh ≤ 10 and fixed +scalar field potential parameter λ = 1 for the characteristic functions presented previously is visualized in Fig. (7). +As qeq increases, the potential wells of the scalar field potential recede from each other and become more pronounced. +All quantities (with minor exceptions) seem to be monotonically growing (monotonically decreasing in the case of +the negative valued quantity min(m)) with qeq. The dependence of the different characteristic quantities of SHBH +solutions on qeq is not so easily deducible. Nevertheless it seems that, with exception of large horizon radius values, +the dependence on qeq may be factorized from that on �rh. +The local scan of the parameter space of solutions has revealed that the space of solutions is effectively only two +dimensional with the dependence on the two parameters factorizing by a monotonically increasing function of qeq to +good approximation within the represented parameter space area. +The topic of the following section will be a stability analysis of SHBH solutions due to perturbations of the q-field +solution. +VII. +Stability of the SHBH solutions +An important question to ask is whether SHBHs are stable. Do perturbation modes of the q-field and the metric +which grow exponentially in the SHBH spacetime of our q-theory model exist? The question of classical instability + +value of the shooting parameter +ADM-massoftheblackholesolution +0.95 +14 +入=4 + ^=1 +13 + ^=1 +0.50 +12 +0.B5 +(4)o%b +0.B0 +1f +●入=4 +^=1 +0.75 + 入=1 +8: + 入= +0.70 - +2.0 +1.5 +-1.0 +0.5 +0.0 +0.5 +1'D +2.0 +1.5 +1.0 +0.5 +0.D +0.5 +1D +fogiatVArn) +logia(VArn) +relative location of the global minimum of m +globalminimum ofm. +●^=4 +-2 +●^=4 +225 ++ ^=1 +← ^=1 +入=1 +200 +4 - + 入= 4 +175 +→ =2 +6 +150 +125 +1D0 +-10 +0.75 +0.50 +-12 +0.25 +1.00 +0.50 +0.25 +o.bo +0.25 +0.50 +0.75 +1D0 +2.0 +1.5 +1.0 +d.5 +0.D +0.5 +1'D +Iogia(rh? +ogia( VArh?17 +Figure 7. The variation of several characteristic quantities of SHBH solutions for q-theory with respect to the scalar field +potential parameter qeq is shown for a horizon radius of rh = 1 and remaining q-field potential parameter λ = 1. +These +quantities include shooting parameter, ADM mass, scalar-hair mass, relative location of the global minimum of the mass +function as well as absolute value of the global minimum of the mass function. +has been discussed for spherically symmetric, time dependent perturbations of the metric and scalar field (here the +q-field) in [29]. Following a similar notation to that introduced in [29] (see Eqs. (17)-(19) therein) we define the +s-wave perturbations by +˜q(t, r) = q(r) + δq(r, t) , +(84) +˜f(r, t) = f(r)(1 − h1(r, t)) , +(85) +˜h(r, t) = h(r)(1 − h2(r, t)) +(86) +in generalized Schwarzschild coordinates. The functions q(r), f(r) and h(r) represent the unperturbed solutions for +q-theory in the generalized Schwarzschild coordinates, while δq(r, t), h1(r, t) and h2(r, t) denote small perturbations +to the non-perturbed solution. Note that in Eq. (86) h2 is defined as a small correction to h with grr = 1/h in (41), +whereas in Eq. (19) in [29] it is defined as a small correction to grr directly with a different sign. The two relations +are equivalent, as is easily checked by substituting h(1 − h2) for h in grr = 1/h and then expanding. +After linearization of the Einstein equations (45-47) it can be shown that [29] +∂rh1(r, t) = ∂rh2(r, t) − 16πr ( ∂r q(r) ) (∂r δq(r)) , +h2(r, t) = 8πr ( ∂r q(r) ) δq(r). +(87) +As such the metric perturbations are expressible in terms of the q-field perturbation. The latter may be determined +by solving a one-dimensional Schrödinger equation of the form [29] +� 1 +2m(−i∂r∗)2 + V ∗ +eff(r∗) +� +ψ(r∗) = +1 +2m(i∂t)2ψ(r∗), +dr∗(r) +dr += exp(−δ(r)) +h(r) +(88) +where ψ(r∗(r)) ≡ rδq(r) and r∗ is the “tortoise” coordinate. The scalar field mass m is obtained as follows. Insert +˜q(t, r) into the action functional and expand in δq(t, r) around q(r). Far away from the event horizon when q(r) is +close to its equilibrium value qeq, the kinetic cross terms as well as the linear potential term in δq are negligible and + +value of the shooting parameter +ADM-massoftheblackholesolution +22.5 +0.98 +中 +24.0 +O qeg = 0.223 +0.96 +★ qeg = 0.193 +17.5 + qeg = 0.173 ++ qeg = 0.158 +60 +15.0 +(4.jb +(usjnavw + qeg = 0.129 ++qeg=0.112 +12.5 +0.92 +14.0 +qeg = 0.223 +0.90 +qeg = 0.193 + qeo = 0.173 +7.5 +qeg = 0.158 +0.8 - +★ qeg =0.129 +5.D +→ q= 0.112 +1.00 +0.50 +0.25 +0.00 +0.25 +0.50 +0.75 +LDo +1.00 +0.50 +0.25 +0.0 +0.25 +0.50 +0.75 +LDO +fogia(rh) +fogia(rh) +relative location of the global minimum of m +globalminimumofm. +-qeg=0.223 +0+ +25 ++ qe = 0.193 + qeg = 0.173 +→ qea = 0.158 +★ qeg = 0.129 +DOT- +2D +← qeg= 0.112 +Jogia(rmin/rh) +O qeg =0.223 +tusjujuu + Qeg = 0.193 +200 + qeg = 0.173 +15 + qea = 0.158 +★ qea = 0.129 +← qeg=0.112 +300 +1D +0.5 +400 +1.00 +0.50 +.25 +0.00 +0.25 +0.50 +0.75 +LDO +1.00 +0.50 +±.25 +0.DO +0.25 +0.50 +0.75 +LDO +fogia(rh) +Iogia(rh?18 +the action with dynamical field (perturbation) δq has a proper kinetic term with at least quadratic potential terms. +The quadratic term yields m in the same way as does the real Klein Gordon action with (self-)interactions. The so +obtained value for m reads +m = +� +d2ρ(q) +dq2 +������ +q=qeq += +� +3 +2λq2eq. +(89) +It will formally not be needed in the following but has been introduced in order to provide a properly normalized +Schrödinger problem. The effective potential reads [29] (note that the expression for Veff in this work is defined with +a factor of +1 +2m compared to Eq. (22) in [29]) +Veff(r∗(r)) = +1 +2mh(r) exp(2δ(r))[h(r) +r +(δ′(r) + h′(r) +h +) +− 8πrh(r)(q′(r))2(δ′(r) + h′(r) +h(r) + 1 +r ) ++ 16πrq′(r)dρ +dq (q(r)) + d2ρ +dq2 (q(r))]. +(90) +The separation ansatz ψ(r∗(r)) = ξ (r∗(r)) exp +� +±i +√ +2mEt +� +leads to the following stationary Schrödinger equation +for ξ (r∗(r)) +� 1 +2m(−i∂r∗)2 + V ∗ +eff(r∗) +� +ξ(r∗) = Eξ(r∗) +(91) +with energy eigenvalue E. A sufficient condition for static, spherically symmetric configurations to be unstable [37] +is that the differential operator on the left-hand side of (91), is negative in the Hilbert space L2(M), where M is the +spacetime manifold on which the metric is defined. Accordingly, a sufficient condition for unstable solutions is the +existence of a bound state E < 0 in the Schrödinger problem. In the same manner the existence of bound states +that correspond to E < 0 in the Schrödinger problem with potential Veff is equivalent to the existence of unstable +perturbations of q-theory solutions. The exponential factor of the perturbation becomes of order unity after a time +of order +τ = +1 +√−2mEmin +. +(92) +The subscript min on E signifies the lowest bound state energy which is of most importance for giving the scale of +τ (for a discrete and finite bound state spectrum, as is the case here). This time can be seen as the lifetime of the +SHBH in the presence of these perturbations. +In order to determine the value Emin, the lowest eigenvalue of bound state energies of the effective Schrödinger +equation (91), we solve the eigenvalue problem numerically in the variable r for horizon radii in the range 1 ≤ rh ≤ 1000 +by approximating the equation on a grid of finite size. The differential operator is approximated by a central point +stencil method accurate to fourth order of the grid spacing. The Runge Kutta solver shares this level of accuracy. +We employ vanishing boundary conditions for the eigenfunctions in the limits r → 0 as well as r → ∞. To get an +impression of the shape of the eigenfunction to the eigenvalue Emin, we replace the effective potential, illustrated +in Fig. (8) for different horizon radii, by an auxiliary potential. This auxiliary potential is a parabola fitted to the +negative potential well of the effective potential in the region shown in red in the figure (where Veff < 0). Outside +it is set to zero. This yields a finite depth harmonic oscillator potential. The solutions of the wave equation may be +determined exactly in this auxiliary potential. Finding the bound state energy eigenvalues is then identical to the +quantum harmonic oscillator except for them being finite in amount, while the eigenfunctions are compromised due +to the vanishing of the potential. Nevertheless, as is in concordance with the results in [29], the expectation is that +the lowest energy bound state eigenfunction is close to a Gaussian in shape which is the exact eigenfunction in this +case for the quantum harmonic oscillator. That this expectation is also fulfilled in our case is illustrated in Fig. (9). +We plot the normalized eigenfunctions corresponding to the lowest bound state energy eigenvalues for several horizon +radii and scalar field potential parameters λ = 1 and qeq = 0.158. A comparison with Fig. (8) shows that the peaks +of the eigenfunctions are situated almost exactly at the minimum of the effective Schrödinger potential. It becomes +apparent here and has been observed that for increasing horizon radii the Gaussians loose their shape and tend to +disappear. In this regime the lowest eigenvalues approach zero very quickly from below. This inspires the conclusion + +19 +Figure 8. The effective potential of the scalar perturbation mode ξ(r∗) is shown for different horizon radii and scalar field +potential parameters λ = 1 and qeq = 0.158. It comprises a negative valued well where the wave function of bound states are +predominantly located. The well is highlighted in red. A zoom makes this region visible for the horizon radii rh = 100 and +rh = 1000. +Figure 9. Normalized eigenfunction solutions to the lowest bound state eigenvalue for the scalar perturbation mode ξ(r∗) are +shown for different horizon radii and scalar field potential parameters λ = 1 and qeq = 0.158. They are almost perfect Gaussians +in shape which is the case for the quantum harmonic oscillator. +that large SHBHs are indeed stable and opposes that found in [29]. In Fig. (10) the lifetime τ = τ(rh) of SHBHs +as a function of the horizon radius and scalar field potential parameters λ = 1 and qeq = 0.158 is shown. We fit the +obtained lifetimes τ = τ(rh) corresponding to the solutions for the lowest eigenvalues e = e(rh) to the function +f(rh) = a · (log10(rh))ntan(c · log10(rh) − b) + d +(93) +parameterized by the scale parameters a and c, the horizontal shift parameter b, the vertical shift parameter d and +the power parameter n with optimal parameters popt and covariance matrix pcov given by +popt = +� +� +� +� +� +a +b +c +d +n +� +� +� +� +� = +� +� +� +� +� +1.53 +1.14 +1.73 +5.31 +1.23 +� +� +� +� +� , +pcov = +� +� +� +� +� +0.044 +0.082 0.053 0.037 −0.007 +0.081 +0.179 0.115 0.085 +0.014 +0.053 +0.115 0.074 0.055 +0.009 +0.037 +0.085 0.055 0.043 +0.010 +−0.007 0.014 0.009 0.010 +0.028 +� +� +� +� +� . +(94) +The lifetime then becomes infinite at the finite horizon radius r0 +h = π+2b +2c +and the SHBHs are therefore stable beyond +the threshold r0 +h for the chosen scalar field potential parameters. + +effective Schrodinger potential for the q-field perturbation +0.125 +0.002 +(u)"PA +000'0 +0.05 +0.002 +0.025 +0.105 +0.110 +0.115 +log 1o(r/rn) +0.00 +rn = 1 +In = 5 +.025 +F = 10 +Tn = 25 +.050 +rn = 100 +rn= 1000 +0.0 +0.5 +1D +15 +2D +25 +3.D +fogia(rtrh)eigenfunctionwith corresponding eigenvalue E-E +0.6 + T = 1 +- rh= 5 +0.5 +- rn = 25 +. +0.4 +!! +tts)* +*3 +0.3 +!! +:1 +0.2 +0.1 +0.D . +0.D +0.5 +LD +15 +20 +25 +3.D +logia(r/rn?20 +Figure 10. The lifetime of SHBHs due to s-wave q-field and metric perturbations represented by the scalar perturbation mode +ξ(r∗) is shown as a function of the horizon radius with scalar field potential parameters λ = 1 and qeq = 0.158. The lifetime is +finite for small SHBHs below a certain radius or mass threshold value, whereas SHBHs are stable beyond this threshold. The +threshold is highlighted in red. +The stability analysis has revealed that SHBH are unstable due to classical s-wave perturbations of both the q- +field and the metric below a certain size or equivalently mass threshold and stable beyond (at least for the chosen +parameters). +VIII. +Conclusion +In the present paper we consider q-theory comprising a scalar field q minimally coupled to gravity. +The q-field +describes a dynamical gravitating vacuum. According to the estimates proposed in [24], this theory may contain BH +solutions that resemble that of a gravastar, i.e. a configuration with energy concentrated inside a thin spherical shell. +Contrary to the conventional gravastar, the state proposed in [24] contains an event horizon. Using direct numerical +calculations, we confirm the existence of similar BH configurations, with some reservations, though. Namely, inside +the event horizon space - time resembles the interior of the Schwarzschild BH, and does not contain the de Sitter - like +domain. Besides, the mentioned thin shell is located outside the event horizon, not inside. Furthermore, there should +exist a region in space, where the energy density is negative. This is required to satisfy the “no-hair” theorems. As +a result, the thin shell situated just outside of the horizon contains both a piece of negative energy and a piece with +positive energy density. The integration of the energy density inside the shell yields a total positive energy resulting +in the ADM mass perceived by the distant observer. +According to Eq. (30) the energy density is proportional to the derivative of the mass function m(r). The latter +function is represented within Fig. 2. One can see that for the given example solution, the spherical shell of finite +thickness exists and is situated just outside the horizon. Inside this shell the essential variation of m(r) is localized. +Close to but outside the event horizon, the energy density is negative, then it passes through zero, and becomes +positive in the second piece of the shell. The shell ends where m(r) exponentially approaches its asymptotic value, +the constant that represents the black hole mass seen by the infinitely distant observer. +The results of section VI demonstrate that the spherical shell approaches the horizon relative to its size when the +horizon radius is increased. In the limit of very large BHs, virtually the entire energy due to the q-field is localized +in the thin shell situated outside the horizon and close to it. It does not approach the event horizon ifinitely close, +but stops, such that the energy density sign transition is positioned around 7 +5rh, where rh denotes the event horizon +radius. +A stability analysis with respect to the s-wave metric and q-field perturbations shows that the BH solutions of the +type considered in the present paper may be classically unstable . However, the corresponding configurations are +stable for sufficiently large BHs. We therefore claim that stable heavy SHBHs do exist. +We do not discuss here questions related to the stability of the considered configurations on the quantum level. +This issue remains outside of the scope of the present paper. +In conclusion, in this work we confirm the supposition of [24] about the existence of BH solutions in q-theory that +look similar to gravastars. These states escape the conditions of the no-hair theorem, due to the region in space with +negative energy density. At spatial infinity these solutions approach the Schwarzschild solution, but differ from it + +lifetime of aSHBH duetoperturbations +120 +fit +threshold +data +140 +8+ +40 +24 +0.+ +0.D +0.5 +1b +15 +2D +25 +3.D +logiatrh?21 +Figure 11. The effect of different choices of E(∞) on the SHBH solution for q-theory outside the event horizon is shown for a +horizon radius of rh = 1, q-field potential parameters λ = 1 and qeq = 0.158, a grid size of 106 and shift parameter ϵ = 10−6. +The inward and outward massless particle velocities are shown for both Schwarzschild spacetime and q-theory spacetime with +Schwarzschild mass parameter M = mq−theory(rh). +essentially close to the horizon. Inside the horizon, the vacuum density is negative and changes sign very close to the +center. The singularity of curvature at r = 0 is the same as that of the Schwarzschild solution. +The authors are grateful to G.E.Volovik for the proposition to consider the given problem, and for useful discussions +during the initial stage of the work. +Appendix A. Test particle characteristics. +We discuss here the properties of test particles contained in the free parameter E(∞) = limr→∞ E(r). It is related +to the total energy per unit rest mass of a test particle e by E(∞) = (e2 − 1)/2 which moves towards the SHBH +horizon starting at infinity with initial velocity v(∞) = +� +2E(∞) where v(∞) = limr→∞ v(r). Different values for +E(∞) correspond to different initial kinetic energies of a test particle. The choice E(∞) = 0 corresponds to a particle +at rest, while E(∞) > 0 corresponds to a particle initially moving towards the SHBH. E(∞) < 0 is not possible, since +then v(r) would become imaginary while approaching asymptotically flat infinity. The motivation for the introduction +of generalized Painlevé-Gullstrand coordinates as well as their relation to test particle motion are presented in more +detail in [34]. +The numerical solution presented in Fig. 2 for different initial values of the free parameter E(∞) is shown in Fig. 11. +The function E(r) is monotonically decreasing with r as is v(r) in q-theory, whereas E(r) is constant for Schwarzschild +spacetime while v(r) is also decreasing. This is in concordance with the increase of the kinetic energy of a test particle +as it moves towards the event horizon of the black hole. The inward and outward velocities of a massless particle +( dr +dt )in/out are monotonically increasing with r, as the gravitational pull of the black hole decreases by further recession +from the horizon. This is true for both Schwarzschild and q-theory spacetime with one exception in q-theory. In the +region of large change of the q-field the velocity of outward moving massless particles has a small dip before increasing +again. In this region the energy density of the q-field becomes positive. As expected, the outward motion tends to + +energy function +velocity function +3.5 +6 +E() = 0 +E() = 0 +3.D +E(∞) = 0.01 +E() = 0.01 +E(∞) = 0.05 +5 +E() = 0.05 +25 +E() = 0.1 +E() = 0.1 ++2E(r) +E() = 0.5 +. +E() = 0.5 +2D +E(∞) = 1 +- E(∞) = 1 +foge(1 +15 +LD +2 - +0.5 - +1 - +00 +0 + +0.D +0.5 +15 +2D +25 +3.0 +0.D +0.5 +15 +2D +25 +3.D +Iogia(rfrh? +logia(r/rh? +massless particle outward velocity (Schwarzschild) +massless particle inward velocity (Schwarzschild) +LD +1.0 +0.B +1.5 +0.6 +E(∞) = 0 +2.0 +E(∞) = 0.01 +0.4 +E() = 0.05 +E(∞) = 0.1 +E(o) = 0 +2.5 +E(∞) = 0.5 +0.2 +E() = 0.01 +E() = 1 +E() = 0.05 +E() = 0.1 +3.0 +0.D +E(∞) = 0.5 +E(∞) = 1 +0.2 +3.5 +0.D +0.5 +15 +2D +25 +3.D +0.0 +0.5 +1D +15 +2D +25 +3.D +Iogia(rfrh) +Iogia(rfrh) +massless particle outward velocity (q-theory) +massless particle inward velocity (g-theory) +LD +E() = 0 +E() = 0 +0.B +E(∞) = 0.01 +E() = 0.01 +E() = 0.05 +E(∞) = 0.05 +E(∞) = 0.1 +E(∞) = 0.1 +0.6 +E() = 0.5 +E(∞) = 0.5 +E(∞) = 1 +E(∞) = 1 +0.4 +0.2 +日 +0.0 +-10 +0.2 +0.0 +0.5 +15 +2D +25 +3.D +0.0 +0.5 +LD +15 +2D +25 +3.D +Iogia(rfrh) +logia(rfrh)22 +Figure 12. The effect of different shift parameters on SHBH solutions for q-theory outside the event horizon is shown for a +horizon radius of rh = 1, q-field potential parameters λ = 1 and qeq = 0.158 and a grid size of 106. +zero as the event horizon is approached. We chose the Schwarzschild mass parameter M = mq−theory(rh). Choosing +M = mq−theory(∞) instead would have implied that the graphs for ( dr +dt )out in the Schwarzschild case cross zero +already outside the event horizon as set by q-theory. Close to the event horizon both inward and outward massless +particle velocities are smaller for q-theory spacetime as compared to Schwarzschild spacetime. This suggests that +black hole absorptivity is greater for q-theory spacetime as compared to Schwarzschild spacetime. +Appendix B. Accuracy estimates. +The numerical solution presented in Fig. (2) is approximate both because of finite grid size and finite shift parameter. +The solution with rh = 1 and q-field potential parameters λ = 1 and qeq = 0.158 is analyzed in Fig. (12) and Fig. +(13) with respect to variations of the shift parameter and grid size, respectively. +The plots in (12) show that absolute differences of q-fields and mass functions for different neighboring shift parameters +ϵ are almost exactly coincident with the absolute size of the imaginary part of the q-fields and mass functions for +the larger shift parameter present in the corresponding absolute difference plots. The maximal value of the absolute +size of the imaginary parts of the represented q-fields and mass functions shrinks by one order of magnitude for each +decrease of the shift parameter by one order of magnitude as expected. It is about two order of magnitude smaller +than the shift parameter for the q-fields and about one order of magnitude larger than that of the shift parameter +for the mass functions, though. We choose a shift parameter of ϵ = 10−6 in most of our plots, as it is seen to be +negligibly small to have any effect. The same choice argument will be applied for the grid size to which we now turn. +The plots in (13) show the absolute differences of q-fields and mass functions for different neighboring grid sizes as well +as relative error estimates for the functions q′, q′′ and m′ analogous to the lowermost left plot in Fig. (2) for different +grid sizes. The expectation that the differences between the q-fields and mass functions as well as the error estimates +decrease with increasing grid size are fulfilled. The differences of the q-fields and mass function decrease by about +one order of magnitude for a grid size increase of one order of magnitude. The error estimates are comparable for +the different grid sizes close to the horizon. This indicates that no mayor improvement may be achieved with further +increase of the grid size. Further away from the horizon the error estimates indeed decrease visibly with increasing +grid size. +[1] Albert Einstein. Cosmological Considerations in the General Theory of Relativity. Sitzungsber. Preuss. Akad. Wiss. Berlin +(Math. Phys. ), 1917:142–152, 1917. +[2] M. Bronstein. Über den spontanen Zerfall der Photonen. Phys. Z. Sowjetunion, 10(4):686–688, 1936. +[3] Lev Davidovich Landau and M. Bronstein. On the Second Law of Thermodynamics and the Universe. Phys. Z. Sowjetunion, +4, 1933. +[4] Y. B. Zeldovich. Cosmological Constant and Elementary Particles. JETP Lett., 6:316, 1967. + +difference of q-fields with respect to +absolute size of im(qtr) with respect to +4 +4 +Jog1alqs (r) - qz (r3) +6 +fog1a(im(qe(r)l) +8 +-10 +-10 +-12 +=10-8+=10-9 +-12 + =10-8 +E=10-7 +E= 10-8 + = 10~7 +-14 +E= 10-°+E= 10-7 +-14 +++++. + = 106 +-16 + = 10-5 = 10-6 +-16 + = 10~5 +-1B += 10-4+= 10-5 +-1B += 10-4 +E=10-3 += 10-4 + = 10~3 +20 +LD +12 +20 +0.D +0.2 +0.4 +0.6 +0.B +14 +16 +0.0 +0.2 +0.4 +0.6 +0.B +LD +12 +14 +16 +Iogia(rfrh) +Iogia(rfrh) +difference of the mass functions with respect to 2 +absolute size of im(mtr) with respect to z +2 +-2 +JogialIme, (r) - me, (rl) +logia(m(me(r)) +-6 +6 +-8 +-10 +=10-8+=10 + =10-8 +-10 += 10-7 +-12 +E = 106 + +E= 10° +-12 += 10-6 +14 +14 + = 10-5 +16 +E=10-4+=10 + = 10-4 +=10-3+=10-4 +~16 += 10~3 +0.D +0.4 +0.6 +0.B +12 +14 +16 +00 +0.2 +t0 +0.6 +0.B +1D +12 +14 +16 +Iogia(rfrh) +Iogia(rfrh)23 +Figure 13. The effect of different grid sizes on SHBH solutions for q-theory outside the event horizon are shown for a horizon +radius of rh = 1, q-field potential parameters λ = 1 and qeq = 0.158 and shift parameter ϵ = 10−6. +[5] Steven Weinberg. Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity. John Wiley +and Sons, New York, 1972. +[6] M. J. G. Veltman. Quantum Theory of Gravitation. Conf. Proc. C, 7507281:265–327, 1975. +[7] P. de Bernardis et al. A Flat universe from high resolution maps of the cosmic microwave background radiation. Nature, +404:955–959, 2000. +[8] G. Hinshaw et al. +Three-year Wilkinson Microwave Anisotropy Probe (WMAP) observations: temperature analysis. +Astrophys. J. Suppl., 170:288, 2007. +[9] Adam G. Riess et al. New Hubble Space Telescope Discoveries of Type Ia Supernovae at z>=1: Narrowing Constraints +on the Early Behavior of Dark Energy. Astrophys. J., 659:98–121, 2007. +[10] Steven Weinberg. The Cosmological Constant Problem. Rev. Mod. Phys., 61:1–23, 1989. +[11] Varun Sahni and Alexei A. Starobinsky. +The Case for a positive cosmological Lambda term. +Int. J. Mod. Phys. D, +9:373–444, 2000. +[12] T. Padmanabhan. Cosmological constant: The Weight of the vacuum. Phys. Rept., 380:235–320, 2003. +[13] Stefan Nobbenhuis. Categorizing different approaches to the cosmological constant problem. Found. Phys., 36:613–680, +2006. + +difference of q-fields with respect to grid size +4 +Iu)Yob- +-6 +grid size = 5 10° +grid size = 2.5 · 104 +grid size = 10§ + grid size = 5 - 104 +grid size = 2.5 - 105 + grid size = 105 +10 +grid size = 5 · 10° ++ grid size = 2.5 · 105 +-12 +grid size = 10° + grid size = 5 - 105 +-14 +-16 +0.D +0.2 +0.4 +0.6 +0.B +1D +12 +14 +16 +logiatrfrn? +relative size of the solution error (5-p.s.m.) for q' with respect to grid size +00 +-2.5 +grid size = 5 · 104 +Jog1a(/Agiel) +5.0 +grid size = 105 +grid size = 2.5 · 105 +7.5 + grid size = 5 · 105 +10.0 +grid size = 106 +grid size = 2.5 - 106 +12.5 +15.0 +0 +0.2 +t0 +0.6 +0.B +LD +12 +14 +16 +ogiatr/rh? +relative size of the solution error (5-p.s.m.) for q" with respect to grid size +0.D +-2.5 +grid size = 5 · 104 +Jog1a(/Agtel) +0'5- +grid size = 105 +grid size = 2.5 · 105 +-7.5 +grid size = 5 · 105 +10.0 +grid size = 106 +grid size = 2.5 · 100 +-12.5 +15.0 +00 +0.2 +0.4 +0.6 +0.B +LD +12 +14 +16 +logia(rfrn) +difference of the mass functions with respect to grid size +m(.(rl) +4 +grid size =5. 10° + grid size = 2.5 - 104 +-6 : +grid size = 105 +grid size = 5 - 104 +Jogia(Im(ms(r) - +grid size = 2.5 - 105 +grid size = 105 +.- grid size = 5 - 105 + grid size = 2.5 - 105 +grid size = 10° + grid size = 5 - 105 +-10 +-12 +0.D +0.2 +0.4 +0.6 +0.B +LD +12 +14 +16 +fogia(rfrh) +relative size of the solution error (5-p.s.m.) for m' with respect to grid size +00 +-2.5 +grid size = 5 · 104 +fog1a(/Amiel) +grid size = 105 +5.0 +grid size = 2.5 · 105 +7.5 +grid size = 5 . 105 +10.0 +grid size = 10° +grid size = 2.5 · 106 +12.5 +15.0 +0.D +0.2 +0.4 +0.6 +0.B +1D +12 +14 +16 +logia(rfrh?24 +[14] Joseph Polchinski. The Cosmological Constant and the String Landscape. In 23rd Solvay Conference in Physics: The +Quantum Structure of Space and Time, pages 216–236, 3 2006. +[15] F. R. Klinkhamer and G. E. Volovik. Self-tuning vacuum variable and cosmological constant. Phys. Rev. D, 77:085015, +2008. +[16] F. R. Klinkhamer and G. E. Volovik. Dynamic vacuum variable and equilibrium approach in cosmology. Phys. Rev. D, +78:063528, 2008. +[17] G. E. Volovik. Fermi-point scenario of emergent gravity. In From Quantum to Emergent Gravity: Theory and Phenomenol- +ogy, 9 2007. +[18] C. D. Froggatt and H. B. Nielsen. Derivation of Poincare invariance from general quantum field theory. Annalen Phys., +517:115, 2005. +[19] James Bjorken. Emergent gauge bosons. In 4th Workshop on What Comes Beyond the Standard Model?, 11 2001. +[20] Gian Francesco Giudice. Theories for the Fermi scale. J. Phys. Conf. Ser., 110:012014, 2008. +[21] S. W. Hawking. +Particle Creation by Black Holes. +Commun. Math. Phys., 43:199–220, 1975. +[Erratum: +Com- +mun.Math.Phys. 46, 206 (1976)]. +[22] Farshid Soltani, Carlo Rovelli, and Pierre Martin-Dussaud. +End of a black hole’s evaporation. II. +Phys. Rev. D, +104(6):066015, 2021. +[23] G. Chapline, E. Hohlfeld, R. B. Laughlin, and D. I. Santiago. Quantum phase transitions and the breakdown of classical +general relativity. Int. J. Mod. Phys. A, 18:3587–3590, 2003. +[24] G. E. Volovik. Type-II Weyl Semimetal versus Gravastar. JETP Lett., 114(4):236–242, 2021. +[25] Pawel O. Mazur and Emil Mottola. Gravitational condensate stars: An alternative to black holes. 9 2001. +[26] Abhay Ashtekar, Alejandro Corichi, and Daniel Sudarsky. Hairy black holes, horizon mass and solitons. Class. Quant. +Grav., 18:919–940, 2001. +[27] S. W. Hawking. The Cosmological Constant Is Probably Zero. Phys. Lett. B, 134:403, 1984. +[28] Antonio Aurilia, H. Nicolai, and P. K. Townsend. Hidden Constants: The Theta Parameter of QCD and the Cosmological +Constant of N=8 Supergravity. Nucl. Phys. B, 176:509–522, 1980. +[29] Ulises Nucamendi and Marcelo Salgado. Scalar hairy black holes and solitons in asymptotically flat space-times. Phys. +Rev. D, 68:044026, 2003. +[30] Remo Ruffini and John A. Wheeler. Introducing the black hole. Phys. Today, 24(1):30, 1971. +[31] Markus Heusler. No hair theorems and black holes with hair. Helv. Phys. Acta, 69(4):501–528, 1996. +[32] P. Painlevé. C. R. Acad. Sci., 173, 1921. +[33] Allvar Gullstrand. Allgemeine Lösung des statischen Einkörperproblems in der Einsteinschen Gravitationstheorie, volume +16,8 of Arkiv för matematik, astronomi och fysik. Almqvist & Wiksell, Stockholm, 1922. +[34] Yuki Kanai, Masaru Siino, and Akio Hosoya. Gravitational collapse in Painleve-Gullstrand coordinates. Prog. Theor. +Phys., 125:1053–1065, 2011. +[35] D. Sudarsky. A Simple proof of a no hair theorem in Einstein Higgs theory,. Class. Quant. Grav., 12:579–584, 1995. +[36] Alejandro Corichi, Ulises Nucamendi, and Marcelo Salgado. Scalar hairy black holes and scalarons in the isolated horizons +formalism. Phys. Rev. D, 73:084002, 2006. +[37] Robert M. Wald. On the instability of the n=1 Einstein Yang-Mills black holes and mathematically related systems. J. +Math. Phys., 33:248–255, 1992. + diff --git a/KdE1T4oBgHgl3EQfGgND/content/tmp_files/load_file.txt b/KdE1T4oBgHgl3EQfGgND/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e61b2057206538bef9c460f6acc8a15dbac44191 --- /dev/null +++ b/KdE1T4oBgHgl3EQfGgND/content/tmp_files/load_file.txt @@ -0,0 +1,1257 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf,len=1256 +page_content='Gravastar-like black hole solutions in q-theory M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Selch, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Miller, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='Zubkov Ariel University, Ariel, 40700, Israel (Dated: January 10, 2023) We present a stationary spherically symmetric solution of the Einstein equations, with a source generated by a scalar field of q-theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' In this theory Riemannian gravity, as described by the Einstein - Hilbert action, is coupled to a three - form field that describes the dynamical vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Formally it behaves like a matter field with its own stress - energy tensor, equivalent to a scalar field minimally coupled to gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The asymptotically flat solutions obtained to the field equations represent black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' For a sufficiently large horizon radius the energy density is localized within a thin spherical shell situated just outside of the horizon, analogous to a gravastar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The resulting solutions to the field equations, which admit this class of configurations, satisfy existence conditions that stem from the Black Hole no - hair theorem, thanks to the presence of a region in space in which the energy density is negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Introduction General Relativity (GR) is based on the axiom that the gravitational field is encoded by the geometry of spacetime: in a perfect vacuum spacetime is flat, while massive objects distort the surrounding spacetime from being otherwise flat to having a curved geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Test particles move along geodesics of the background spacetime in a way such that the trajectory depends on the geometry of the spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' In flat space the trajectory is a straight line, while in curved space it gets accelerated away from straight line motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' This is the equivalence principle: the gravitational field is equivalent to acceleration in the background spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' In this manner Newton’s ‘action at a distance’ theory of gravity is replaced by Einstein’s field theory approach, in which the field is the very geometry of the spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The vacuum Einstein field equations contain a cosmological constant [1], which relates the large-scale expansion of the universe in cosmological models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' In turn the cosmological constant is related to the vacuum energy density [2–6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Its value according to astronomical observations lies at a typical energy scale of the order of 10−3 eV [7–9], while its range of values as inferred from theoretical models is much larger [10–14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Volovik and Klinkhamer have suggested [15, 16] that the smallness of the observed vacuum energy density can be explained on the basis of a thermodynamic argument by which the vacuum energy density is exactly canceled in equilibrium (perfect quantum vacuum).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' q-theory appears as a low-energy effective theory [17–20] as opposed to a purely fundamental theory [10–14], and as a result contributions to the vacuum energy density become suppressed at macroscopic scales and reside in small perturbations of the equilibrium state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' In the Klinkhamer Volovik model, the effective vacuum energy density that enters the low-energy field equations is described by a vacuum field variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' At high energies it may be described by the 3-form field Aβγδ, which is antisymmetric with respect to permutation of indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' From this tensor the scalar q-field, the low energy vacuum variable, is composed as q2 = − 1 24FαβγδF αβγδ, where Fαβγδ = ∇[αAβγδ] is the field strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The equilibrium value of q alters if the vacuum is perturbed towards a new equilibrium state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' More details can be found in §II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Black holes (BHs) are unstable due to several effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' They may evaporate gradually resulting in Hawking radiation [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Alternatively a BH may undergo a transition to a white hole [22], through quantum mechanical tunneling from inside a trapped region to an anti-trapped region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Another possible outcome is the formation of a vacuum star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' In this model the event horizon takes the form of a boundary between different phases of the quantum vacuum [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' There are a number of similar phenomena between semi-metals and BHs, for example the event horizon emerging on the boundary between type I and type II Weyl semimetals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Volovik [24] has discussed an analogous process that occurs in Dirac and Weyl semimetals that suggests the viability of the formation of a gravastar or vacuum star after vacuum reconstruction, once Hawking radiation has been ended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' According to [24] (and unlike conventional gravastars [25]) the gravastar admits three distinct regions: the vacuum inside the Cauchy horizon with the de Sitter metric, the vacuum inside the thin shell between the Cauchy horizon and the event horizon, and the vacuum outside the event horizon with the ordinary Schwarzschild metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Even classically, BHs may be unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Regarding it as an ‘excited state’, decay due to exponential growth of the metric (plus soliton) perturbations becomes possible [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The geometry of the spacetime in a neighbourhood of a gravastar can be described using the action of a q-field coupled to gravity, where the q-field, as mentioned above, is related to the energy density of the quantum vacuum [27, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The goal of this work is to show (by numerical means) that (static and spherically symmetric) “scalar-haired” BHs exist within q-theory induced by the scalar q-field minimally coupled to gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The solutions are discussed thoroughly and are interpreted within the context of gravastars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' A similar (numerical) calculation by Nucamendi and Salgado can be found in refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' [29], in which solutions to the field equations are derived for the case of a scalar field arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='02914v1 [gr-qc] 7 Jan 2023 2 coupled minimally as well, in a generic static, spherically symmetric and asymptotically flat spacetime very similar but not identical to our considerations within q-theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' There, “scalar-hair” BH solutions were shown to exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' As mentioned above, such BH solutions admit non-trivial “hair” associated with the scalar field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' A general BH “no-hair” conjecture was originally proposed by Ruffini and Wheeler [30] (see also Hawking 1975 [21] for a thorough pedagogical overview).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' A set of conditions arise from no-hair theorems [31] for the existence of a solution to the Einstein equations with a scalar field source, one of which is the the no-hair integral, defined in (60), of the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' It is shown in §IV that these criteria are satisfied for the solutions obtained in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' For a given spacetime the presence of an event horizon can be inferred from analyzing ingoing null trajectories, as explained in §III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' A set of coordinates convenient for both tracking null trajectories and describing the whole neighborhood of an event horizon are Painlevé-Gullstrand (PG) coordinates, originally suggested in [32, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Below can be found a brief description of how they are derived and the manner in which PG coordinates describe null trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' More complex spacetime tensors including the stress-energy and Einstein tensors in PG coordinates are given in §III A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' PG coordinates were originally suggested as an alternative to Schwarzschild coordinates for describing radial null trajectories in Schwarzschild spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The unique solution of the Einstein equation that is spherically-symmetric, stationary, non-spinning with no net charge is the Schwarzschild metric with the form ds2 = −fdt2 + 1 f dr2 + r2dΩ2, (1) where dΩ2 = dθ2 +sin2 θdϕ2 is the line-element of a unit two-sphere and f = 1−2M/r is the Schwarzschild term with M being the mass of the background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The four-velocity U µ = dxµ/dτ ≡ ˙xµ (τ being proper time along the worldline) satisfies the normalization condition −1 = gµνU µU ν = −f ˙t2 + f −1 ˙r2 = U 2 � −f + f −1(dr/dt)2� where U ≡ dt/dτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The quantity ε = −gµνξµU ν = fU is a constant of motion, since ξµ = (∂/∂t)µ is a timelike Killing field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Accordingly, the four velocity of a radially outgoing or ingoing spherically-symmetric worldline is U µ = � ε f , − � ε2 − f, 0, 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' (2) In PG coordinates, the time coordinate, denoted tp to distinguish it from the t in Schwarzschild coordinates, is the proper time along the worldline of the geodesic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' As such the four velocity now has the more natural form U µ p = � ˙tp, ˙r, ˙θ, ˙φ � = � 1 , − � ε2 − f, 0, 0 � ≡ (1, −v, 0, 0) (3) where a ‘dot’ refers to a derivative with respect to tp, and v = � ε2 − f (4) is the radial component of the velocity on the free-falling trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The PG time coordinate tp is related to the Schwarzschild time coordinate as dtp = εdts + � ε2 − f f dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' (5) After writing the Schwarzschild metric (1) in PG coordinates the Painlevé-Gullstrand metric is obtained with the form ds2 = −dtp 2 + 1 ε2 (dr + vdtp)2 + r2dΩ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' (6) Note that as pointed out in [34], the form of the metric in (6) is somewhat analogous to the conserved Newtonian energy E = 1 2 � dr dtp �2 + Φ(r) (7) where Φ(r) = −M/r is a Newtonian type potential and E = (ε2 − 1)/2 is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' If the particle falls from rest at infinity, ε = 1, E = 0, such that (6) reduces to the standard Painlevé Gullstrand metric ds2 = −dtp 2 + � dr + � 2M r dtp �2 + r2dΩ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' (8) 3 Both forms of the metric in (6) and (8) are regular at the horizon r = 2M, ergo the spacetime geometry inside and outside the horizon of a black hole can be related without the emergence of any singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' As explained in [34], the Newtonian type energy motivates the following ansatz for the metric in the generalized Painlevé-Gullstrand form: ds2 = −dtp 2 + 1 1 + 2E(tp, r) � dr + v(tp, r)dtp �2 + r2dΩ2, (9) where v(tp, r) = � 2E(tp, r) + 2m(tp, r) r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' (10) Here E and M are not constant values, rather they are functions of tp and r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The metric in (10) is the one used in this paper, but without an explicit dependence on tp, namely only stationary solutions are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The metric signature is taken to be (−1, 1, 1, 1), and we use natural units, namely c = ℏ = 1 is assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' In the opening section we defer setting the Newtonian constant G = 1 for the purpose of offering clarity in our calculations, but later it will be set to unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' This paper is structured in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' In section II we introduce our model for q-theory, which is effectively a scalar field theory with a double-well potential interaction, minimally coupled to Einstein gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' In section III the Einstein equations are given for the case of a static spherically symmetric spacetime, in two different sets of coordinates: generalized Painlevé-Gullstrand, and those that shall be referred to as generalized Schwarzschild coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' In section IV the restrictions on solutions to the field equations are explain, which arise due to the no- hair theorems that hold for scalar field theories minimally coupled to gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Section V contains a detailed discussion of one specific static and spherically symmetric q-theory solution to the field equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Section VI builds on section V, containing a local scan of the space of solutions around the solution considered in section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' In section VII instabilities of the obtained solutions are addressed, both due to classical perturbations as well as due to Hawking radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' We end our work in VIII with a conclusion of our findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The model under consideration In this paper we consider a gravitating dynamical vacuum of the type introduced in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' [15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' One way to describe such a system is through a 3-form field Aβγδ, antisymmetric with respect to permutation of indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' From a stand point, this system can be considered to be matter described by a field Aβγδ, interacting with the gravitational field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The secondary scalar field q, which is the effective degree of freedom at low energies, is composed of a three form field Aβγδ as q2 = − 1 24FαβγδF αβγδ, Fαβγδ = ∇[αAβγδ], (11) Fαβγδ = ±q√−gεαβγδ, F αβγδ = ±q 1 √−g εαβγδ , (12) where εαβγδ and εαβγδ are completely antisymmetric, namely ε0123 = 1 and ε0123 = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The square brackets denote antisymmetrization of indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' From these relations it follows that δq δgαβ = 1 2qgαβ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' (13) The action of the model has the form S = � d4x√−g � R 16πG − ϵ(q) − 1 2gαβ∇αq∇βq � (14) where G is Newton’s constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' ϵ is a polynomial function in q that has the form ϵ(q) = λ 4 � q4 − 1 aGq2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' (15) λ and a are real numbers assumed to be O(1)-parameters with λ, a > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The scale associated with the potential function is due to G and, therefore, it is the Planck scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' 4 Variation of the action with respect to the metric results in the Einstein equations: Rαβ − 1 2gαβR = −8πG � gαβ � ρ(q) + 1 2∇αq∇αq � − ∇αq∇βq + 2□q δq δgαβ � (16) where ρ(q) = ϵ(q) − dϵ dq �1 2gµν δq δgµν � = ϵ(q) − dϵ dq q (17) follows directly from (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The function ρ enters the Einstein equations in the same way as the cosmological constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The shift from ϵ to ρ, as well as the final term on the right hand side of the Einstein equations, follow from the relation q = q(gµν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' A self-sustained quantum vacuum fulfills 0 = P = −ρ (18) in thermodynamic equilibrium, where P refers to pressure and ρ is the energy density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' This yields, in our example, the equilibrium values qeq = 0, ± 1 √ 3aG, which satisfy ρ(qeq) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' By further analogy with thermodynamics, a chemical potential µ may be defined (up to a constant) as µ = dϵ dq ���� q=qeq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' (19) Given that the original potential function is even with respect to q, and assuming that the vacuum has a non-trivial equilibrium configuration, it shall be assumed that qeq = 1 √ 3aG from now on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The energy density function is then given by ρ(q) = ϵ(q) − µq = λ 4 � q4 − 1 aGq2 + 2 (3aG) 3 2 q � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' (20) It has a local minimum at q = qmin = −( 1 √ 3 + 1) 1 √ 4aG, a local maximum at q = qmax = (− 1 √ 3 + 1) 1 √ 4aG and another local minimum at the equilibrium value q = qeq = 1 √ 3aG with ρ(q = qeq) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Consequently, the equilibrium value is also a double root of ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The other roots are located at q = 0 and q0 = − 2 √ 3aG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Vacuum stability requires the vacuum compressibility χvac to be positive : χ−1 vac = � q2 d2ϵ dq2 � ���� q=qeq = λ 6a2G2 ≥ 0 , (21) fulfilled for λ, a > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The energy density function for λ = 1 and different values of a is plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' (1) with the black line marking the level of vanishing energy density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The potential has the same basic characteristic of two wells: the well containing the equilibrium value of the q-field as a minimum with zero energy density, and the well that is deeper and hence allows for negative energy densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The region of negative energy densities starts at q = q0 and ends at q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The action in (14) lacks higher derivative terms of the q-field, which usually appear in the effective field theory description without fine tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' These terms are (in the absence of an intermediate high-energy physics scale) suppressed by the Planck energy scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' As long as the q-field varies slowly (q′ ≪ 1 in units with G = 1), these contributions are negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' This is the approach adopted in the following part of the discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Variation of the action with respect to the three - form field Aαβγ yields the generalized Maxwell equation ∇α(√−g � −dϵ(q) dq δq δFαβγδ + □q δq δFαβγδ ) � = 0 (22) ⇔ ϵαβγδ∇α � −dϵ(q) dq + □q � = 0 (23) ⇔ dϵ(q) dq − □q = µ (24) Here µ is the integration constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Inserting this into the Einstein equations (16) yields Rαβ − 1 2gαβR = −8πG � gαβ(ϵ(q) − µq + 1 2∇αq∇αq) − ∇αq∇βq � (25) 5 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The energy density of the q-field for λ = 1 and different values of a is depicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' While λ sets the absolute scale, a determines the depth and separation width of the potential wells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' which comprises both the gravitational field and the matter (generalized Maxwell) equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The fact that the q-field is not fundamental, but rather only an effective degree of freedom leads to the effective replacement of ϵ with ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' In equilibrium, the scalar-field value corresponds to a minimum of the energy density ρ, which is located at the value zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' This shows that the problem reduces to that of solving the (modified) Einstein equations given by (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' It is equivalent to a scalar field theory minimally coupled to gravity with a scalar field potential ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Static and spherically symmetric solutions of the Einstein equations The discussion is now about solving the static, spherically symmetric Einstein equations in order to find asymptotically flat solutions that describe a BH with a non-trivial q-field behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The q-field, as well as all the other functions, depend on a single coordinate only, which we choose to be the standard radial coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The q-field is expected to relax to the equilibrium value at large values of the radial coordinate, and to deviate from the equilibrium value approaching smaller and smaller values for the radial coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' In the following, two different ansätze are used for the metric in order to solve the Einstein equations: Gα β = 8πG(Tq)α β, Gα β = Rα β − 1 2gα βR (26) where the Einstein tensor is Gα β and the q-field energy-momentum tensor is (Tq)α β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Primes above symbols label derivatives with respect to the radial variable, throughout this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Generalized Painlevé coordinates The first ansatz is a generalized Painlevé-Gullstrand metric with the form ds2 = −dt2 + 1 1 + 2E(r)(dr + v(r)dt)2 + r2dΩ2 (27) and v(r) = � 2E(r) + 2Gm(r) r (28) while dΩ2 = dθ2 + sin2(θ)dφ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' (29) energy density p(q) energy density pgi,zoomed 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='0 a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='1 a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='1 a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='2 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='5 a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='2 a=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='5 a=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='5 50 a=l a=l a=2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='D a=2 40 a= 5 a= 5 25 (b)d 30 (b)d 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='D 24 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='0 0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='5 10 2 L- 2 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='0 4 E- 0 1 3 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='D 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content="5 1D 1'5 2D a q6 As explained in the introductory remarks, the terms E and v in the metric are related to the kinematical quantities of a particle in motion in the background." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' As elucidated in [34], v(r) may be interpreted as the velocity of a freely falling test particle as it falls in towards a (spherically symmetric) gravitating object from infinity, while E(r), at least asymptotically, can be related to the normalized total energy of a test particle (E(∞) = (e2−1) 2 , where e represents the total energy per unit rest mass of a test particle at infinity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' This motivates labeling v a velocity function, and E an energy function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' For the first metric ansatz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' the non-vanishing Einstein tensor components are Gt t = −2Gm′ r2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Gt r = −2E′ r(1 + 2E) � 2E + 2Gm r ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' (30) Gr r = −2rE′ + 4E′Gm − 4EGm′ − 2Gm′ (1 + 2E)r2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' (31) Gθ θ = Gφ φ = (3r2(1 − 2Gm r )(E′)2 + 3r(1 + 2E)E′Gm′ − (r + m)(1 + 2E)E′ (32) − r2(1 − 2Gm r )(1 + 2E)E′′ − r(1 + 2E)2Gm′′)/((1 + 2E)2r2) The energy momentum tensor (Tq)α β for the q-field takes the form (Tq)α β = −(gα β(ρ(q) + 1 2gαβ∇αq∇βq) − gαγ∇γq∇βq) (33) with non-vanishing components (Tq)t t = (Tq)θ θ = (Tq)φ φ = −(ρ(q) + 1 2(1 − 2Gm r )(q′)2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' (34) (Tq)t r = � 2E + 2Gm r (q′)2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' (35) (Tq)r r = −(ρ(q) − 1 2(1 − 2Gm r )(q′)2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' (36) The Einstein equations can thus be brought into the following form Gm′ = 4πGr2(ρ(q) + 1 2(1 − 2Gm r )(q′)2)) (37) 2E′ 1 + 2E = −8πGr(q′)2 (38) q′′ = (1 − 2Gm r )−1(−2q′ r + 2Gmq′ r2 + 8πGrρq′ + dρ dq ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' (39) The first two equations are obtained from the radial and time components of the Einstein equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Using these to simplify the equation due to the angular components leads to the third equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The second equation can be solved for E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' With the definition F = ln(1 + 2E) and E(∞) = lim r→∞ E(r) the result is F(r) = ln(1 + 2E(∞)) + 8πG � ∞ r s(q′(s))2 ds .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' (40) This leaves the first and third equations, which are solved numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Generalized Schwarzschild coordinates An analogous procedure for the second ansatz yields the metric in the form ds2 = −f(r)dt2 + 1 h(r)dr2 + r2dΩ2, (41) 7 which shall be referred to as generalized Schwarzschild coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' From this form of the metric the following non-vanishing components of the Einstein tensor are derived: Gt t = rh′ + h − 1 r2 , Gr r = rhf ′ + hf − f r2f , (42) Gθ θ = Gφ φ = −1 4(rh(f ′)2 − 2rfhf ′′ − 2fhf ′ − (rff ′ + 2f 2)h′)/(rf 2) (43) The non-vanishing components of the corresponding energy momentum tensor are (Tq)t t = (Tq)θ θ = (Tq)φ φ = −(ρ(q) + 1 2h(q′)2), (Tq)r r = −(ρ(q) − 1 2h(q′)2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' (44) The Einstein equations can finally be brought into the following form 1 − h − rh′ = 8πGr2(ρ + 1 2h(q′)2) (45) f ′ f − h′ h = 8πGr(q′)2 (46) q′′ = 1 h dρ dq − h′ h q′ − 2 r q′ − 4πGr(q′)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' (47) The first two equations are obtained from the radial and time components of the Einstein equations, and subsequently used to simplify the relation for the angular components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' This results in the third equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The second equation can be solved and reads, with k = ln(f) and the assumption k(r = ∞) = 0, as k(r) = − � ∞ r �h′(s) h(s) + 8πGs(q′(s))2) � ds .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' (48) The connection between the two parameterizations is provided by the relation h(r) = 1 − 2Gm(r) r (49) according to which the first and third equations of the two ansätze are identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' For convenience and for the purpose of being consistent with other literature on this topic, we introduce the notation f(r) = h(r)e2δ(r) (50) and ˜δ(r1, r2) = � r2 r1 4πGs q′(s)2 ds .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' (51) C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Black hole spacetime characteristics In §V we will start from the assumptions that there exists an event horizon at a certain radial coordinate value r = rh, and that spacetime is asymptotically flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' It then follows that δ(r) = −˜δ(rh, ∞) + ˜δ(rh, r), F(r) = ln(1 + 2E(∞)) + 2˜δ(r, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' (52) In order to check for the existence of an event horizon, radial null geodesics in generalized Painlevé-Gullstrand coordinates or generalized Schwarzschild coordinates need to be discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The requirement ds2|θ=θ0,φ=φ0 = 0 leads to −dt2 + (dr + vdt)2 1 + 2E = 0 ⇔ dr dt = ± √ 1 + 2E − v (53) in generalized Painlevé-Gullstrand coordinates and −fdt2 + 1 hdr2 = 0 ⇔ dr dt = ± � fh (54) 8 in generalized Schwarzschild coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The plus sign is associated with outward motion, while the minus sign corresponds to inward motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' If dr dt < 0 for both signs and every r < r0, then r0 marks an event horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' More precisely, it marks the locus of an apparent horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' In a static (or more generally in a stationary) spacetime, the apparent and the event horizon coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Generalized Schwarzschild coordinates are only valid on one side of the event horizon, since they become singular at an event horizon (with at least either f = 0 or h = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' In the vicinity of an event horizon, h(r) ≪ 1, and the outward velocity of a massless particle may be approximated by (dr dt )out = � 1 + 2E(r) − v(r) = � 1 + 2E(r) − � 2E(r) + 2Gm r (55) = � 1 + 2E(r) − � 2E(r) + 1 − h(r) = h(r) 2 � 1 + 2E(r) + O(h(r)2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' As a result, r = rh marks an event horizon if h(rh) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Taking a look back at the Einstein equations in the generalized Painlevé-Gullstrand ansatz (39), it can be observed that, although the coordinates are regular on the horizon, the inclusion of a scalar field in Einstein gravity leads to singular behavior as the horizon is approached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' This is manifest in the third Einstein equation for both ansätze.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Of particular interest about solutions in q-theory is the distribution of energy density inside and outside the event horizon, as well as the question of whether or not out solutions are singular at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' To answer these questions it is convenient to define −(Tq)t t = V (r) + T(r), V (r) = ρ(q(r)), T(r) = 1 2h(r)(q′(r))2 (56) such that the energy density into a potential part V and a kinetic part T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The latter quantity may be deduced from the Kretschmann invariant K(r) = RµνρσRµνρσ (57) as r → 0, where Rµνρσ is the Riemann curvature tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' For generalized Schwarzschild coordinates it takes the explicit form Kq−theory(r) = 1 r4 [4r4h2(r)(δ′(r))4 + 8r4h2(r)(δ′(r))2δ′′(r) + 4r4h2(r)(δ′′(r))2 + 8r2h2(r)(δ′(r))2 + r4(h′′(r))2 + (9r4(δ′(r))2 + 4r2)(h′(r))2 + 4(3r4h(r)(δ′(r))3 + 3r4h(r)δ′(r)δ′′(r) + 2r2h(r)δ′(r))h′(r) + 2(2r4h(r)(δ′(r))2 + 2r4h(r)δ′′(r) + 3r4δ′(r)h′(r))h′′(r) + 4(h(r) − 1)2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' (58) For Schwarzschild spacetime with h(r) = f(r) = 1 − 2GM r and Schwarzschild mass parameter M the Kretschmann scalar reads KSchwarz(r) = 48G2M 2 r6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' (59) Black hole spacetimes with minimally coupled scalar fields underlie a series of criteria to be fulfilled as dictated by the black hole no-hair theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' These criteria are the topic of the next chapter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' From now on we will work in units with G = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Black hole no-hair theorems In this section the BH no-hair theorems are explained, that restrict the set of allowed solutions of scalar hair black holes (SHBH’s) in curved spacetime, which is the same class of solutions considered in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The BH no-hair theorems, as discussed in [31, 35], assert the following: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' In the absence of event horizons there exist no non-trivial, regular scalar soliton solutions, that satisfy the dom- inant energy condition but violate the strong energy condition, at every point in asymptotically flat spacetimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' In the presence of an event horizon in a static, spherically symmetric and asymptotically flat spacetime, no non-trivial regular scalar-field solution exists outside the event horizon, if the dominant energy condition holds but the strong energy condition is violated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' 9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Any SHBH solution must necessarily have V (rh) < 0 where rh denotes the radial location of the event horizon, and V is the potential energy density of the scalar field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The vicinity of the event horizon is enveloped by a region of negative scalar-field energy density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The strong energy condition is defined by the requirement that Tαβkαkβ ≥ 0, where Tαβ is the covariant energy momentum tensor and kα is an arbitrary null vector field, and the requirement that −T α βpβ must be a future pointing causal vector field whenever pα is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The strong energy condition stipulates that for every timelike vector field uα, the condition (Tαβ − 1 2Tµνgµνgαβ)uαuβ ≥ 0 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' If in the first case, spherical symmetry is assumed with a positive scalar field potential, hence the dominant energy condition does not need to be imposed in order to infer the absence of non-trivial scalar field solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' From the first two criteria, it is immediate that within the domain of outer communications (the region from the event horizon to asymptotically flat infinity), the scalar field energy density must have negative values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The third criterion then specifies where this region of negative energy density must be located.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' This leaves as the only possibility for a non-trivial SHBH solution in the considered setup a q-field which asymptot- ically relaxes to its equilibrium value but sweeps over field values corresponding to negative potential energy density as the horizon is approached, for sure in the horizon proximity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Both SHBH’s, and scalar solitons, have to fulfill an integral equation as a necessary condition for existence, as derived from a scaling argument [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' These conditions are written below in our notation convention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' A necessary condition for the existence of a scalar soliton (scalaron) in curved spacetime, in a non BH geometry is � ∞ 0 4πr2 exp (2δ(r)) � Eflat kin (r) + 3V (r) � dr = 0 (60) It comprises the flat space kinetic energy density Eflat kin (r) = 1 2(q′(r))2, as well as the potential energy density V (r) = ρ(q(r)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Analogously, a necessary condition for the existence of a SHBH solution is � ∞ rh 4πr2 exp(2δ(r)) �2rh r � 1 − m(r) r � − 1 � Eflat kin (r) + �2rh r − 3 � Vpot(r))dr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' (61) where rh is the radial coordinate of the event horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The fulfillment of this latter condition is taken as a tool to fine-tune the shooting parameter q(rh) (introduced and discussed in section V), in finding SHBH solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' In the limit rh → 0 (60) is recovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' For future convenience we introduce the function nhf(r) = � r rh s2 exp(2δ(s)) �2rh s (1 − m(s) s ) − 1 � Eflat kin (s) + �2rh s − 3 � V (s)) ds (62) which is then supposed to fulfill lim r→∞ nhf(r) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' A representative SHBH solution for q-theory The existence of SHBH solutions has been known for some time and was first considered numerically in [29] for a scalar field minimally coupled to gravity in a double well scalar field interaction potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Subsequently, these solutions have been discussed in the framework of the of isolated horizons [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The latter formalism has been treated in detail in [26] and references cited therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The difference between this work and [29] is in the more restrictive potential of q-theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' There are only two free parameters, the absolute scale provided by λ and the depth or separation of the wells as parameterized by a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' A shift in the q-field, accompanied by a corresponding change in the scalar energy density function, does not lead to a further quantitative change in the solution as induced by the shift in the q-field itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' It can be seen as fixed by the condition ρ(q = 0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The scalar potential in [29] has three free parameters and allows for adjusting the positions of the wells, independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Consequently, the solutions found and presented within that work cannot be used for q-theory, since they lie outside the parameter space spanned by λ and a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' In the following, we replace the parameter a by the location of the minimum qeq = 1 √ 3a of the shallow potential well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' We use Python for plotting and numerically solving the Einstein equations for the minimally coupled q-field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The solver is non-adaptive and makes use of a refined (fourth order) Runge-Kutta method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Refined means that the grid size close to the horizon is smaller than further away.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' This is due to the observation that the equations become singular at the horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' We will nevertheless employ a prescription of how to start “close” to the horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' This comes about since only one of the three boundary conditions of the differential equations (q(r = rbound), q′(r = rbound) and m(r = rbound) at radial coordinate boundary position r = rbound) is free for an asymptotically flat black hole spacetime regular at the horizon which we are searching for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' This freedom resides in the radial 10 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' A representative SHBH solution for q-theory outside the event horizon is shown for a horizon radius of rh = 1, q-field potential parameters λ = 1 and qeq = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='158, a grid size of 106 and shift parameter ϵ = 10−6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' coordinate location of the event horizon for a fixed scalar field potential function, as is the case for Schwarzschild spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The dependencies of the boundary values on the horizon radius are known precisely only at the horizon except for one parameter, the “shooting” parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' How the singular behavior at the horizon is circumvented is explained further below, together with an account of numerical errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The results have been confirmed by an adaptive routine of the NDsolve-method in Mathematica, which is the most accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' We will still stick with Python in the following discussion, since the difference between the solving routines is negligibly small for tiny grid sizes (of the order ∼ 105 − 106), as used in Python.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Both a discussion of how to numerically avoid the singularity in the third Einstein equation at the horizon, as well as a discussion of numerical errors are lacking in [29], but will be provided in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' We shall start with a qualitative discussion of a representative solution in this section before focusing on the space of solutions in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Outside the event horizon The method of finding solutions for q-theory, makes use of previously discovered solutions, insofar as they are used as a starting point in an interpolation of solutions between the respective parameter spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' For a horizon radius of rh = 1, q-field potential parameters λ = 1 and qeq = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='158, a grid size of 106 and shift parameter ϵ = 10−6 (to be introduced further below) the solution is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' (2) for the region outside the event horizon (also termed the domain of outer communications).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' It is in qualitative agreement with that of [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The q-field starts with a negative value such that q0 < q(r = rh) < 0, which is equivalent to V (r) = ρ(q(r)) < 0, as is required by the no-hair theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' More specifically we have qmin < q(r = rh) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The q-field starts in the right half of the deep potential well shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' (1) with the characteristic points of the double well potential named as discussed in section I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' It increases into the positive energy density domain, passes the local maximum and relaxes asymptotically to the equilibrium value qeq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=" The metric function m(r), termed mass function, starts from the horizon with m(rh) = 1 2rh, initially decreases as q-field potential and kinetic energy densities versus q' 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='02 q (r) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='1 - Vr vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' T(r) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=" q'(r) 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='D1 T(r) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='. 50V(r) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='D q(r) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='3 15 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='5 1b 20 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='D 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='0 S0 15 2D 25 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='D logia(r/rh?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' logia(r/rh?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' evolution of the mass function metric functions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='D h(r) 25 f(r) 5 - (μ)g 20 m(r) 0 5- LD OT- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='5 15 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='D 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='5 1D 15 2D 25 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='D 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='D 15 20 25 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='D logia(r/rh?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' logia(rfrh?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' relabive ermor of the q-field solution no-hair function 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='00 0 f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=',5-p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='m for g f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=',5-p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='m for q 175 2 Jogia of the relative 4 f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=',5-p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='m for m 150 6 125 1LDo oT- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='75 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='50 - 14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='25 16 00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='B 12 16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='DO 14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='D 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='5 15 20 25 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='D logia(r/rh?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' logia(r/rh?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='11 ρ(q) < 0 until reaching a global minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' It is located at a radial coordinate value shortly advancing that of q = 0, since the kinetic energy density T(r) = 1 2h(r)(q′(r))2 is positive everywhere outside the horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The mass function increases until finally approaching its asymptotic value, the ADM mass of the spacetime, from below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The potential and kinetic energy densities, as introduced in equation (56), are shown together with the derivative of the q-field on the upper right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The potential energy density is seen to be of minor size as compared to the kinetic part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Further shown are the metric functions in generalized Schwarzschild coordinates in the central plot on the right hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' As is required by asymptotic flatness we have the limits lim r→∞ f(r) = 1, lim r→∞ h(r) = 1, lim r→∞ δ(r) = 0, (63) while limr→∞ E(r) remains a free parameter for the generalized Painlevé-Gullstrand coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' This free parameter is mostly irrelevant for the discussion of SHBH solutions and only of interest for test particle motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Its variation will be discussed in Appendix A but is of no crucial importance elsewhere in the discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The lowermost plot on the left shows the error sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' We calculate the relative error of the solution by taking the first numerical derivative (f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=') of q, q′ and m and comparing it with the solution as given by q′, q′′ and m′ and obtained from the non-adaptive, refined (fourth order) Runge-Kutta algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The first numerical derivative is calculated using a central 5-point stencil method (5-p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' On a formal level this method approximates the derivative of a five times differentiable function accurately up to O((∆r)4)-corrections where ∆r represents the grid spacing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The grid size is 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The first four peaks mark the radial coordinate values where the grid size changes abruptly and are artificial, since the 5-p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' formula we employed is based on equal grid size spacing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The final peaks for the relative errors are due to the appearance of the local extrema of q′ as well as m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Apart from these points the relative error can be seen to be smaller than 10−7 throughout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The lowermost plot on the right shows the function nhf(r) introduced in the last section in equation (62) which indeed fulfills limr→∞ nhf(r) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' This latter property was taken to fine-tune the shooting parameter q(rh), though, which will be discussed shortly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' It can be seen that the plot of the relative errors do not reach as far out as the other plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' An asymptotic analysis of the third Einstein equation with δq(r) = q(r) − qeq yields the linear approximation (δq)′′ = −2(δq)′ r + ρ′ = −2(δq)′ r + λ 2aδq , (64) which has the solution δq(r) = c1 1 r exp � − � λ 2ar � + c2 1 r exp � + � λ 2ar � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' (65) The asymptotic behavior of the relevant functions can be deduced from the first Einstein equation and the condition limr→∞ m(r) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' A first order Taylor expansion of the potential energy density and its derivatives around q = qeq is well justified for δq ≪ 1 √a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The approximation of the third Einstein equation is well justified for 2m(r) r , 4πr2ρ(q(r)) ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Taken together we then obtain (64).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The asymptotic solution has an exponentially growing and an exponentially depleting part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' In the search of a solution it is found that when the q-field and the mass function approach their asymptotic values they leave them again after some critical value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' This can not be avoided and is an artifact of the numerical approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' It induces a non-vanishing coefficient c2 and therefore exponential growth due to numerical uncer- tainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Therefore the exponentially depleting part is fitted onto the functions q, q′ and m after they tend to change very slowly by approaching their asymptotic values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' This fitting of the free parameter c1 takes place at those radial coordinate values where the relative error plots end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' At this point δq(r) qeq , m(∞)−m(r) m(∞) , nhf(r) ≪ 1 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' We now turn to the near horizon region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' We search for solutions regular at the horizon by imposing limr→rh q′′(r) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The third Einstein equation as well as the assumptions of the existence of an event horizon and of asymptotic flatness then reduce the freedom of the boundary conditions of q, q′ and m to one parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' This parameter may be declared as the horizon radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' After fixing the scalar field potential parameters and thereby the theory, the space of solutions is one dimensional and parameterized by rh as is Schwarzschild spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Up to one degree of freedom, the boundary values are known at the horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The event horizon condition h(r) = 0 on the one hand and limr→rh q′′(r) < ∞ on the other hand imply m(rh) = rh 2 , q′(rh) = rh dρ dq (q(rh)) (1 − 8πr2 hρ(q(rh))) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' (66) 12 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' A representative SHBH solution for q-theory inside the event horizon is shown for a horizon radius of rh = 1, q-field potential parameters λ = 1 and qeq = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='158, a grid size of 106 and shift parameter ϵ = 10−6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The remaining freedom then resides in the value of q(rh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' It is adjusted so as to yield ("shoot" towards) an asymptot- ically flat solution and therefore termed shooting parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' By choosing it more and more accurately the approach of q, q′ and m to their asymptotic values may be improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' This suggests that there exists exactly one shooting pa- rameter which is appropriate for ensuring asymptotic flatness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' We can only approach it to within a certain numerical accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' As soon as the solutions to the first and third Einstein equations are obtained, the horizon values of E(r) and δ(r) may be deduced by integration of the second Einstein equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' They are given by E(rh) = ((1 + 2E(∞)) exp( � ∞ rh r(q′(r))2dr) − 1)/2, δ(rh) = −˜δ(rh, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' (67) The singular behaviour of the third Einstein equation at r = rh is avoided by the prescription r → r(1 + iϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' We call ϵ the shift parameter and choose it such that 0 < ϵ ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The quantities plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' (2) are then understood as the real parts of the functions of the (total, complex valued) solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The accuracy of the numerical calculations due to finiteness of grid size as well as shift parameter is analyzed in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Inside the event horizon In extension of the plots of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' 2, the solution for a horizon radius of rh = 1, q-field potential parameters λ = 1 and qeq = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='158, a grid size of 106 and shift parameter ϵ = 10−6 is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' 3 for the region inside the event horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' In distinction to the corresponding figure for the outside region the lowermost plot on the right hand side highlights the energy function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The relative error size remains below 10−5 for all of the functions q′, q′′ and m′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The peaks mark again those radial coordinate values where the grid size changes abruptly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Close to the event horizon it has been chosen smaller as in the case of the region outside the event horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Of interest is the behaviour in the limit where the radial coordinate tends to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The solution is found to be singular in the q-field in this limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The mass function, in contrary, tends to a constant value of about limr→0 m(r) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='503.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' An expansion of the third Einstein q-field potential and kinetic energy densities versus q 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='19B 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='200 V(r) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' T(r vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=" q'(r) log 1o(T(r) log 1o(V(r) 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='202 因 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='204 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='206 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='20B 10 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='210 5 4 2 L- 4 2 0 logia(r/rh?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Iogia(rfrh) evolution of the mass function metric-functions 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content="504 log 1o(h(r) EOS'0 4 2 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='502 (sjuu 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='501 azis 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='50 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='499 6 + 60 5 4 E- 2 1 4 E- 2 1 logia(r/rh?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' logia(rfrn) relabive ermor of the q-field solution energy function 0 fogia of the relative error f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=',5-p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' for q 2 f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=',5-p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' for q f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=',5-p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' for m Jogia(E(r) - E(rh)) 6 9- ot- 12 714 16 3 4 E- 2 1 logia(r/rh?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Iogia(rfrh?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='13 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The Kretschmann invariant for a representative SHBH solution for q-theory inside and outside the event horizon is shown for a horizon radius of rh = 1, q-field potential parameters λ = 1 and qeq = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='158, a grid size of 106 and shift parameter ϵ = 10−6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' equation for r ≪ 1 yields the approximate equation q′′ = −q′ r (68) with solution q(r) = d1 log10(r) + d2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' (69) It is found that d1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='0010±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='0001 and d2 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='2010±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='0001 by a straight line fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' This implies the approximations F(r) ≈ ln(1 + 2E(rh)) + 8π � rh r0 r(q′(r))2dr + 8πd2 1(ln(r0) − ln(r)) (70) = C + 8πd2 1ln(1 r ) ⇔ E(r) ≈ exp(C) 2 r−8πd2 1 − 1 2 δ(r) ≈ δ(rh) − � rh r0 4πr(q′(r))2dr − 4πd2 1(ln(r0) − ln(r)) (71) = D − 4πd2 1ln(1 r ) ⇔ exp(2δ(r)) ≈ exp(2D)r8πd2 1, h(r) ≈ 1 − 2m(0) r (72) for r < r0 ≪ 1 with for the further discussion irrelevant constants C and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Consequently the limits lim r→0 q(r) = −∞, lim r→0 q′(r) = ∞ (73) for the q-field and its derivative and lim r→0 E(r) = ∞, lim r→0 h(r) = −∞, lim r→0 δ(r) = −∞ (74) for the metric functions follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Since d1 ≪ 1 √ 8π, E(r) and exp(2δ(r)) vary very slowly as compared to h(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Therefore the behavior of the metric as r → 0 will asymptote that of Schwarzschild spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Especially, very close to the center the energy density will change sign again and becomes positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The metric can not be continued to the radial coordinate origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' There is a curvature singularity in the limit r → 0 as is well known for Schwarzschild spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The Kretschmann invariants for q-theory and Schwarzschild spacetime are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' While they differ visibly asymptotically far from the horizon where both tend to zero, they show the same 1 r6 -divergence as r → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' After discussing one solution in detail we proceed with a local scan of the space of solutions around that just presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Both grid size and shift parameter will no longer be mentioned from now on and chosen to be very large in the former case and negligibly small in the latter as has been done within this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Kretschmann invariant 30 K5chaz(r) 25 21 ((u)xjDTbo) 15 1 5 - 5- 10 E- 2 1 1 fogia(rfrh?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='14 VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The parameter space of SHBH solutions for q-theory The qualitative features of the representative solution presented in the previous section are common to all SHBH solutions in q-theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The space of solutions is parametrized by the horizon radius rh as well as the q-field potential parameters λ and qeq (or as well a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' In order to understand the differences between solutions we perform a local scan around the representative solution in the space of solutions and extract different quantities for each individual solution, in part in analogy with [29, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The ADM mass of a (static and spherically symmetric) solution is given by MADM(rh, λ, qeq) = lim r→∞ mrh,λ,qeq(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' (75) We decompose it into two contributions MADM(rh, λ, qeq) = M Schwarz ADM (rh) + Mhair(rh, λ, qeq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' (76) The first contribution is the mass parameter of a Schwarzschild black hole of horizon radius rh, M Schwarz ADM (rh) = rh 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' It is independent of the scalar field potential parameters, as it describes a (static and spherically symmetric) black hole in vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The dressing by the non-constant q-field outside the event horizon yields the non-vanishing additional contribution Mhair(rh, λ, qeq) = − � ∞ rh 4πr2T t tdr (77) = � ∞ rh 4πr2(ρ(q(r)) + 1 2h(r)(q′(r))2)dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' (78) Further quantities of interest are the shooting parameter as well as the radial coordinate location relative to rh and absolute depth of the global minimum of the mass function as functions of the parameters of the space of solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The relative radial coordinate location of the global minimum of the mass function is important insofar as it marks the region where both the q-field and the mass function vary significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Dependence on the horizon radius Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' 5 illustrates the characteristic quantities introduced above for a horizon radius range 10−3 < rh < 103 and scalar field potential parameters coincident with those of the representative solution presented in section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The lower limit has been chosen such that the asymptotic dependence on rh is visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The upper bound is due to lack of precision of fitting the asymptotic part of the q-field and mass function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The asymptotic plateau becomes difficult to identify, since both q-field and mass function behave less and less smooth in the fitting region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The asymptotics for large horizon radii seem to be deducible from the plots as well, though.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' We will assume that the plots show the true asymptotics in both extreme situations rh ≪ 1 and rh ≫ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' We may then draw the following conclusions: 1) The shooting parameter qshoot(rh) = q(rh) varies significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Its asymptotic values are limrh→0 qshoot(rh) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='913qmin and limrh→∞ qshoot(rh) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='500qmin, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' 2) For rh ≪ 1 the ADM mass and scalar hair mass converge towards the value limrh→0 log10(MADM(rh)), log10(Mscalar hair(rh)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='817.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' For rh ≫ 1 both ADM mass and scalar hair mass increase linearly with the horizon radius and may be parameterized by log10(MADM(rh)) = c1 log10(rh) + c2 (79) log10(Mscalar hair(rh)) = c3 log10(rh) + c4 (80) with c1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='0031 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='0001, c2 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='1985 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='0001, c3 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='0136 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='0001 and c4 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='8747 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='0001 obtained by a straight line fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The q-theory ADM mass is a monotonically increasing function of the horizon radius and everywhere larger than the corresponding Schwarzschild spacetime ADM mass (which coincides with the mass parameter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' 3) The region outside the event horizon where the q-field and mass function vary significantly approaches the event horizon relative to its size for rh ≫ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' So it seems that in this regime the q-field behaves non-trivially only just 15 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The variation of several characteristic quantities of SHBH solutions for q-theory with respect to the horizon radius rh is shown for the q-field potential parameters λ = 1 and qeq = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='158.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' These quantities include shooting parameter, ADM mass, scalar-hair mass, relative location of the global minimum of the mass function as well as the absolute value of the global minimum of the mass function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' outside the horizon and relaxes to its equilibrium value very quickly in the near horizon regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' This region of significant change does not approach the horizon infinitely close but relaxes to the value limrh→∞ rmin(rh)/rh = 1 + e−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='916 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' In the opposite limit rh ≪ 1 the region of significant change of both q-field and mass function gets pushed further and further away from the horizon region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' This implies that for the very limit rh → 0 no scalar soliton (scalaron) exists (contrary to the conclusions drawn in [29]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Rather the q-field remains constant outside the event horizon with a value of qshoot(0) = limrh→0 qshooot(rh) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='913qmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The corresponding energy density is negative resulting in a Schwarzschild-anti de Sitter spacetime with cosmological constant Λ = 8πGρ(qshoot(0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' 4) The size of the global minimum of the mass function converges to a constant for rh ≪ 1 with value limrh→0 log10(−min(m)(rh)) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' For rh ≫ 1 it decreases linearly and faster than the negative ADM mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' It may be parametrized by log10((−min(m)(rh)) = d1 log10(rh) + d2 (81) with d1 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='0268 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='0001 and d2 = −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='5790 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='0002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Dependence on the parameters of the scalar field potential We now consider changes in the scalar field potential parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The effect of a variation of the scalar field potential parameter λ for a horizon radius range 1 10 ≤ rh ≤ 10 and fixed scalar field potential parameter qeq = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='158 for the characteristic functions presented previously in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' (5) is visualized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The radial parameters and masses have been rescaled in a particular way following [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' It can be seen that the rescaled quantities seem to depend only value of the shooting parameter ADM-massoftheblackholesolution 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='9 2 MSchwa2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='B 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='7 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='6 2 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='5 2 1 0 i 2 3 E- 2 1 0 i 2 3 Iogia(rh) fogia(rh) relative location of the global minmum of m global minimum of m(rh) log 1o(rmin/rn) log 1o(rmin rn)/rn) Jogia(Fmin/rh) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Jogia(Fmin - Fh)/rh) 5 - 3 logia(-min(m(rh))) 3 1 2 0 1 1 0 1 E- 2 1 0 1 2 3 E- 2 1 0 1 2 3 logia(rh?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Iogia(rh?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='16 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The variation of several characteristic quantities of SHBH solutions for q-theory with respect to the scalar field potential parameter λ is shown for a horizon radius of rh = 1 and remaining q-field potential parameter qeq = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='158.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' These quantities include shooting parameter, ADM mass, scalar-hair mass, relative location of the global minimum of the mass function as well as absolute value of the global minimum of the mass function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' on two instead of three parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' To be more precise about the parameter dependencies define �rh = √ λrh, � rmin = √ λrmin, ˆ MADM = √ λMADM, � min(m) = √ λmin(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' (82) It then follows that � rmin = 1, ˆ MADM = ˆ MADM( �rh, qeq), � min(m) = � min(m)( �rh, qeq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' (83) The effect of a variation of the scalar field potential parameter qeq for a horizon radius range 1 10 ≤ rh ≤ 10 and fixed scalar field potential parameter λ = 1 for the characteristic functions presented previously is visualized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' As qeq increases, the potential wells of the scalar field potential recede from each other and become more pronounced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' All quantities (with minor exceptions) seem to be monotonically growing (monotonically decreasing in the case of the negative valued quantity min(m)) with qeq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The dependence of the different characteristic quantities of SHBH solutions on qeq is not so easily deducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Nevertheless it seems that, with exception of large horizon radius values, the dependence on qeq may be factorized from that on �rh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The local scan of the parameter space of solutions has revealed that the space of solutions is effectively only two dimensional with the dependence on the two parameters factorizing by a monotonically increasing function of qeq to good approximation within the represented parameter space area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The topic of the following section will be a stability analysis of SHBH solutions due to perturbations of the q-field solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Stability of the SHBH solutions An important question to ask is whether SHBHs are stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Do perturbation modes of the q-field and the metric which grow exponentially in the SHBH spacetime of our q-theory model exist?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The question of classical instability value of the shooting parameter ADM-massoftheblackholesolution 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='95 14 入=4 ^=1 13 ^=1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='50 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='B5 (4)o%b 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='B0 1f 入=4 ^=1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='75 入=1 8: 入= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='70 - 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content="5 1'D 2." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='D 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='5 1D fogiatVArn) logia(VArn) relative location of the global minimum of m globalminimum ofm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' ^=4 2 ^=4 225 + ^=1 ← ^=1 入=1 200 4 - 入= 4 175 → =2 6 150 125 1D0 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='50 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='25 o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='bo 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='75 1D0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='0 d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='D 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content="5 1'D Iogia(rh?" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' ogia( VArh?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='17 Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The variation of several characteristic quantities of SHBH solutions for q-theory with respect to the scalar field potential parameter qeq is shown for a horizon radius of rh = 1 and remaining q-field potential parameter λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' These quantities include shooting parameter, ADM mass, scalar-hair mass, relative location of the global minimum of the mass function as well as absolute value of the global minimum of the mass function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' has been discussed for spherically symmetric, time dependent perturbations of the metric and scalar field (here the q-field) in [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Following a similar notation to that introduced in [29] (see Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' (17)-(19) therein) we define the s-wave perturbations by ˜q(t, r) = q(r) + δq(r, t) , (84) ˜f(r, t) = f(r)(1 − h1(r, t)) , (85) ˜h(r, t) = h(r)(1 − h2(r, t)) (86) in generalized Schwarzschild coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The functions q(r), f(r) and h(r) represent the unperturbed solutions for q-theory in the generalized Schwarzschild coordinates, while δq(r, t), h1(r, t) and h2(r, t) denote small perturbations to the non-perturbed solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Note that in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' (86) h2 is defined as a small correction to h with grr = 1/h in (41), whereas in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' (19) in [29] it is defined as a small correction to grr directly with a different sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The two relations are equivalent, as is easily checked by substituting h(1 − h2) for h in grr = 1/h and then expanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' After linearization of the Einstein equations (45-47) it can be shown that [29] ∂rh1(r, t) = ∂rh2(r, t) − 16πr ( ∂r q(r) ) (∂r δq(r)) , h2(r, t) = 8πr ( ∂r q(r) ) δq(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' (87) As such the metric perturbations are expressible in terms of the q-field perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The latter may be determined by solving a one-dimensional Schrödinger equation of the form [29] � 1 2m(−i∂r∗)2 + V ∗ eff(r∗) � ψ(r∗) = 1 2m(i∂t)2ψ(r∗), dr∗(r) dr = exp(−δ(r)) h(r) (88) where ψ(r∗(r)) ≡ rδq(r) and r∗ is the “tortoise” coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The scalar field mass m is obtained as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Insert ˜q(t, r) into the action functional and expand in δq(t, r) around q(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Far away from the event horizon when q(r) is close to its equilibrium value qeq, the kinetic cross terms as well as the linear potential term in δq are negligible and value of the shooting parameter ADM-massoftheblackholesolution 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='98 中 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='0 O qeg = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='223 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='96 ★ qeg = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='193 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='5 qeg = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='173 + qeg = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='158 60 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='0 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='jb (usjnavw qeg = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='129 +qeg=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='112 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='92 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='0 qeg = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='223 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='90 qeg = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='193 qeo = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='173 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='5 qeg = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='158 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='8 - ★ qeg =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='129 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='D → q= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='112 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='75 LDo 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='75 LDO fogia(rh) fogia(rh) relative location of the global minimum of m globalminimumofm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' qeg=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='223 0+ 25 + qe = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='193 qeg = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='173 → qea = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='158 ★ qeg = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='129 DOT- 2D ← qeg= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='112 Jogia(rmin/rh) O qeg =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='223 tusjujuu Qeg = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='193 200 qeg = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='173 15 qea = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='158 ★ qea = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='129 ← qeg=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='112 300 1D 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='5 400 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='50 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='75 LDO 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='50 ±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='DO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='75 LDO fogia(rh) Iogia(rh?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='18 the action with dynamical field (perturbation) δq has a proper kinetic term with at least quadratic potential terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The quadratic term yields m in the same way as does the real Klein Gordon action with (self-)interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The so obtained value for m reads m = � d2ρ(q) dq2 ������ q=qeq = � 3 2λq2eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' (89) It will formally not be needed in the following but has been introduced in order to provide a properly normalized Schrödinger problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The effective potential reads [29] (note that the expression for Veff in this work is defined with a factor of 1 2m compared to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' (22) in [29]) Veff(r∗(r)) = 1 2mh(r) exp(2δ(r))[h(r) r (δ′(r) + h′(r) h ) − 8πrh(r)(q′(r))2(δ′(r) + h′(r) h(r) + 1 r ) + 16πrq′(r)dρ dq (q(r)) + d2ρ dq2 (q(r))].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' (90) The separation ansatz ψ(r∗(r)) = ξ (r∗(r)) exp � ±i √ 2mEt � leads to the following stationary Schrödinger equation for ξ (r∗(r)) � 1 2m(−i∂r∗)2 + V ∗ eff(r∗) � ξ(r∗) = Eξ(r∗) (91) with energy eigenvalue E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' A sufficient condition for static, spherically symmetric configurations to be unstable [37] is that the differential operator on the left-hand side of (91), is negative in the Hilbert space L2(M), where M is the spacetime manifold on which the metric is defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Accordingly, a sufficient condition for unstable solutions is the existence of a bound state E < 0 in the Schrödinger problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' In the same manner the existence of bound states that correspond to E < 0 in the Schrödinger problem with potential Veff is equivalent to the existence of unstable perturbations of q-theory solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The exponential factor of the perturbation becomes of order unity after a time of order τ = 1 √−2mEmin .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' (92) The subscript min on E signifies the lowest bound state energy which is of most importance for giving the scale of τ (for a discrete and finite bound state spectrum, as is the case here).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' This time can be seen as the lifetime of the SHBH in the presence of these perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' In order to determine the value Emin, the lowest eigenvalue of bound state energies of the effective Schrödinger equation (91), we solve the eigenvalue problem numerically in the variable r for horizon radii in the range 1 ≤ rh ≤ 1000 by approximating the equation on a grid of finite size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The differential operator is approximated by a central point stencil method accurate to fourth order of the grid spacing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The Runge Kutta solver shares this level of accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' We employ vanishing boundary conditions for the eigenfunctions in the limits r → 0 as well as r → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' To get an impression of the shape of the eigenfunction to the eigenvalue Emin, we replace the effective potential, illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' (8) for different horizon radii, by an auxiliary potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' This auxiliary potential is a parabola fitted to the negative potential well of the effective potential in the region shown in red in the figure (where Veff < 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Outside it is set to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' This yields a finite depth harmonic oscillator potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The solutions of the wave equation may be determined exactly in this auxiliary potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Finding the bound state energy eigenvalues is then identical to the quantum harmonic oscillator except for them being finite in amount, while the eigenfunctions are compromised due to the vanishing of the potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Nevertheless, as is in concordance with the results in [29], the expectation is that the lowest energy bound state eigenfunction is close to a Gaussian in shape which is the exact eigenfunction in this case for the quantum harmonic oscillator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' That this expectation is also fulfilled in our case is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' We plot the normalized eigenfunctions corresponding to the lowest bound state energy eigenvalues for several horizon radii and scalar field potential parameters λ = 1 and qeq = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='158.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' A comparison with Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' (8) shows that the peaks of the eigenfunctions are situated almost exactly at the minimum of the effective Schrödinger potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' It becomes apparent here and has been observed that for increasing horizon radii the Gaussians loose their shape and tend to disappear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' In this regime the lowest eigenvalues approach zero very quickly from below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' This inspires the conclusion 19 Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The effective potential of the scalar perturbation mode ξ(r∗) is shown for different horizon radii and scalar field potential parameters λ = 1 and qeq = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='158.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' It comprises a negative valued well where the wave function of bound states are predominantly located.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The well is highlighted in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' A zoom makes this region visible for the horizon radii rh = 100 and rh = 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Normalized eigenfunction solutions to the lowest bound state eigenvalue for the scalar perturbation mode ξ(r∗) are shown for different horizon radii and scalar field potential parameters λ = 1 and qeq = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='158.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' They are almost perfect Gaussians in shape which is the case for the quantum harmonic oscillator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' that large SHBHs are indeed stable and opposes that found in [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' (10) the lifetime τ = τ(rh) of SHBHs as a function of the horizon radius and scalar field potential parameters λ = 1 and qeq = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='158 is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' We fit the obtained lifetimes τ = τ(rh) corresponding to the solutions for the lowest eigenvalues e = e(rh) to the function f(rh) = a · (log10(rh))ntan(c · log10(rh) − b) + d (93) parameterized by the scale parameters a and c, the horizontal shift parameter b, the vertical shift parameter d and the power parameter n with optimal parameters popt and covariance matrix pcov given by popt = � � � � � a b c d n � � � � � = � � � � � 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='53 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='14 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='73 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='31 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='23 � � � � � , pcov = � � � � � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='044 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='082 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='053 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='037 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='007 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='081 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='179 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='115 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='085 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='014 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='053 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='115 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='074 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='055 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='009 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='037 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='085 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='055 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='043 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='010 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='007 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='014 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='009 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='028 � � � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' (94) The lifetime then becomes infinite at the finite horizon radius r0 h = π+2b 2c and the SHBHs are therefore stable beyond the threshold r0 h for the chosen scalar field potential parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' effective Schrodinger potential for the q-field perturbation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='002 (u)"PA 000\'0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='105 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='110 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='115 log 1o(r/rn) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='00 rn = 1 In = 5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='025 F = 10 Tn = 25 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='050 rn = 100 rn= 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='5 1D 15 2D 25 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='D fogia(rtrh)eigenfunctionwith corresponding eigenvalue E-E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='6 T = 1 rh= 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='5 rn = 25 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='4 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' tts)* 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='3 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' :1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='D .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='D 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='5 LD 15 20 25 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='D logia(r/rn?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='20 Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The lifetime of SHBHs due to s-wave q-field and metric perturbations represented by the scalar perturbation mode ξ(r∗) is shown as a function of the horizon radius with scalar field potential parameters λ = 1 and qeq = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='158.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The lifetime is finite for small SHBHs below a certain radius or mass threshold value, whereas SHBHs are stable beyond this threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The threshold is highlighted in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The stability analysis has revealed that SHBH are unstable due to classical s-wave perturbations of both the q- field and the metric below a certain size or equivalently mass threshold and stable beyond (at least for the chosen parameters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Conclusion In the present paper we consider q-theory comprising a scalar field q minimally coupled to gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The q-field describes a dynamical gravitating vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' According to the estimates proposed in [24], this theory may contain BH solutions that resemble that of a gravastar, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' a configuration with energy concentrated inside a thin spherical shell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Contrary to the conventional gravastar, the state proposed in [24] contains an event horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Using direct numerical calculations, we confirm the existence of similar BH configurations, with some reservations, though.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Namely, inside the event horizon space - time resembles the interior of the Schwarzschild BH, and does not contain the de Sitter - like domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Besides, the mentioned thin shell is located outside the event horizon, not inside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Furthermore, there should exist a region in space, where the energy density is negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' This is required to satisfy the “no-hair” theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' As a result, the thin shell situated just outside of the horizon contains both a piece of negative energy and a piece with positive energy density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The integration of the energy density inside the shell yields a total positive energy resulting in the ADM mass perceived by the distant observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' According to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' (30) the energy density is proportional to the derivative of the mass function m(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The latter function is represented within Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' One can see that for the given example solution, the spherical shell of finite thickness exists and is situated just outside the horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Inside this shell the essential variation of m(r) is localized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Close to but outside the event horizon, the energy density is negative, then it passes through zero, and becomes positive in the second piece of the shell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The shell ends where m(r) exponentially approaches its asymptotic value, the constant that represents the black hole mass seen by the infinitely distant observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The results of section VI demonstrate that the spherical shell approaches the horizon relative to its size when the horizon radius is increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' In the limit of very large BHs, virtually the entire energy due to the q-field is localized in the thin shell situated outside the horizon and close to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' It does not approach the event horizon ifinitely close, but stops, such that the energy density sign transition is positioned around 7 5rh, where rh denotes the event horizon radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' A stability analysis with respect to the s-wave metric and q-field perturbations shows that the BH solutions of the type considered in the present paper may be classically unstable .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' However, the corresponding configurations are stable for sufficiently large BHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' We therefore claim that stable heavy SHBHs do exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' We do not discuss here questions related to the stability of the considered configurations on the quantum level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' This issue remains outside of the scope of the present paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' In conclusion, in this work we confirm the supposition of [24] about the existence of BH solutions in q-theory that look similar to gravastars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' These states escape the conditions of the no-hair theorem, due to the region in space with negative energy density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' At spatial infinity these solutions approach the Schwarzschild solution, but differ from it lifetime of aSHBH duetoperturbations 120 fit threshold data 140 8+ 40 24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='+ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='D 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='5 1b 15 2D 25 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='D logiatrh?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='21 Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The effect of different choices of E(∞) on the SHBH solution for q-theory outside the event horizon is shown for a horizon radius of rh = 1, q-field potential parameters λ = 1 and qeq = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='158, a grid size of 106 and shift parameter ϵ = 10−6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The inward and outward massless particle velocities are shown for both Schwarzschild spacetime and q-theory spacetime with Schwarzschild mass parameter M = mq−theory(rh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' essentially close to the horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Inside the horizon, the vacuum density is negative and changes sign very close to the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The singularity of curvature at r = 0 is the same as that of the Schwarzschild solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The authors are grateful to G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='Volovik for the proposition to consider the given problem, and for useful discussions during the initial stage of the work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Test particle characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' We discuss here the properties of test particles contained in the free parameter E(∞) = limr→∞ E(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' It is related to the total energy per unit rest mass of a test particle e by E(∞) = (e2 − 1)/2 which moves towards the SHBH horizon starting at infinity with initial velocity v(∞) = � 2E(∞) where v(∞) = limr→∞ v(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Different values for E(∞) correspond to different initial kinetic energies of a test particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The choice E(∞) = 0 corresponds to a particle at rest, while E(∞) > 0 corresponds to a particle initially moving towards the SHBH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' E(∞) < 0 is not possible, since then v(r) would become imaginary while approaching asymptotically flat infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The motivation for the introduction of generalized Painlevé-Gullstrand coordinates as well as their relation to test particle motion are presented in more detail in [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The numerical solution presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' 2 for different initial values of the free parameter E(∞) is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The function E(r) is monotonically decreasing with r as is v(r) in q-theory, whereas E(r) is constant for Schwarzschild spacetime while v(r) is also decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' This is in concordance with the increase of the kinetic energy of a test particle as it moves towards the event horizon of the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The inward and outward velocities of a massless particle ( dr dt )in/out are monotonically increasing with r, as the gravitational pull of the black hole decreases by further recession from the horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' This is true for both Schwarzschild and q-theory spacetime with one exception in q-theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' In the region of large change of the q-field the velocity of outward moving massless particles has a small dip before increasing again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' In this region the energy density of the q-field becomes positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' As expected, the outward motion tends to energy function velocity function 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='5 6 E() = 0 E() = 0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='D E(∞) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='01 E() = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='01 E(∞) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='05 5 E() = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='05 25 E() = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='1 E() = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='1 +2E(r) E() = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' E() = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='5 2D E(∞) = 1 E(∞) = 1 foge(1 15 LD 2 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='5 - 1 - 00 0 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='D 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='5 15 2D 25 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='D 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='5 15 2D 25 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='D Iogia(rfrh?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' logia(r/rh?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' massless particle outward velocity (Schwarzschild) massless particle inward velocity (Schwarzschild) LD 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='B 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='6 E(∞) = 0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='0 E(∞) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='4 E() = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='05 E(∞) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='1 E(o) = 0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='5 E(∞) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='2 E() = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='01 E() = 1 E() = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='05 E() = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='D E(∞) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='5 E(∞) = 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='D 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='5 15 2D 25 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='D 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='5 1D 15 2D 25 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='D Iogia(rfrh) Iogia(rfrh) massless particle outward velocity (q-theory) massless particle inward velocity (g-theory) LD E() = 0 E() = 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='B E(∞) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='01 E() = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='01 E() = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='05 E(∞) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='05 E(∞) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='1 E(∞) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='6 E() = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='5 E(∞) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='5 E(∞) = 1 E(∞) = 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='2 日 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='0 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='5 15 2D 25 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='D 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='5 LD 15 2D 25 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='D Iogia(rfrh) logia(rfrh)22 Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The effect of different shift parameters on SHBH solutions for q-theory outside the event horizon is shown for a horizon radius of rh = 1, q-field potential parameters λ = 1 and qeq = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='158 and a grid size of 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' zero as the event horizon is approached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' We chose the Schwarzschild mass parameter M = mq−theory(rh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Choosing M = mq−theory(∞) instead would have implied that the graphs for ( dr dt )out in the Schwarzschild case cross zero already outside the event horizon as set by q-theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Close to the event horizon both inward and outward massless particle velocities are smaller for q-theory spacetime as compared to Schwarzschild spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' This suggests that black hole absorptivity is greater for q-theory spacetime as compared to Schwarzschild spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Accuracy estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The numerical solution presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' (2) is approximate both because of finite grid size and finite shift parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The solution with rh = 1 and q-field potential parameters λ = 1 and qeq = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='158 is analyzed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' (12) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' (13) with respect to variations of the shift parameter and grid size, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The plots in (12) show that absolute differences of q-fields and mass functions for different neighboring shift parameters ϵ are almost exactly coincident with the absolute size of the imaginary part of the q-fields and mass functions for the larger shift parameter present in the corresponding absolute difference plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The maximal value of the absolute size of the imaginary parts of the represented q-fields and mass functions shrinks by one order of magnitude for each decrease of the shift parameter by one order of magnitude as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' It is about two order of magnitude smaller than the shift parameter for the q-fields and about one order of magnitude larger than that of the shift parameter for the mass functions, though.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' We choose a shift parameter of ϵ = 10−6 in most of our plots, as it is seen to be negligibly small to have any effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The same choice argument will be applied for the grid size to which we now turn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The plots in (13) show the absolute differences of q-fields and mass functions for different neighboring grid sizes as well as relative error estimates for the functions q′, q′′ and m′ analogous to the lowermost left plot in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' (2) for different grid sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The expectation that the differences between the q-fields and mass functions as well as the error estimates decrease with increasing grid size are fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The differences of the q-fields and mass function decrease by about one order of magnitude for a grid size increase of one order of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The error estimates are comparable for the different grid sizes close to the horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' This indicates that no mayor improvement may be achieved with further increase of the grid size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Further away from the horizon the error estimates indeed decrease visibly with increasing grid size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' [1] Albert Einstein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Cosmological Considerations in the General Theory of Relativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Sitzungsber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Preuss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Akad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Wiss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Berlin (Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' ), 1917:142–152, 1917.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' [2] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Bronstein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Über den spontanen Zerfall der Photonen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Sowjetunion, 10(4):686–688, 1936.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' [3] Lev Davidovich Landau and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Bronstein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' On the Second Law of Thermodynamics and the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Sowjetunion, 4, 1933.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' [4] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Zeldovich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Cosmological Constant and Elementary Particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' JETP Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=', 6:316, 1967.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' difference of q-fields with respect to absolute size of im(qtr) with respect to 4 4 Jog1alqs (r) - qz (r3) 6 fog1a(im(qe(r)l) 8 10 10 12 =10-8+=10-9 12 =10-8 E=10-7 +E= 10-8 = 10~7 14 E= 10-°+E= 10-7 14 ++++.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' = 106 16 = 10-5 = 10-6 16 = 10~5 1B = 10-4+= 10-5 1B = 10-4 E=10-3 += 10-4 = 10~3 20 LD 12 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='D 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='B 14 16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='B LD 12 14 16 Iogia(rfrh) Iogia(rfrh) difference of the mass functions with respect to 2 absolute size of im(mtr) with respect to z 2 2 JogialIme, (r) - me, (rl) logia(m(me(r)) 6 6 8 10 =10-8+=10 =10-8 10 = 10-7 12 E = 106 + E= 10° 12 = 10-6 14 14 = 10-5 16 E=10-4+=10 = 10-4 =10-3+=10-4 ~16 = 10~3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='D 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='B 12 14 16 00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='2 t0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='B 1D 12 14 16 Iogia(rfrh) Iogia(rfrh)23 Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The effect of different grid sizes on SHBH solutions for q-theory outside the event horizon are shown for a horizon radius of rh = 1, q-field potential parameters λ = 1 and qeq = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='158 and shift parameter ϵ = 10−6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' [5] Steven Weinberg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' John Wiley and Sons, New York, 1972.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' [6] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' G.' metadata={'source': 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Telescope Discoveries of Type Ia Supernovae at z>=1: Narrowing Constraints on the Early Behavior of Dark Energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=', 659:98–121, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' [10] Steven Weinberg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The Cosmological Constant Problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=', 61:1–23, 1989.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' [11] Varun Sahni and Alexei A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Starobinsky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The Case for a positive cosmological Lambda term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' D, 9:373–444, 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' [12] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Padmanabhan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Cosmological constant: The Weight of the vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Rept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=', 380:235–320, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' [13] Stefan Nobbenhuis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Categorizing different approaches to the cosmological constant problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=', 36:613–680, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' difference of q-fields with respect to grid size 4 Iu)Yob- 6 grid size = 5 10° +grid size = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='5 · 104 grid size = 10§ + grid size = 5 - 104 grid size = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='5 - 105 + grid size = 105 10 grid size = 5 · 10° ++ grid size = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='5 · 105 12 grid size = 10° + grid size = 5 - 105 14 16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='D 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='B 1D 12 14 16 logiatrfrn?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' relative size of the solution error (5-p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=") for q' with respect to grid size 00 2." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='5 grid size = 5 · 104 Jog1a(/Agiel) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='0 grid size = 105 grid size = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='5 · 105 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='5 grid size = 5 · 105 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='0 grid size = 106 grid size = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='5 - 106 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='2 t0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='B LD 12 14 16 ogiatr/rh?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' relative size of the solution error (5-p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=') for q" with respect to grid size 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='D 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content="5 grid size = 5 · 104 Jog1a(/Agtel) 0'5- grid size = 105 grid size = 2." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='5 · 105 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='5 grid size = 5 · 105 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='0 grid size = 106 grid size = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='5 · 100 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='0 00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='B LD 12 14 16 logia(rfrn) difference of the mass functions with respect to grid size m(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' (rl) 4 grid size =5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' 10° + grid size = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='5 - 104 6 : grid size = 105 +grid size = 5 - 104 Jogia(Im(ms(r) - grid size = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='5 - 105 +grid size = 105 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='- grid size = 5 - 105 + grid size = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='5 - 105 grid size = 10° + grid size = 5 - 105 10 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='D 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='B LD 12 14 16 fogia(rfrh) relative size of the solution error (5-p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=") for m' with respect to grid size 00 2." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='5 grid size = 5 · 104 fog1a(/Amiel) grid size = 105 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='0 grid size = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='5 · 105 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='5 grid size = 5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' 105 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='0 grid size = 10° grid size = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='5 · 106 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='D 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='B 1D 12 14 16 logia(rfrh?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content='24 [14] Joseph Polchinski.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' The Cosmological Constant and the String Landscape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' In 23rd Solvay Conference in Physics: The Quantum Structure of Space and Time, pages 216–236, 3 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' [15] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Klinkhamer and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Volovik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Self-tuning vacuum variable and cosmological constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' D, 77:085015, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' [16] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Klinkhamer and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Volovik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Dynamic vacuum variable and equilibrium approach in cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' D, 78:063528, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' [17] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Volovik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Fermi-point scenario of emergent gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' In From Quantum to Emergent Gravity: Theory and Phenomenol- ogy, 9 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' [18] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' D.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' [19] James Bjorken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Emergent gauge bosons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' In 4th Workshop on What Comes Beyond the Standard Model?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=', 11 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' [20] Gian Francesco Giudice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Theories for the Fermi scale.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' D, 104(6):066015, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' [23] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Chapline, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' Hohlfeld, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfGgND/content/2301.02914v1.pdf'} +page_content=' B.' 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+1,2199 @@ +Astronomy & Astrophysics manuscript no. SOAP_GPU +©ESO 2023 +January 12, 2023 +SOAP-GPU: Efficient Spectral Modelling of Stellar Activity Using +Graphical Processing Units +Yinan Zhao1 and Xavier Dumusque1 +Department of Astronomy of the University of Geneva, 51 chemin de Pegasi, 1290 Versoix, Switzerland +e-mail: yinan.zhao@unige.ch +January 12, 2023 +ABSTRACT +Context. Stellar activity mitigation is one of the major challenges for the detection of earth-like exoplanets in radial velocity mea- +surements. Several promising techniques are now investigating the use of spectral time-series, to differentiate between stellar and +planetary perturbations. In this context, developing a software that can efficiently explore the parameter space of stellar activity at the +spectral level is of great importance. +Aims. The goal of this paper is to present a new version of the Spot Oscillation And Planet (SOAP) 2.0 code that can model stellar +activity at the spectral level using graphical processing units (GPUs). +Methods. We take advantage of the computational power of GPUs to optimise the computationally expensive algorithms behind the +original SOAP 2.0 code. For that purpose, we developed GPU kernels that allow to model stellar activity on any given wavelength +range. In addition to the treatment of stellar activity at the spectral level, SOAP-GPU also includes the change of spectral line +bisectors from center to limb, and can take as input PHOENIX spectra to model the quiet photosphere, spots and faculae, which allow +to simulate stellar activity for a wider space in stellar properties. +Results. Benchmark calculations show that for the same accuracy, this new code improves the computational speed by a factor of 60 +compared with a modified version of SOAP 2.0 that generates spectra, when modeling stellar activity on the full visible spectral range +with a resolution of R=115’000. Although the code now includes the variation of spectral line bisector with center-to-limb angle, +the effect on the derived RVs is small. We also show that it is not possible to fully separate the flux from the convective blueshift +effect when modeling spots, due to their lower temperature and thus the appearance of molecular absorption in their spectra. Rather +negligible for the Sun, this degeneracy between the flux and convective blueshift effect become more important when we move to +cooler stars, however, this issue does not impact the estimation of the total effect (flux plus convection), and therefore users can trust +this output. +Conclusions. The publicly available SOAP-GPU code allows to efficiently model stellar activity at the spectral level, which is essential +to test further stellar activity mitigation techniques working at the level of spectral timeseries not affected by other sources of noise. +Besides a huge gain in performance, SOAP-GPU also includes more physics and is able to model different stars than the Sun, from +F to K dwarfs, thanks to the use of the PHOENIX spectral library. We however note that due to the limited understanding of stellar +convection and activity on other stars than the Sun, the more we go away from the solar case, the more the output of the code should +be taken with care. +Key words. Methods: data analysis – Techniques: radial velocities – Techniques: spectroscopic - Stars: activity +1. Introduction +The radial velocity (RV) method has been proven to be one of +the most successful method to detect exoplanets since the dis- +covery of the first exoplanet orbiting a solar-type star (Mayor +& Queloz 1995). In order to detect earth-like planets orbiting in +the habitable zone of its parent star, a precision of a few dozens +of cms−1 must be reached. Although the state-of-the-art spec- +trographs such as ESPRESSO, and EXPRESS are not far from +that precision (50 and 58 cms−1, respectively Pepe et al. 2021; +Brewer et al. 2020), the main limitation to detect Earth-like plan- +ets with the RV technique is stellar activity. Two major physical +processes dominating stellar activity on a time scale of the host +star’s rotational period are the flux imbalance due to the temper- +ature difference and therefore contrast between active and quiet +regions (hereafter flux effect. (e.g. Saar & Donahue 1997; Du- +musque et al. 2014; Donati et al. 2017)) and the inhibition of +convective blueshift (hereafter CB effect). The CB effect is due +to the presence of strong local magnetic fields inside active re- +gions, which suppress the CB inside those regions and leads to +positive RV variations (e.g. Cavallini et al. 1985a; Meunier et al. +2010). +Many methods have been proposed to mitigate activity- +induced variations using photometric and spectroscopic time se- +ries. In the one-dimensional time series space, many parametric +models based on analytic forms or different Gaussian process +(GP) frameworks have been developed to model stellar activ- +ity using photometry or spectroscopic activity indicators (e.g. +Aigrain et al. 2012; Rajpaul et al. 2015; Aigrain et al. 2016; +Gilbertson et al. 2020b; Barragán et al. 2022). Jointly modeling +the data with Keplerians to model planets in addition to a GP to +model stellar activity may significantly reduce the stellar activity +but may also lead to overfitting when the GP kernel or priors are +not wisely set. This is particularly dangerous when the planetary +properties are not constrained from transit observations. +Due to inherent problems in modeling stellar activity in one- +dimensional time series, the community is now shifting toward +modeling it in a two-dimensional space. Collier Cameron et al. +Article number, page 1 of 17 +arXiv:2301.04259v1 [astro-ph.SR] 11 Jan 2023 + +A&A proofs: manuscript no. SOAP_GPU +(2021) calculated the autocorrelation function (ACF) of cross- +correlation function (hereafter CCF Baranne et al. 1996), to iso- +late Doppler shift from shape shift variations and applied prin- +ciple component analysis (PCA) on the obtained ACFs to model +shape changes related to stellar activity. A planet signal of am- +plitude ∼ 40 cm/s can be recovered when the algorithm is ap- +plied to the HARPS-N solar data (Dumusque et al. 2015; Col- +lier Cameron et al. 2019; Dumusque et al. 2021). Zhao et al. +(2022a) projected CCFs time series onto the Fourier basis func- +tions and modelled line variability using different basis. Results +on simulated data show a 48% reduction in RV rms. de Beurs +et al. (2022) trained a convolutional neural network (CNN) on +both simulated CCFs and HARPS-N solar CCFs and were able +to significantly reduce stellar activity effects. +The idea behind building the CCF is to extract with the best +precision the RV information contained in a spectrum. How- +ever, key variations at the spectral level related to stellar activ- +ity may be lost when performing the dimensionality reduction +imposed by the CCF. Therefore, several methods have been pro- +posed to disentangle stellar activities at the spectral level. Davis +et al. (2017) applied PCA to simulated spectral time series and +demonstrated that eigen-vectors are spectral line dependent. Ra- +jpaul et al. (2020) used GP to directly derive RV information +from spectral time series. Jones et al. (2017) also applied mul- +tivariate GP to model stellar activity on PCA-reduced spectral +dataset. Cretignier et al. (2022), based on the knowledge that the +impact of stellar activity is line-depth dependant (Cretignier et al. +2020a), used PCA to model stellar activity in the flux-flux gra- +dient space (named the “shell” space) and results on HD10700 +(τ Ceti) and HD12861 (α Cen B) indicates the method can +successfully remove variations from non-Doppler origin. Last +by not least, Binnenfeld et al. (2020, 2022) are developing the +unit-sphere representation periodogram (USuRPER), to seper- +ate Doppler from other RV variations. This technique is based +on representing spectra as unit vectors in a multidimensional hy- +perspace. +The spectral time series used to evaluate the performance of +the algorithms developed to mitigate stellar activity at the spec- +tral level are either obtained from simulated data or real obser- +vations. The major issue with simulations, is that most of them +only model the RV activity effect at the CCF level (e.g. Du- +musque et al. 2014; Herrero et al. 2016) due to computational +inefficiency. A few other libraries of simulated spectra affected +by stellar activity exist, but generating them takes hours to run, +which is not convenient when exploring the parameter space in +stellar activity and properties (e.g. Gilbertson et al. 2020a; Du- +musque 2016). Regarding real observations, solar data obtained +by the HARPS-N solar telescope (Collier Cameron et al. 2019; +Dumusque et al. 2021), HELIOS on HARPS1 and more recently +the solar feed of NEID (Lin et al. 2022) are the best we can get, in +terms of S/N and sampling. However, those spectra corresponds +for the most part to quiet activity phases of the Sun (end of cy- +cle 24 end beginning of cycle 25) and can only used to mitigate +stellar activity for star very similar to the Sun. When moving to +stellar observations, the recent Extreme precision Spectrograph +(EXPRES) Stellar Signals Project (ESSP) shared some valuable +data. However, due to the small number of stars and the rather +small number of spectra available, it was rather difficult to com- +pare different activity mitigation techniques together (Zhao et al. +2022b). As a conclusion of this discussion, it is essential for the +community to have access to a code that can simulate efficiently +1 https://www.eso.org/public/announcements/ann18033/ +stellar activity at the spectral level, and for a wide range of stellar +properties. +In this paper, we present a new code, Spot Oscillation And +Planet Graphical Process Unit (SOAP-GPU) based on GPU +computation that can efficiently model simplified and realistic +stellar activity at the spectral level. In Sect 2, we revisit the ar- +chitecture of the SOAP 2.0 code it is based on (Dumusque et al. +2014) and discuss about its limitations. The algorithms behind +SOAP-GPU are presented in Sect 3. In Sect 4, we explore the +physical parameters of stellar activity and simulation of different +cases are presented. Finally, we draw our conclusion in Sect 5. +The SOAP-GPU code is publicly available on Github and Zen- +odo2 along with a brief manual and some examples. +2. Revisiting SOAP 2.0 +In this section, we revisit the code Spot Oscillation And Planet +(SOAP 2.0 Dumusque et al. 2014). This code aims at modeling +both the flux effect and the CB effect of active regions affecting +RV measurements. Although the public version of the SOAP 2.0 +code can only simulate stellar activity at the level of the CCFs, +modeling the effect at the spectral level follow the same ideas. In +this section, we first discuss the basic algorithms behind SOAP +2.0 and demonstrate the limit of the code, in terms of computa- +tional efficiency, when we want to model stellar activity at the +spectral level. +Fig. 1. The stellar disk is initialized with velocity and intensity fields. +Left: The intensity in each cell is computed depending on a limb dark- +ening law. Right: The velocity in each cell is computed considering ro- +tational period, stellar inclination and radius of the star. As we can see, +iso-velocity lines are not vertical as we implemented differential rota- +tion in SOAP-GPU, which was not the case in SOAP 2.0. +2.1. The structure of SOAP 2.0 +SOAP 2.0 first computes the “quiet” (without any active region) +emission spectrum of the star. To do so, a 2-dimension stellar +disk containing N × N cells is initialized (N being the resolu- +tion of the disk, the same parameter called “grid” in Boisse et al. +2012). Velocity and intensity of all disk cells are computed based +on the physical configuration of the star (rotational period, stel- +lar inclination and radius of the star) and a limb darkening law +(as shown in Fig. 1). In each cell, the quiet photosphere spectrum +is injected, weighted by the cell intensity (limb-darkening), and +Doppler-shifted to the projected velocity of that cell (rotation). +Linear interpolation is applied at this step to project the Doppler- +shifted spectrum into the original wavelength grid, to make sure +that spectra in different cells are on a common wavelength grid. +We note that the public version of SOAP 2.0 was using the CCF +2 code available here https://github.com/YinanZhao21/SOAP_ +GPU and https://doi.org/10.5281/zenodo.7499461 +Article number, page 2 of 17 + +Light ratio field +Velocity field (km/s) +300 +1.0 +300 +2.0 +1.5 +250 +0.9 +250 +1.0 +200 +0.8 +200 +0.5 +0.7 +150 +150 +0.0 +0.6 +-0.5 +100 +100 +-1.0 +0.5 +50 +50 +-1.5 +0.4 +0 + +0 + +2.0 +0 +50 +100 +150 +200 +250 +300 +0 +50 +100 +150 +200 +250 +300Yinan Zhao et al.: SOAP-GPU +of the high-resolution Kitt Peak Observatory Fourier Transform +Spectrograph (FTS) quiet photosphere spectrum (S quiet(λ), Wal- +lace et al. 1998) as approximation of the quiet Sun to increase +computational speed. However, injecting the original spectrum is +possible, with the only difference that the dimension of the input +is ∼500000, compared to 400 for the CCF, and that we will need +to apply a Doppler-shift each time we want to change the veloc- +ity of this spectrum, while a simple translation was sufficient in +the case of the CCF. After injecting the quiet photosphere spec- +trum in each cells, the integrated quiet solar spectrum is obtained +by summing the content of all the cells together. All those pro- +cesses are summarized in the pseudo code below (Algorithm 1). +Algorithm 1 Quiet spectrum integration +1: for Xlocation = 1, 2, . . . N do +2: +for Ylocation = 1, 2, . . . , N do +3: +Shift S quiet(λ) with velocity velX,Y. S quiet(λ) +→ +S quiet(λ +′). +4: +Do linear interpolation to project the spectrum back +to the original wavelength grid. S quiet(λ +′) → S +′ +quiet(λ) +5: +Weight S +′ +quiet(λ) by limb-darkening intensity IX,Y and +integrate spectra along disk surface. Squiet+ = IX,Y ×S +′ +quiet(λ) +6: +end for +7: end for +The next step consists in initializing the active regions us- +ing the following parameters: the number of active regions, their +size, their corresponding latitudes and longitudes, their types (ei- +ther spot or faculae) and the resolution of the active region con- +tour. An active region spectrum is also needed at this step to +model the CB effect. The original SOAP 2.0 code uses the CCF +of the observed spot spectrum in the visible obtained from the +Kitt Peak Observatory FTS (S active(λ) with λ the same as for +S quiet(λ) Wallace et al. 2005). The spectrum used for faculae re- +gions is the same, with the difference that the contrast of such +a region follow what is observed in the Sun (e.g. Fig. 3 in Me- +unier et al. 2010), thus brighter than the quiet sun and with a +center-to-limb brightening. Other groups use synthetic spectra at +different temperature to model the quiet photospehere, spots and +faculae, and include the effect of CB using results from magneto- +hydrodynamical simulations (e.g. the STARSIM 2 code3 Herrero +et al. 2016). We note that injecting observed or synthetic spectra +have their advantages and drawbacks. Using observed spectra +allows to better model the inhibition of convection inside ac- +tive regions, but we note that if we use the observed spectra of +a spot to model a facula (because an observed spectrum of a +facula across the entire visible spectral range does not seem to +exist), molecular features will be present in the facula spectrum +despite the temperature being higher. Using synthetic spectra on +the contrary allows to better model the temperature, and there- +fore spectral features, of the injected spectra. The choice of the +spectra will be addressed in the later sections. +In order to simulate a spectral time series, we need to calcu- +late the disk location of active regions at each timestamp. As +shown in Equation 1 and Equation 2 of (Boisse et al. 2012), +active regions are first put in the center of the disk and their +initial configuration is obtained using a rotation matrix. Next, +to get the position of those active regions as a function of +time, another rotation matrix is used. At each timestamp, the +code evaluates which active regions are visible, and which ones +3 code available here https://github.com/rosich/starsim-2 +are hidden behind the star. This is performed by the function +Localize(lat, long, i, ph), where lat and long are the latitude and +longitude of the active region center, i is the inclination angle of +the stellar disk and ph is the rotational phase. The output of this +function is a binary; one if visible, zero otherwise. If a region is +visible, the code proceed to estimate the difference between the +quiet solar spectrum and active spectrum at the location of the +active regions. +The difference for the flux effect, the CB effect and the com- +bination of the two (total) in each cell can be calculated using +the following equations: +∆S +′ +flux(X, Y) = S +′ +quiet(X, Y) − Iratio × S +′ +quiet(X, Y), +(1) +∆S +′ +bconv(X, Y) = S +′ +quiet(X, Y) − S +′ +active(X, Y), +(2) +∆S +′ +tot(X, Y) = S +′ +quiet(X, Y) − Iratio × S +′ +active(X, Y), +(3) +where Iratio is the contrast of the spot or faculae region. The code +then integrates over all the cells covered by active regions to get +final difference between the quiet spectrum and active spectrum. +The final spectrum of each effect at each timestamp can be cal- +culated by: +Sintegrated, final = Sintegrated,quiet − ∆Sintegrated,quiet−active. +(4) +Once a spectrum for each timestamp is obtained, the code then +lowers the resolution of the integrated spectrum to match the +resolution provided in the configuration file. The pseudo code +that describes how active regions are included, and how the final +integrated spectra is obtained is summarized below. +2.2. The limitation of SOAP 2.0 +The structure of SOAP 2.0 provides an efficient way to estimate +stellar activities on spectroscopic measurement by simulating +CCFs at different timestamps. The major drawback when chang- +ing the input from CCFs to spectra is the dimension of the data. +The dimension of the input CCFs in SOAP 2.0 was 400 in veloc- +ity space while the input high-resolution spectra we want to use +have a dimension of ∼ 500000 in the wavelength domain. From +Algorithm 1 and Algorithm 2, we clearly see that the linear in- +terpolation is repeatedly called when injecting the spectrum in +each cell, which is computationally expensive for an array with +dimension of ∼ 500000. For example, SOAP 2.0 takes ∼ 800 +seconds to calculate an integrated quiet sun spectrum using a +300×300 disk-grid. Another issue is how the code handles multi- +ple active regions. Each active region is modeled independently, +without information from other regions. This algorithm cannot +handle the case in which some active regions overlap with each +other. From real observations, we know that some active regions +have complicated configurations. For example, most of active re- +gions are a combination of a large faculae presenting a small spot +in its center. In this context, a more computationally efficient and +generalized algorithm is needed. +3. Description of SOAP-GPU +In the previous section we’ve demonstrated the limitation of +SOAP 2.0 when modeling stellar activity at the spectral level. +Here, we present a new version of SOAP, based on Graphical +Processing Unit (GPU) computing, that is much more efficient +in term of computational speed, but also that adds some physical +complexity. +Article number, page 3 of 17 + +A&A proofs: manuscript no. SOAP_GPU +Algorithm 2 Active region updates +1: for nregion = 1, 2, . . . N do +2: +for ttimestep = 1, 2, . . . , T do +3: +Localize(lat, long, i, ph). +4: +if Localize = 1 then +5: +Shift S quiet(λ) with velocity velX,Y. S quiet(λ) → +S quiet(λ +′). +6: +Do linear interpolation to project the spectrum +back to the original wavelength space. S quiet(λ +′) → S +′ +quiet(λ) +7: +Weight S +′ +quiet(λ) by limb-darkening intensity IX,Y +8: +Shift S active(λ) with velocity velX,Y. S active(λ) → +S active(λ +′). +9: +Do linear interpolation to project the spec- +trum back to the original wavelength space. S active(λ +′) → +S +′ +active(λ) +10: +Weight S +′ +active(λ) by limb-darkening intensity +IX,Y +11: +Use Equations 1 to 3 to calculate the difference +of each effect +12: +end if +13: +Compute summation of ∆S +′(X, Y) for each effect. +14: +end for +15: end for +16: for ttimestep = 1, 2, . . . , T do +17: +for nregion = 1, 2, . . . N do +18: +Use Equation 4 to update final spectrum at tT. +19: +Lower the resolution of final spectrum at tT to match +the HARPS-N observation. +20: +end for +21: end for +3.1. The basic concept of GPU computing +The popularity of artificial intelligence has in recent year sig- +nificantly increased due to the programmability of graphic hard- +wares. GPU computing uses graphical card as a co-processor for +parallel computing. Compared with CPU, GPU solves problems +by breaking them into separate tasks and processing them simul- +taneously. The basic computational unit that can independently +perform simple calculation in a graphic card is called a thread. +A group of threads that communicate and share memory with +each other is called a block. +The new version of SOAP presented here, SOAP-GPU, +is written using the Compute Unified Device Architecture +(CUDA). CUDA is a compiler and toolkit for programming +NVIDIA GPUs, and is an extension of the C/C++ programming +language. CUDA invokes kernel functions by using the syntax of +<<< Nblocks, Nthreads >>>. This syntax allows the user to define +the thread hierarchy before launching in parallel the same pro- +gram function called kernel to many threads. In order to launch +the computation at the level of the GPU, a host function defined +in CPU controls the data transfer between CPU and GPU and +can execute the kernel function inside the GPU. +Threads in the same block can be accessed as 1D, 2D or +3D structures. In order to perform the thread level calculation, +the index of individual thread and block need to be accessed. +The index of each thread in the same block can be expressed +as threadIdx. If the block is launched as the 1D structure, each +thread in the same block can be accessed as threadIdx.x. The +number of the treads used in each 1D block can be obtained +as blockDim.x. Grid is a group of blocks. It can be either 1D, +2D or 3D. For the 1D grid, the index of each block in the +grid can be expressed as blockIdx.x. Since the input spectra +of quiet sun and active region are both 1D, we used the con- +figuration of 1D grid with 1D block and the global index is +index = blockIdx.x ∗ blockDim.x + threadIdx.x. +3.2. Fast linear interpolation with GPU +As mentioned in previous sections, the major limitation in SOAP +2.0 is the way it handles linear interpolation in each disk cell. A +GPU provides thousands of cores which can be implement for +linear interpolation for large data array. Considering that both +quiet sun and spot spectra are evenly sampled in the wavelength +domain, then the input wavelength can be described as: +λn = λ0 + nk, +(5) +where n is the pixel number and k is the step size. When a +Doppler shift is applied, the wavelength array is modified as fol- +low: +λ +′ +n = λn + λn f(β), +(6) +where f(β) = − +� +1 − +� +(1+β) +(1−β) +� +and β = v/c. The variable v is the +velocity for each cell and c is the speed of light. Since we need +to project the shifted spectrum back to the original wavelength +space S (λ +′) → S +′(λ), we have to find the index m which satisfies +λ +′ +n < λm < λ +′ +n+1. For the left side, we have: +λ +′ +n < λm, +λ +′ +n = λn + λn f(β) = λ0 + nk + λ0 f(β) + nk f(β) < λ0 + mk, +so we have: +n(1 + f(β)) + f(β)λ0 +k +< m. +(7) +For the right side, we have: +m < (n + 1)(1 + f(β)) + f(β)λ0 +k +. +(8) +Once the integer m is known, we can estimate the flux for λm +using the spectrum derivative: +S +′ +m = +∆S n +(λ +′ +n+1 − λ +′ +n) × (λm − λ +′ +n) + S n, +(9) +where S n = S (λn) and ∆S n = S n+1 − S n. +Equations 7 to 9 can be parallelised using GPU. We launch +1D grid of 1D blocks with <<< Nblocks, Nthreads >>> to per- +form the linear interpolation mentioned above and the number of +blocks and threads satisfies Diminput_spectrum = Nblocks × Nthreads. +The pseudo code for this part is summarised in Algorithm 3 and +the quiet sun spectra integration can be rewritten as Algorithm +4. +Algorithm 3 Fast interpolation with GPU +1: index = blockIdx.x ∗ blockDim.x + threadIdx.x +2: indextarget = ceil(index ∗ (1 + f(β)) + f(β) ∗ λ0/k) +3: S +′ +indextarget = +∆S index +(λ′ +index+1−λ′ +index) × (λindextarget − λ +′ +index) + S index. +Article number, page 4 of 17 + +Yinan Zhao et al.: SOAP-GPU +Algorithm 4 Quiet spectrum integration with GPU +1: for Xlocation = 1, 2, . . . N do +2: +for Ylocation = 1, 2, . . . , N do +3: +Apply Doppler shift with velocity velX,Y and derive +S +′ +quiet(λ) using GPU fast interpolation +4: +Weight S +′ +quiet(λ) by limb-darkening intensity IX,Y and +integrate spectra along disk surface. Squiet+ = IX,Y ×S +′ +quiet(λ) +5: +end for +6: end for +3.3. Active region updates +As addressed in the previous section, one of the disadvantage of +SOAP 2.0 is that each active region is modeled independently, +which makes the code unable to handle complicated active re- +gion configurations: some active regions may overlap with each +other; spots may be surrounded by facualae regions. Here we +propose a revised algorithm to update active regions: an empty +disk map called In foMap is allocated in the GPU first. At each +timestamp, A list of active regions with their properties is up- +loaded and the code calculates the location of active regions pro- +jected on the disk map. If some regions are visible, we update +the corresponding pixels with their active region types in the +information map. For example, if a faculae region is visible at +(xn, yn), In foMap(xn, yn) = 1.0. If there are multiple regions +with the same type overlapping with each other, the overlapping +region in the information map will remain the same. This will +avoid the over-calculation for the overlapping region issue in the +SOAP 2.0 since each acitve region is calculated independently. +This algorithm can also simulate complicated active region con- +figurations. For example, a spot surrounded by a large faculae +can be simulated by updating the information map with a fac- +ulae first. If the spot region is embedded inside the faculae, the +overlapping region in the information map will be updated with +the type of the spot. The pseudo code of this part is summarised +in Algorithm 5. +Algorithm 5 Active region updates with GPU +1: for ttimestep = 1, 2, . . . , T do +2: +for nregion = 1, 2, . . . N do +3: +Localize(lat, long, i, ph). +4: +if Localize = 1 then +5: +Updating InfoMap(xn, yn) = the type of active +region. +6: +end if +7: +end for +8: +Inject velocity velX,Y with GPU fast interpolation for the +active regions in the information map. and derive S +′ +active(λ). +9: +Weight S +′ +active(λ) by limb-darkening intensity IX,Y +10: +Use Equations 1 to 3 to calculate the difference of each +effect +11: +Compute summation of ∆S +′(X, Y) for each effect. +12: +Use Equation 4 to derive the final spectrum at tT. +13: +Lower the resolution of final spectrum at tT to match the +HARPS-N observation. +14: end for +3.4. Differential rotation +In the original SOAP 2.0 code, there is no differential rotation +implemented. In order to better model stellar activity, differential +rotation is included when the stellar disk is initialized according +to the equation ω = ω0 + ω1 sin2(θ), where ω0 = 14.371◦/day +and ω1 = −2.587◦/day for the Sun (Borgniet et al. 2015). To +generalise this for other stars, the user can select in the configu- +ration of SOAP-GPU a rotation period and a differential rotation +rate. ω0 is then equal to 360/PROT and ω1 to DIFF_ROT*PROT +(PROT=25.05 and DIFF_ROT=-0.18 for the solar case to repro- +duce the above equation). +4. Results +Fig. 2. The computation speed comparison between SOAP 2.0 and +SOAP GPU: The integrated quiet sun spectrum is calculated with dif- +ferent disk resolutions. When disk resolution is below 10, SOAP 2.0 +is faster than SOAP-GPU since the communication between CPU and +GPU in SOAP-GPU is time-consuming. For resolutions above 10, +SOAP-GPU is significantly faster than SOAP 2.0. With a typical res- +olution value of 300, the quiet disk spectrum integration in SOAP-GPU +is 100 times faster than in SOAP 2.0 +4.1. Performance and precision comparison with SOAP 2.0 +We examined the performance of SOAP-GPU in two aspects: +computational speed and accuracy. SOAP-GPU code is executed +on a Nvidia RTX-3090 card while we run the modified SOAP 2.0 +that generates spectra in a MacBook Pro with 2.6 GHz 6-Core +Intel Core i7. We analysed the speed performance of SOAP- +GPU by calculating the time it takes to obtain an integrated quiet +sun spectrum. The input quiet sun spectrum has a dimension of +547840, thus the kernel function fast interpolation is launched +with <<< Nblocks, Nthreads >>>=<<< 1070, 512 >>>. We note +that Nthreads is fixed to 512 and Nblocks is an adaptive number +based on the dimension of the input. SOAP 2.0 is executed with +the same simulation configuration on a single CPU. We com- +puted the integrated quiet sun spectrum with different disk res- +olution and their computational time is show in Figure 2. When +the disk resolution is very low, smaller than 10, SOAP 2.0 is +faster than SOAP-GPU. This is not surprising since the data +transfer between GPU and CPU in SOAP-GPU is the dominating +factor. When the disk resolution increases, SOAP-GPU is signif- +icantly faster than SOAP 2.0. When the resolution is above 100, +the quiet sun spectrum integration of SOAP-GPU is 100 times +faster than SOAP 2.0 and both computational curves linearly in- +crease in log-log space. +Article number, page 5 of 17 + +103 +SOAP 2.0 +SOAP GPU +time (seconds) +102 +101 +Comuputation +100 +10-1 +100 +101 +102 +Disk resolution (N)A&A proofs: manuscript no. SOAP_GPU +Boisse et al. (2012) found no significant change in their re- +sults with resolution beyond 300, therefore, we used this disk +resolution for the following of the paper. For the typical disk +resolution of 300, a spot at disc center with an area of 1% of the +entire disk will be contained in a grid of 34×34 cells. Due to the +small size of the grid for such a configuration, the fast interpo- +lation algorithm (see Algorithm 3) is only able to gain a factor +of ∼10 in computation time. If the spot size increases to 9% of +the entire disk, the simulation can then gain almost the full speed +boost from fast interpolation (100 time faster). Fast interpolation +at the level of the active region modelisation makes therefore sig- +nificant improvements in computational speed when considering +high-resolution simulations or simulations with large active re- +gions. +Fig. 3. Comparison of the RVs derived from the simulated spectra mod- +eled by SOAP2.0 and SOAP-GPU. A single equatorial spot with 1% +area of the entire disk surface is simulated. It took 1749.3 seconds to +simulate those spectra with SOAP 2.0 while only 27.9 seconds with +SOAP-GPU on a Nvidia RTX-3090 card. The computation speed is im- +proved by a factor of 63. +We also examined the accuracy of SOAP-GPU. We simu- +lated a single equatorial spot with a 1% area of the entire disk +surface using a disk resolution of 300. It took 1749.3 seconds to +simulate those spectra with SOAP 2.0 for 100 timestamps while +only 27.9 seconds using SOAP-GPU, which corresponds to a +gain of a factor 63. The modeled RVs relative to the flux effect, +the CB effect and the total effect are derived from the simulated +spectra by cross-correlating them with the same mask originally +used in SOAP 2.0, and measuring the RV as the mean of a Gaus- +sian profile fitted to the obtained CCFs. Figure 3 illustrates that +the simulated spectra from SOAP-GPU provides the same RVs +as the spectra from SOAP 2.0. +4.2. Exploration of active region properties +The dynamics of active regions plays an important role for un- +derstanding the stellar activity-induced RVs. Most of previous +study aimed at investigating these effects with real observations. +For example, Meunier et al. (2010) derived the stellar activity +induced RVs by using Michelson Doppler Imager/Solar and He- +liospheric Observatory (MDI/SOHO) magnetograms images. At +the simulation level, Gilbertson et al. (2020a) investigated the ef- +fect of spot evolution on the long-term and at the spectral level, +using a modified version of SOAP 2.0. However, they only con- +sidered spots, and only their decaying phase. In order to illustrate +the effects of active region dynamics, we discuss in this section +the photometric and RV variations observed when an active re- +gion changes in size, when different number of active regions are +present and when the active region configuration changes. +4.2.1. The size evolution of active region +To explore different active region evolution scenarios, we de- +veloped and included an evolution module in SOAP-GPU. This +module can model evolution in three different ways: i) a linear +growing phase, ii) a linear decaying phase or iii) a growing and +decaying phase modeled by an an asymmetric Gaussian func- +tion (Muraközy et al. 2014). Other user-defined functions can be +added to this module if desired. We show in Fig. 4 the impact +of active region evolution on the light-curve and on the differ- +ent RVs derived (flux, CB and total effects). For the asymmet- +ric Gaussian evolution phase, the maximum size is set to 10000 +millionths of solar hemisphere (MSH) equivalent to 1% of the +visible hemisphere, the FWHM to 10 days and an asymmetry +factor of 0.09. For the growing only, or decaying only evolution +phases, the initial size is set to 10000 MSH and the growth or +decay rate is set to 400 MSH/day. We found that both flux and +CB effects are sensitive to the evolution of active regions. +4.2.2. Complex active regions +SOAP-GPU also allows users to simulate complex active region +configurations. From the observational point of view, facluae and +spots are not independent from each other. The facula distribu- +tion is based on the spot distribution. This leads to a complex +configuration in which spots may overlap faculae (Borgniet et al. +2015; Chapman et al. 2001). In order to model such a configu- +ration, the SOAP-GPU config file allows users to define the dis- +tribution of active regions, as a sequence of spots and faculae +with given properties (size, initial longitude, initial latitude). For +example, in order to simulate a spot surrounded by a facula, the +user can define the location and the size of the large facula first +and then define a smaller spot at the same location. The region of +overlap will be replaced by the spot as mentioned in Algorithm +5. A simulation of this case is illustrated in Figure 5, with a 1% +spot surrounded by a 9% facula. Since the spot has a higher con- +trast than the faculae, the light curve and RVs of the flux effect +is dominated by the spot while the RVs of the CB effect is domi- +nated by facula. Overall, the CB RV effect induced by the facula +dominates all the other contributions, and thus the total RVs is +affected mainly by the facula, as it was already demonstrated in +several studies (e.g. Meunier et al. 2010; Dumusque et al. 2014; +Milbourne et al. 2019). +4.3. Exploration of spectral properties +In this section, we explore the input spectra properties and +demonstrate how the derived RV behaves depending on the +wavelength domain. +4.3.1. Chromatic effects of different wavelength coverage +To explore the effect induced by different wavelength coverage, +we injected into SOAP-GPU only the red or only the blue part of +the quiet sun and spot spectra (see Fig. 6). The red and blue parts +have the same dimension of 204800, which is different from the +full spectra. As the fast interpolation kernel function depends +on the dimensions of the input spectra, the code automatically +configures the kernel with the option <<< 400, 512 >>>. +Article number, page 6 of 17 + +CPU Tot +CPU FIux +CPU Bconv +4 +GPU Tot +GPU FIuX +GPUBconv +2 +[s/u] +RV +0 +-2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +PhaseYinan Zhao et al.: SOAP-GPU +Fig. 4. SOAP-GPU simulation of different active region evolution curves: A single spot with a latitude of 30◦and longitude of 180◦is simulated. +Three spot size evolution types are demonstrated: i) a fast growth and slow decay evolution is shown in the first column, ii) a linear decay evolution +curve in the second column and iii) a linear growth curve in the third column. The evolution curves and the simulated light curves are shown in the +first two rows. The RVs of the total effect, the flux effect and the CB effect are present in the rest of the rows. The simulation of an non-evolving +spot (red dashed line ) is also shown in each figure for comparison. +The measured RVs are shown in Figure 7. The RVs of the CB +effect are different between the blue and red parts. One notable +thing is that the RVs of the CB effect simulated from the red +inputs goes below zero, while we expect the CB effect to only +be positive, as it corresponds to an inhibition of CB. To confirm +that nothing was wrong at the level of the code, we injected for +the spotted region the same spectrum as the quiet Sun, but we +red-shifted it by 300 m/s to model at first order the inhibition of +CB. In this idealist case, the CB effect does not provide negative +values. After further investigation, those negative values comes +from the fact that the spot temperature is lower than the quiet +photosphere. Thus, spectral lines will change in depth, which +will induce a flux effect even when not considering the contrast +of the active regions when estimating the CB effect (see Eq. 2). +In the case of the Sun, this flux effect seen in the CB derived RVs +is mainly coming from the red part of the spectrum due to molec- +ular absorption that can be seen in the spot spectrum, but not in +the quiet photosphere spectrum. As we will see in Sect. 4.3.3, +we do not obtain negative values when injecting PHOENIX solar +equivalent spectra for the quiet and active Sun instead of the Kitt +Peak solar quiet and active atlases, however we still see a slight +asymmetry in the derived CB RV effect, pointing toward a small +flux effect contribution. This is likely because PHOENIX spectra +are not able to model all the absorptions coming from molecular +bands, and thus the flux effect seen in the CB estimation only is +stronger for the real solar spectra than for the synthetic spectra. +This issue prevent us of fully separating the flux from the CB +effect, however, we note that the total effect (flux + CB) should +be modeled properly. We note that this feature was not visible in +SOAP 2.0, as after computing the CCF for the quiet and actives +regions, we were renormalising them. +We note that in SOAP 2.0, we used a fixed contrast to model +the flux effect of active regions. This contrast was derived by +comparing the Planck function of the quiet Sun effective tem- +perature and of the spot or facula temperature4, at the average +wavelength of the input spectra 5293 Å. Now that we use spectra +4 the config file in SOAP 2.0 allows to give an effective temperature for +the quiet photosphere, 5778 K for the Sun, and a temperature difference +with respect to this former value for the spot spectrum (663 K as the +default value in SOAP 2.0). For a facula, the temperature is dependent +on the center-to-limb angle and was following what is observed on the +Sun (e.g. Fig. 3 in Meunier et al. 2010). +Article number, page 7 of 17 + +20000 +Gauss +Decay +Growth +Non +Non +Non +area +15000 +Active region +10000 +5000 +0 +1.000 +Light +0.996 +0.994 +Non +Non +Non +Gauss +Decay +Growth +7.5 +Non +Non +Non +5.0 +Gauss +Decay +Growth +RV (m/s) +2.5 +0.0 +Tot +2.5 +-5.0 +Non +Non +Non +4 +Gauss +Decay +Growth +RV (m/s) +2 +0 +Flux +-2 +-4 +-6 +4 +Non +Non +Non +Gauss +Decay +Growth +0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Phase +Phase +PhaseA&A proofs: manuscript no. SOAP_GPU +Fig. 5. Three different active region configurations are simulated. A sin- +gle spot located at the latitude of 30◦and the longitude of 180◦with a +fixed size of 1% of the entire solar disk is present in black. A single fac- +ula with the same coordinates and a fixed size of 9% of the entire solar +disk is shown in blue line. The 1% spot region surrounded with 9% fa- +clua region is labeled in red dashed line. The top panel demonstrates the +light curves of different configurations and the rest of the panels shows +the RVs of the total effect, the flux effect and the CB effect. +Fig. 6. Injecting different size of spectra in SOAP-GPU input. Three +different sets of quiet sun and spot spectra, with different wavelength +ranges are used as input to SOAP-GPU: The entire spectra with length +547840 is labeled in black. The blue and red parts of the spectra with +length 204800 are over plotted in blue and red, respectively. +as input, and not CCFs, we implemented a contrast that is wave- +length dependant to model the chromatic effects of stellar activ- +ity. To do so, we introduced a new GPU kernel function called +SOAPcontrast <<< Nblocks, Nthreads >>>. This new kernel al- +lows to perform the wavelength dependent contrast calculation: +each wavelength pixel is first accessed by the global index of the +kernel. Next on each thread, it derives the contrast by calculating +the ratio of two Planck functions at two different effective tem- +peratures. The absolute value of the contrast in the blue part of +the spectrum is higher than in the red part. This implies that the +Fig. 7. The chromatic effect of RVs. SOAP-GPU is initialized with three +different spectra: entire wavelength coverage, and only red and blue +spectral parts (see Sect. 6). A single spot with a size of 1%, a latitude of +30◦and a longitude of 180◦is modeled by the code. The measured RVs +with different input spectra are labeled with black, red and blue, respec- +tively. Top: The measured RVs of the total effect. Middle: The measured +RVs of the flux effect. An offset of 1 ms−1 is added in the red and blue +RVs. Bottom: The measured RVs of the CB effect. +flux effect for the blue part is stronger than in the red part, which +can be seen in the middle panel of Fig. 7. +4.3.2. Convection as a function of center-to-limb angle +Solar spectral line profiles become asymmetric due to convective +motions varying with physical depth inside the solar photosphere +(e.g. Dravins et al. 1981; Gray 2009). This effect also leads to a +change in shape of the bisector of spectral lines from disk-center +to the limb, as photons are coming from different physical depths +(e.g. Cavallini et al. 1985b). In order to better model the effect of +convection in SOAP-GPU, we derived this effect from very-high +spatial and spectral resolution observations of the Sun (Löhner- +Böttcher et al. 2018; Stief et al. 2019; Löhner-Böttcher et al. +2019). +We note that the varying shape of spectral line with center- +to-limb µ angle is also modelled in the STARSIM 2 code (Her- +rero et al. 2016) by fitting a fourth-order polynomial function +on magneto-hydrodynamic CIFIST 3D models (Ludwig et al. +2009). However, this fifth-order polynomial is only valid for line +depth as strong as ∼ 0.5 (see Fig. 6 in Herrero et al. 2016)5. This +was enough to model the shape change of the CCFs in STAR- +SIM 2, which does not go deeper. However, when working at the +spectral line level, this polynomial will give completely wrong +estimate for the core of deep lines, due to extrapolation. We +therefore used the quiet sun observations at different µ angles +provided in Löhner-Böttcher et al. (2019). We first measured the +bisectors of all the available iron deep lines at different µ angle in +5 The parameters mentioned here are derived from the published code +( https://github.com/rosich/starsim-2) +Article number, page 8 of 17 + +Light curve +1.002 +1.000 +0.998 +Spot only +Faculae only +0.996 +Combined +Spot only +Tot RV (m/s) +20 +Faculae only +Combined +10 +0 +Flux RV (m/s) +4 +Spot only +Faculae only +2 +Combined +-2 +Bconv RV (m/s) +Spot only +20 +Faculae only +Combined +10 +0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Phase1.0 +Iflux +0.8 +Normalized +0.6 +0.4 +0.2 +Quiet +Quiet_blue +1:8: +Quiet_red +I flux +0.8 +Normalized +0.6 +0.4 +0.2 +Spot +Spot_blue +0.0 +Spot_red +4000 +4500 +5000 +5500 +6000 +6500 +Wavelength (A)5.0 +Full +Blue +Tot RV (m/s) +2.5 +Red +0.0 +-2.5 +-5.0 +4 +Full +Flux RV (m/s) +Blue +2 +Red +0 +2 +-4 +4 +Full +Bconv RV (m/s) +Blue +2 +Red +0 +-2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +PhaseYinan Zhao et al.: SOAP-GPU +Löhner-Böttcher et al. (2019)6 , and fitted them using polynomial +functions. For µ angle smaller then 0.5, we used a straight line +to fit the bisectors of the selected deep lines, to prevent strong +divergence when extrapolating the fit towards very large depths. +For µ angle larger or equal to 0.5, 3rd order polynomial func- +tions are used to capture the curvature of the bisectors around +disk center. The final bisectors, shown in Fig. 8 are obtained by +interpolating and extrapolating those bisectors from depth 0 to +1. +To obtain the dependency of the spectral line bisector as a +function of µ in an active region, we use the observations pre- +sented in Cavallini et al. (1985a). We parameterised the bisec- +tors of the FeI at 6301.5008Å, for the disk center (µ = 1) and +different center-to-limb angles (µ = 0.82, 0.66 and 0.44). The +bottom part of the bisectors, below 0.5, is fitted using a straight +line, the upper part for which we have data, using a 5th-order +polynomial. Rather than extrapolating the fitted polynomial to- +wards very shallow depths, which can give unrealistic redshifted +values, we decided to use the more redshifted data value of the +top bisector for extrapolation. We show in Fig. 8 the obtained +active bisectors from depth 0.0 to 1.0. +Once we have our model for line bisectors at different µ an- +gles, we can use the Python module Convec.model to apply those +bisectors to the original spectra, and thus obtain different spec- +tra for different µ angles. Each cell in the stellar disk takes the +bisector that has the closest µ angle. However, the code first has +to remove the original bisector from the spectral lines of the in- +put quiet and active Kitt Peak solar spectra. To do so, we select +the same lines as in Löhner-Böttcher et al. (2019) in the input +spectra and measure their individual bisectors. To model the av- +erage bisector of the lines selected in the quiet spectrum, we use +a second-order polynomial. For the active spectrum, due to the +lower effective temperature, the wings of certain lines fitted are +blended, which strongly impact the bisector measurement. We +therefore rejected bisector points that are significantly off. Then, +we model the average active bisector of the lines by fitting the +regions below and above a depth of 0.5 with two different linear +models. Fitting a higher-order polynomial for those active bi- +sectors was giving unrealistic values when extrapolating to very +small or very large depths. The measured individual bisectors +with our models are shown in Fig. 9. To finally obtain quiet and +active spectra with proper bisector shape as a function of µ an- +gles, we remove the original bisectors of the quiet and active +Kitt peak solar spectra, and then add the bisectors measured for +different µ angles. This is done by shifting each point in those +spectra depending on their normalised depth. +To inject the proper bisectors for different µ angles in our +original spectra, we first remove the original bisector, which +changes any CB difference between the quiet and active solar +spectra. We therefore need to impose a shift between the bisec- +tors of quiet and active regions in order to properly model the in- +hibition of CB inside active regions. We here make two assump- +tions: i) the CB is fully inhibited at µ=0.2 in the quiet Sun, and +ii) it is also fully inhibited for magnetic regions, and this at all µ +angles. Using the first assumption, we measure for the quiet Sun +the maximum shift between the bisector at µ=0.85, which is the +bisector that is the most blueshifted, and the bisector at µ=0.2. +This maximum happens at a depth of 0.58 and equals to 375 m/s. +This value is extremely similar to the 340 m/s CB value derived +from a fit to the data of Liebing et al. (2021) (see Sect. 4.3.3). +6 We use the following lines: FeI 5250.2084Å, FeI 5250.6453Å, FeI +5434.5232Å, FeI 5432.9470Å, FeI 5576.0881Å, FeII 6149.2460Å, FeI +6173.3344Å, FeI 6301.5008Å and FeI 6302.4932Å +To match the CB relation derived in Sect. 4.3.3, we rescale the +maximum difference between the µ = 0.85 and µ = 0.2 to be +340 m/s. We note that this rescaling is negligible in the case of +the Sun, however, it will be really needed in Sect. 4.3.3 when +using PHOENIX spectra as input. Using the second assumption, +we impose that at the same depth of 0.58, the difference in veloc- +ity between the quiet bisector at µ=0.85 and all active bisectors +is also 340 m/s. We show the proper shift between the quiet and +active bisectors in Fig. 8. +We show the RV impact of considering the µ angle depen- +dency on the observed solar spectra in Fig. 10. As we can see, +the impact is not significant when looking at the shape of the +signal as a function of phase. This come from the fact that due +to limb-darkening and the projection of active regions on the +limb, most of the signal comes from larger µ angles (close to +disc center). The only significant difference is for the amplitude +of the CB effect. This is because we forced the CB difference be- +tween the quiet and active sun to be 340 m/s, while the CB differ- +ence between the quiet and active Kitt peak solar spectra is less +than 300 m/s when measuring the average difference between the +quiet and active CCF bissectors (see Fig. 2 in Dumusque et al. +2014). We note that the complexity of modifying the bisectors +depending on the center-to-limb angle is not strongly justified +when using real solar spectra as input due to the small difference +observed in the estimated RVs, however, this step is critical when +working with synthetic spectra that does not include the proper +bisectors, as described in Setc. 4.3.3. +We are conscious that depending on the magnetic field of an +active region, the inhibition of the CB will be different and there- +fore the bisectors more or less redshifted compared to the quiet +Sun, as seen in Fig. 1 in Cavallini et al. (1985a). Also, faculae +tend to have weaker magnetic fields than spots and in our case, +we model those two active regions with the same bisectors and +the same CB inhibition. It is therefore likely that the CB effect +for faculae is slightly overestimated, and this will translate in +larger RV amplitudes when modeling the CB effect for faculae. +In summary, in this subsection we present a framework to +model the Sun but also other stars (see also next subsection). +Different bisectors at different µ are derived from the quiet pho- +totsphere (Löhner-Böttcher et al. 2019) and facuale (Cavallini +et al. 1985b) and are injected into the input spectra for which we +have removed any variation in line bisector from a vertical line. +4.3.3. Simulation based on the PHOENIX spectral database +The implementation of convective motions described in the pre- +ceding section allow us to use synthetic spectra as input, since +the effect of convection can be injected using the Convec.model +module. In order to study stellar activity affecting the data used +in RV, a high resolution spectral library is needed. For SOAP- +GPU, we decided to make it easy for the user to use as input +PHOENIX high-resolution spectra (Husser et al. 2013). We note +however that SOAP-GPU can accept other spectral libraries, but +it might be a little more difficult for the users to properly setup +the inputs since the parameters to remove bisectors of input spec- +tra are only optimized for the PHOENIX spectra and the so- +lar atlas from the Kitt Peak Observatory FTS (Wallace et al. +2005). The PHOENIX library propose a collection of spectra +with the wavelength coverage from 500Å to 5.5µm with reso- +lutions of 500,000 in the optical. The library covers stellar ef- +fective temperature from 2300K to 12000K. Since the spectra +in the PHOENIX library are not normalized, which is critical to +perform the injection of CB described in the preceding section, +Article number, page 9 of 17 + +A&A proofs: manuscript no. SOAP_GPU +Fig. 8. Average bisectors of quiet and active solar regions from the disk center (µ = 1.0) to the limb (µ = 0.2). Continuous lines: Fifth-order poly- +nomial fit to the quiet sun bisectors of the FeI 5250.2084Å, FeI 5250.6453Å, FeI 5434.5232Å, FeI 5432.9470Å, FeI 5576.0881Å, FeI 6149.2460Å, +FeI 6173.3344Å, FeI 6301.5008Å and FeI 6302.4932Å lines as measured by the Laser Absolute Reference Spectrograph (LARS) at the German +Vacuum Tower Telescope (Löhner-Böttcher et al. 2019). Dashed lines: Fit of the bisectors of the FeI 6301.5008Å spectral line inside a faculae +region, as measured by the Fabry-Perot interferometer at the Donati Solar Tower (Cavallini et al. 1985a). Below a depth of 0.5, a linear fit is per- +formed, while a fifht-order polynomial is used to model the top part of the bisector. To prevent unrealistic value when interpolating the polynomial +above a normalised flux of 0.9 where no measurement exists, we selected the most redshifted part of the top bisector, explaining the vertical values +for very shallow depths. The two vertical lines are shifted by 340 m/s which corresponds to the solar convective blueshift value derived from a fit +to the data of Liebing et al. (2021) (see Sect. 4.3.3). The active bisectors at different µ angles are all shifted by those 340 m/s at a depth of 0.58 as +we make the hypothesis that convection is fully suppressed in magnetic regions (see Sect 4.3.2 for more information). +Fig. 9. Bisectors of the FeI lines used in Löhner-Böttcher et al. (2019) and fitted model to account for the CB. Left: Bisectors from the quiet Kitt +Peak solar spectrum. Right: Bisectors from the active Kitt Peak solar spectrum. We rejected the bottom part of the 6301.5008Å bisector because it +was significantly off by 2500 m/s due to strong contamination by other weak lines. +we used the open source package “Rassine” (Cretignier et al. +(2020b)) to perform spectral normalization first. The continuum +or spectral energy distribution (SED) of input spectra derived +from “Rassine” are denoted as SEDquiet and SEDactive respec- +tively. +The inhibition of CB when working with PHOENIX spectra +can be rewritten as: +∆S +′ +bconv(X, Y) = S +′ +quiet,n(X, Y) − S +′ +active,n(X, Y), +(10) +where S +′ +quiet,n and S +′ +active,n are the normalized quiet and active +spectra. Since the contrast between the quiet and active region +Article number, page 10 of 17 + +1.0 +μ = 0.20 +μ=0.30 +0.8 +μ=0.40 +μ=0.50 +μ=0.60 +Normalized flux +0.6 +μ= 0.70 +μ=0.80 +μ=0.85 +μ=0.90 +0.4 +μ= 0.95 +μ = 1.00 +Active region μ =0.44 +Active region μ =0.66 +0.2 +Active region μ =0.82 +Active region μ =l.0 +0.0 +-200 +0 +200 +400 +Doppler velocity (m/s)1.0 +1.0 +0.8 +0.8 +Fit +Binned data += 0.6 +Fel5250.2084A +0.6 +Normalized +Normalized +Fel 5250.6453A +Fel5434.5232A +Fel 5432.9470 A +Fit +0.4 +0.4 +Binned data +Fel5576.0881A +Fel 5250.2084A +Fel 6301.5008A +Fel5250.6453A +0.2 +0.2 +Fel 5434.5232A +Fel5432.9470A +Fel5576.0881A +0.0 +0.0 +Fel 6301.5008 A +-200 +-100 +0 +100 +200 +300 +-300-250-200-150-100 +-50 +0 +50 +Doppler Velocity (m/s) +Doppler Velocity (m/s)Yinan Zhao et al.: SOAP-GPU +Fig. 10. Comparison of the RVs derived using two different configu- +rations for the input spectra. A single equatorial spot with 1% area of +the entire disk surface is simulated in both cases. Red line: RVs derived +from simulated spectra with observed quiet sun and spot spectra without +µ dependent bisector injection. Dotted line: RVs derived from simulated +spectra with µ dependent bisector injection. The input spectra with µ- +angle dependency are generated with the Python module Convec.model. +The RVs of the flux, CB and total effect don’t change significantly when +the µ-dependent CB is introduced. +is naturally included in the continuum of input PHOENIX spec- +tra, the flux effect can be rewritten as: +∆S +′ +flux(X, Y) = S +′ +quiet,n(X, Y) × SEDquiet +− LB(X, Y) × S +′ +quiet,n(X, Y) × SEDactive, +(11) +where LB is the function of limb brightening. For simulation of +spot regions, LB = 1.0 along the disk. For simulation of faculae +regions, SOAP 2.0 was using ∆T f = 250.9 − 407.7µ + 190.9µ2 +to model the limb brightening in temperature domain (Meu- +nier et al. 2010). Since this equation is only valid for the Sun, +we use the empirical equation derived from 3D MHD simula- +tions to model other spectral types. 3D MHD simulations mod- +elling faculae on the Sun can reproduce extremely well the limb- +brightening observed for faculae (Norris et al. 2017). Using sim- +ilar simulations, Johnson et al. (2021a) model what would be +the limb-brightening on other stars (see Figure 3 and Table 1 in +Johnson et al. (2021a)). Using the parametrisation in Johnson +et al. (2021a), we derived the limb brightening curves of faculae +for a G2, K0 and M0 dwarfs. We then linearly interpolated be- +tween the G2 and K0 and K0 and M0 simulations to obtain the +limb-brightening dependence for a G8 and G9 dwarf, and a K2 +dwarf, respectively. To obtain the dependence for a F9 dwarf, we +linearly extrapolated from the G2 and K0 models. The derived +limb brightening curves from F9 to K2 are shown in Figure 11. +Finally, the total effect from flux and convection can be de- +rived using the following equation: +∆S +′ +tot(X, Y) = S +′ +quiet,n(X, Y) × SEDquiet +− LB(X, Y) × S +′ +active,n(X, Y) × SEDactive. +(12) +Convection velocity changes with spectral type, and de- +creases towards cooler stars than the Sun. This is a well known +Fig. 11. Faculae intensity contrast as a function of µ for different spec- +tral types. The limb brightening curves for the G2, K0 and M0 dwarfs +are derived from the parametrisation of MHD simulations for 500G fac- +ulae, shown in Table 1 in Johnson et al. (2021a). The limb brightening +curves for other spectral types are linearly interpolated by using the two +closet limb brightening curves. For spectral types hotter than G2, we ob- +tain limb brightening by performing linear extrapolation using the G2 +and K0 curves. The limb brightening derived from Meunier et al. (2010) +is labeled with orange. +effect, that comes out from observations (e.g. Meunier et al. +2017; Liebing et al. 2021) and magneto hydrodynamic mod- +els of stellar phostophere (e.g. Allende Prieto et al. 2013). As +in Liebing et al. (2021), we show in the top panel of Fig. 12 +that the CB is a cubic function of effective temperature in the +range 4800 to 6300 K. The parametrisation that we obtain is +CBvel = 95.2388 × ((Teff − 4400)/1000)3 + 91.2791. Once we +have this relation to measure the velocity of CB as a function of +effective temperature, we simply have to rescale the difference +between the quiet Sun bisectors for µ = 0.85 and µ = 0.2 at +depth 0.58 to be equal to the value given by our relation, and +we also impose that the active bisectors are shifted by the same +value, at the same depth (see Sect. 4.3.2 for justification). This +allows us to model properly the change of convective velocity as +a function of stellar effective temperature. We note that as in the +case of the Sun, before injecting the proper bisector for the quiet +and active regions, we first have to remove the bisector present +in the PHOENIX spectra that we use. This is a necessary step as +the PHOENIX spectral library is obtained from 1D atmospheric +models and cannot properly reproduce line bisector shape. We +show in the appendix, like for Fig. 9 in the case of the Sun, how +the bisector of the original PHOENIX spectra for the quiet stellar +region, a faculae and a spot, are fitted before being removed. +In Fig. 13, we show the results of a few simulations consid- +ering µ dependent input spectra and a single 1% equatorial spot +(left panel) or a single 1% facula (right panel). We highlight the +RV outputs of SOAP-GPU when using PHOENIX spectra, with +the quiet Sun temperature set to Teff = 5778K, the spot temper- +ature set to Teff = 5115K, 5015K and 5215K and the facula tem- +perature set to Teff = 5928K, 6028K and 6128K. We also show +the result when inputting the Kitt Peak solar quiet and spot or +facula spectra (Teff = 5778K and 5115K or 6028K at disk cen- +ter, respectively) as modified in Sect. 4.3.2, and including (us- +ing Eq. 11, 10 and 12) the SED from corresponding PHOENIX +spectra. As we can see, the amplitude of the RV flux effect in- +creases with an increase in temperature difference between the +quiet photosphere and the spot or facula. The CB effect does not +change in amplitude with temperature difference and thus the +larger amplitude observed for larger difference in temperature +Article number, page 11 of 17 + +6 +With μ-conv +Tot RV (m/s) +Without μ-conv +2 +-2 +-4 +4 +Withμ-conv +Flux RV (m/s) +2 +Without μ-conv +0 +-2 +4 +Withμ-conv +Bconv RV (m/s) +3 +Without μ-conv +2 +1 +0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Phase0.25 +Spectral type: F9 +Spectraltype: G8 +Spectral type: G9 +0.20 +Spectral type: K2 +G2 Meunier et al. 2010 +Intensity Contrast +G2 500G +0.15 +K0 500G +M0500G +0.10 +0.05 +0.00 +0.2 +0.4 +0.6 +0.8 +1.0 +μA&A proofs: manuscript no. SOAP_GPU +Fig. 12. CB velocity as a function of spectral type. Top: Data from +Liebing et al. (2021) and cubic fit to them, giving as relation CBvel = +95.2388 × ((Teff − 4400)/1000)3 + 91.2791. We show with coloured +stars the CB velocity value for different spectral types that we model in +the paper. Bottom: The bisector of the quiet photosphere measured from +Löhner-Böttcher et al. (2019) at µ = 0.85 (thin lines) and the bisector of +an active region measured from Cavallini et al. (1985a) at µ = 1.0 (thick +lines) for different spectral types. We impose that the maximum differ- +ence between the quiet bisector at µ = 0.85 and µ = 0.2 (not shown +here), happening at a depth of 0.58 is equal to the CB velocity given by +the relation found in the top panel. We also force the difference between +the quiet bisector at µ = 0.85 and the active bisectors, independent of +their µ angle, to be equal to the same value at a depth of 0.58. +between quiet and active regions is solely driven by the change +in contrast of the active regions with temperature. While for the +flux effect, the simulation using PHOENIX spectra or observed +solar spectra as input gives the same results, which is not surpris- +ing as we use the same SED, this is not the case for the CB ef- +fect. The amplitude derived show a small discrepancy of ∼20%, +and the maximum has a slight phase shift. This likely comes +from a different flux effect contribution seen in the derived CB +RV, due to molecular absorption not perfectly modeled in the +PHOENIX spectra compared to solar real observations (see dis- +cussion in Sect. 4.3.1). We note that this asymmetry was already +something seen in the original SOAP 2.0 paper (see Fig. 6 in Du- +musque et al. 2014). Something also interesting to note, that we +see when using as input both the PHOENIX and solar spectra is +the bump in the CB RV effect when the active region crosses the +center of the disk (phase=0.5). This is induced by the fact that +CB is maximum at µ = 0.85 and not at disk center, as was shown +in Löhner-Böttcher et al. (2019). +In Fig.14 we show the result of the estimated CB RV effect +for an equatorial 1% spot or facula for stars of different tempera- +ture (i.e. spectral type): 6050 K (F9), 5778 K (G2), 5480 K (G8), +5380 K (G9) and 5100 K (K2). For the spot, we see a positive +only effect for the F9 and G2 simulations, which is expected, but +for later spectral type, we start to see the emergence of a flux +effect. This effect, as already discussed in Sect.4.3.1 comes from +the absorption of molecules, that change significantly over a few +hundreds of Kelvin for the quiet photosphere at 5480 K and a +spot at 4817 K for the G8 simulation for example. Also, we see +that the more we go towards cooler stars, the more the CB ef- +fect show a flux contribution, and this comes from the fact that +molecular absorption is not linear with effective temperature. Al- +though molecular absorption prevent us of clearly separating the +flux effect of spots from the CB effect like in the case of the +F9 or G2 star, the total RV effect including both contributions is +still properly estimated. Therefore, users should be careful when +interpreting the estimated RV induced by the inhibition of con- +vection for spots on stars with spectral type later than the Sun, +however, they can trust the total RV effect estimated. Regarding +the facula, we observe a positive only effect for all spectral type, +and thus the CB effect is properly modeled for this type of active +regions. +4.3.4. Limitation of SOAP-GPU when moving away from +solar twins +On a careful note, we want to warn the user that a lot of the +physics included in SOAP-GPU is based on solar observations, +like for example the variation of the bisector of the quiet photo- +sphere with respect to the center-to-limb angle (Löhner-Böttcher +et al. 2019), or the bisector of an active region extracted from +solar observations (Cavallini et al. 1985a). Therefore, although +we try to correct for some effects, like the variation of CB veloc- +ity as a function of spectral type, the more we go away from the +solar case, the more we should be careful about interpreting the +results coming out of SOAP-GPU. +We note that when modifying the bisector shape of solar or +PHEONIX spectra for the disk center, to account for different +convection velocity across spectral type, we model the effect of +the “third signature” of granulation (Gray 2009). The inherent +shape of the bisector (known as the “second signature” of granu- +lation) and how it varies with µ (Löhner-Böttcher et al. 2019) is +still the one observed for the Sun and it is well known in the liter- +ature that the bisector shape of disc-integrated observations, and +therefore by analogy at disk center, varies significantly among +luminosity class and spectral type (see Figure 17.15 in Gray +2008; Ba¸stürk et al. 2011). In those papers, we see that inside the +small range from early G’s dwarfs to early K’s dwarfs, the bisec- +tor shapes does not change drastically and therefore, although +the integrated spectra for those stars won’t be realist in terms of +line bisector, the way SOAP-GPU models stellar activity should +still be quite realistic as in the end, what counts is the differential +between the activity and quiet phases. As we can see in the pre- +ceding subsection, besides SOAP-GPU not being able anymore +to separate the inhibition of the CB effect from the flux effect +for early K’s, the tests we performed show that the code seems +to behaves quite well in estimating the total RV effect induced +by spots and faculae. However, outside of this small range from +G to K dwarfs, bisectors are completely different, and we warn +the users that the present version of SOAP-GPU might give very +unrealistic stellar integrated spectra and estimation of stellar ac- +tivity. There is perhaps only one exception for dwarfs cooler than +early K’s, for which the velocity of convection decreases to level +that are very difficult to measure. For those stars, stellar activity +is dominated by the flux effect from spot and faculae. Therefore, +for such stars, users should ignore the output from the CB effect +and only consider the flux effect. +To better model stellar activity for stars different than the +Sun, we would require disk-resolved spectra for other stars. +Those could come from 3D MHD simulations (e.g. Johnson et al. +2021b; Dravins et al. 2021), or thanks to spatially resolved spec- +troscopic observation across stellar surfaces thanks to transiting +Article number, page 12 of 17 + +700 +Datafrom Liebing etal.2021 +Cubic fit +600 +Spectral type: F9 +Spectraltype:G2 (Sun) +Spectraltype:G8 +500 +Spectral type: G9 +RV (m/s) +Spectraltype:K2 +400 +300 +CB +200 +100 +4800 +5000 +5200 +5400 +5600 +5800 +6000 +6200 +Teff (K) +1.0 +Spectral type: F9 +Spectral type: G2 (Sun) +Spectraltype:G8 +Spectral type: G9 +0.8 +Spectral type:K2 +I flux +Normalized +0.6 +0.4 +0.2 +0.0 +-400 +-200 +0 +200 +400 +Doppler velocity (m/s)Yinan Zhao et al.: SOAP-GPU +Fig. 13. Simulation of an equatorial 1% spot (left) and 1% facula (right) with different temperatures using PHOENIX spectral library. The quiet +Sun spectrum is extracted from PHOENIX spectral library with log(g) of 4.5 and Teff = 5778K. The effective temperature of the spot spectra +are 5015K, 5115K and 5215K, and for the facula spectra 5928K, 6028K and 6128K. We also show the results of using the observed Kitt Peak +solar spectra including the PHOENIX SEDs (using Eq. 11, 10 and 12). The µ dependent CB is included in the simulations. Left: For the spot we +see that when the difference in temperature between the spot and the quiet photosphere increases, the RV flux effect becomes larger. We note a +rather large discrepancy in the CB effect between the observed solar and PHOENIX input spectra. This is likely due to the fact that the observed +solar spectra are not well normalized (see Sect 4.3.3). Right: For the facula, we note the same problem of negative values for the CB effect, which +expected as the same badly normalised solar active spectrum is used. We also see that considering the corresponding PHOENIX SED when using +the observed solar spectra significantly change the flux contribution (see Sect 4.3.3). Considering the PHOENIX SED gives results much closer to +the PHOENIX simulations. +Fig. 14. Simulation of the CB RV effect of an equatorial 1% spot (top) +and 1% facula (bottom) for different temperature of the quiet photo- +sphere (i.e. different spectral type) using the PHOENIX spectral library. +The temperature difference between the quiet photosphere ∆Teff for spot +and facula is fixed to 663K and 250 K, respectively. Due to the strong +absorption effect of molecules in spots on stars cooler than the Sun, we +see the appearance of a flux effect in the derivation for the CB effect +only (see Sect 4.3.3). +planets (e.g. Dravins et al. 2017). Both approach are challeng- +ing, however could led to a much more realistic modelisation +of stellar activity from other stars than our Sun. We note that +it is rather easy to input different spectra in SOAP-GPU than +the ones provided (the Sun and PHEONIX spectra), and there- +fore if users have access to disk-resolved spectra with more re- +alistic line-bisectors, SOAP-GPU could still be used to obtain +efficiently disk-integrated spectra and model the corresponding +stellar activity effect. We note that the CB injection part is an +individual module in SOAP-GPU that users can easily modify. +5. Conclusions +In this paper, we present a GPU-based improvement to SOAP +2.0, named SOAP-GPU, that allows to efficiently model stellar +activity at the spectral level. With the implementation of GPU in- +terpolation and summation, benchmark calculations demonstrate +that SOAP-GPU improves the computational speed by a factor of +60 when modeling stellar activity on a full visible spectral range +at R=115’000 of resolution, while having the same accuracy. +Beside the huge gain in speed, SOAP-GPU also provides +a more complex modelisation of stellar activity compared to +SOAP 2.0. Complex active region scenarios, with regions over- +lapping is now handled by the code. This is mainly useful when +modeling active phases of a star like the Sun, with hundreds of +active regions. The contrast of the active regions is now wave- +length dependant, and therefore change for each wavelength +of the modeled spectra. The dependence of line bisector with +center-to-limb angle µ, following the work of Löhner-Böttcher +et al. (2019) and Cavallini et al. (1985a), is also now accounted +for for the Kitt Peak observed quiet and active atlases, but also +for PHOENIX spectra. Although the induced RV effect is rather +negligible when using the observed solar spectra as input, includ- +ing the framework to change line bisector is crucial to properly +model convection when injecting PHOENIX spectra as input. +The use of PHOENIX spectra allows us now to model a wide +variety of stars with different stellar and active region properties, +and allows us as well to better model faculae, as the correspond- +ing spectrum now has the proper effective temperature (SOAP +2.0 was using the Kitt Peak sunspot atlas to model faculae). +Article number, page 13 of 17 + +6 +SolarspectrawithPhoenixSED +Phoenix spectra,spot Teff=5015K +4 +Phoenix spectra,spot Teff=5115K +: RV (m/s) +Phoenixspectra,spotTeff=5215K +2 +Tot +0 +-2 +4 +Flux RV (m/s) +2 +0 +-2 +4 +RV (m/s) +3 +2 +Bconv F +1 +0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Phase6 +Solar spectra with Phoenix SED +Phoenixspectra,faculaeTeff=5928K +4 +Phoenix spectra,faculae Teff=6028K +Tot RV (m/s) +Phoenix spectra,faculaeTeff=6128K +2 +0 +-2 +4 +Flux RV (m/s) +2 +0 +-2 +-4 +4 +RV (m/s) +3 +2 +Bconv +1 +0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +PhaseF9Teff=6050K +G2 Teff=5779K +Spot RV (m/s) +G8Teff=5480K +G9Teff=5380K +2 +K2Teff=5100K +0 +4 +Faculae RV (m/s) +3 +2 +1 +0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +PhaseA&A proofs: manuscript no. SOAP_GPU +When modeling the inhibition of CB effect using as input +the solar Kitt Peak quiet and active spectral atlases, we noticed +that the derived RVs go negative, which is not expected. This +comes from molecular absorption that can be seen in the spot +spectrum due to lower temperature compared to the quiet Sun. +Even though we do not include the contrast of the active region +when modeling the RV CB effect only (see Eqs. 2 and 10), the +difference in flux at the level of molecular absorption bands will +show up as a flux effect in the estimated CB RVs. Positive val- +ues will be added to the CB RV effect before the spot crosses +the stellar center, and negative values after, therefore creating an +asymmetry. +When modeling other stars than the Sun using PHOENIX +spectral library, users should be aware that a lot of physics in- +cluded in SOAP-GPU are based on solar observations, and al- +though the code tries to correct for known effects like the vari- +ation of CB velocity as a function of effective temperature (i.e +spectral type, see Sect. 4.3.3), the more we go away from the +Sun, the more the results should be interpreted with caution (see +discussion in Sect. 4.3.4). The modelisation of stellar activity +for other stars than the Sun is currently limited by the knowl- +edge we have about disk-resolved bisectors for such stars. Such +information is very challenging to obtain, however, 3D MHD +simulations (e.g. Johnson et al. 2021b; Dravins et al. 2021) and +resolved spectroscopic observation of other stars due to plane- +tary transits (e.g. Dravins et al. 2017) could significantly help. +Also, when modeling stars of later spectral type than the Sun, +we are not able anymore to separate clearly the inhibition of +CB effect from the flux effect due to the strong absorption of +molecules. However, the output for the total RV effect (flux plus +inhibition of CB effects) should be modeled properly. SOAP- +GPU have been tested up to a K2 star (Teff = 5100 K) with spots +663 K cooler and give satisfactory results. Modeling later spec- +tral type is challenging mainly due to continuum normalisation +of the PHOENIX spectra and injection of spectral line bisector +due to line blending, and users should be very careful about the +interpretation of the results for such stars with the present code. +There are still some improvements that could be made to bet- +ter model the physics at play. Although the spot and facula spec- +trum used when considering PHOENIX spectra as input are of +different temperature, and therefore in spectral content, we still +associate to those regions the same active bisector as measured +for the Sun on a facula (Cavallini et al. 1985a). Spots are induced +by stronger magnetic fields than facula, and thus it is likely that +the bisector of spectral lines will be slightly different. Although +it is possible to know what is the bisector of a few spectral lines +inside a spot at disk center, to our knowledge, no measurement +of spot line bisectors for different µ angles are published. Cav- +allini et al. (1985a) also show in their Fig. 1 that depending on +the facula observed, the bisector shape changes due likely to dif- +ferent magnetic field strength and therefore different level of CB +inhibition. As can be seen in Fig. 10, the effect of inducing µ +dependant spectral line shape in the quiet and active regions is +rather small, and although with more solar data about spots and +faculae we could better model the physics at play, results in terms +of RV derivation would be rather similar. This likely comes from +the fact due to limb-darkening, most of the weight is put on the +disk center, where spectral lines does not change significantly in +shape. +With the performance of SOAP-GPU, it is now possible to +model activity at the spectral level for complex stellar surfaces +with many active regions and for a long period of time. A so- +lar activity simulator, either based on statistical properties of so- +lar active regions (similar to Borgniet et al. 2015) or on the ob- +served distribution of those (similar to e.g. Meunier et al. 2010) +will be published in a forthcoming paper. We encourage any per- +son working on techniques to separate the activity effect from +planetary signals at the spectral level, to test their framework on +SOAP-GPU simulations, where photon-noise, instrumental and +telluric systematics are not perturbing the spectral timeseries. +Acknowledgements. We thank the anonymous referee for the insightful and con- +structive comments on this paper. We thank Michael Crerignier for his help in +normalizing PHOENIX spectra with RASSINE. We also thank Xiang Gao for +the constructive comments on GPU computing. This project has received fund- +ing from the European Research Council (ERC) under the European Union’s +Horizon 2020 research and innovation programme (grant agreement SCORE No +851555). This work has been carried out within the framework of the National +Centre of Competence in Research PlanetS supported by the Swiss National Sci- +ence Foundation. The authors acknowledge the financial support of the SNSF. +References +Aigrain, S., Parviainen, H., & Pope, B. J. S. 2016, MNRAS, 459, 2408 +Aigrain, S., Pont, F., & Zucker, S. 2012, MNRAS, 419, 3147 +Allende Prieto, C., Koesterke, L., Ludwig, H.-G., Freytag, B., & Caffau, E. 2013, +A&A, 550, A103 +Ba¸stürk, Ö., Dall, T. H., Collet, R., Lo Curto, G., & Selam, S. 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SOAP_GPU +Appendix A: Line bisectors of PHOENIX spectra +As discussed in Sects. 4.3.2 and 4.3.3, before injecting the µ +dependant bisector for solar or PHOENIX spectra to properly +model CB and its inhibition close to the limb and in active re- +gions, we need to remove any bisector shape already present +in the input spectra. As PHOENIX spectral library is gener- +ated from 1D spectral synthesis, the line bisectors cannot include +properly the CB effect and therefore should be close to straight. +In Fig. A.1, we show for each simulation of different spectral +types the bisector of a few iron lines that are used in Löhner- +Böttcher et al. (2019). For each spectral type simulated, we show +the bisectors for the quiet photosphere, but also for simulated +spot and faculae, 663 K cooler or 250 K hotter, respectively. As +expected, most of the bisector are close to straight lines. We how- +ever fitted the average bisector with a second order polynomial to +remove the small curvatures observed before injecting the proper +bisectors at different µ angles (see Sect. 4.3.2). It is not clear if +those curvatures are real effect in the spectral synthesis, or sim- +ply due to blends. The correction performed is small compared +to the bisectors that we inject afterward, therefore if only due to +blends, this process does not significantly change the outputs. +Article number, page 16 of 17 + +Yinan Zhao et al.: SOAP-GPU +Fig. A.1. Bisector of PHOENIX spectra. For each input seed spectrum using PHOENIX spectral library, we use five strong iron lines: FeI +5250.2084Å (green), FeI 5250.6453Å (cyan), FeI 5434.5232Å (purple), FeI 6173.3344Å (orange) and FeI 6301.5008Å (yellow) to measure the +average bisector of the input spectra. Bisector outliers outside a window of 0.1Å around each line center are rejected to avoid those points, certainly +affected by line blending, to bias our measurement of line bisector. Each line correspond to a different spectral type, and from left to right, we can +see the bisector of the spectrum used for the quiet photosphere, a spot region (663 K cooler) and a facula region (250 K hotter). We average those +line bisectors at certain depth (as shown by the red dots) and fit the obtained data with a second order polynomial. The fitted bisector is used to +remove the bisector of input seed spectrum. +Article number, page 17 of 17 + +F9 Teff = 6050 K +F9 Teff = 5387K +F9 Teff = 6300 K +1.0 - +1.0 +1.0 +0.8 +0.8 +8'0 +0.6 - +0.6 - +0.6 +0.4 - +0.4 +0.4 +0.2 +0.2 - +0.2 +0.0 - +00 +00 +-200 +-150 +-100 +-50 +50 +100 +150 +200 +-200 +-150 +-100 +-50 +50 +100 +150 +200 +200 +-150 +-100 +-50 +50 +100 +150 +200 +G2 Teff = 5778 K +G2 Teff = 5115 K +G2 Teff = 6028 K +1.0 +1.0 +1.0 +0.8 +0.8 +8'0 +0.6 - +0.6 +0.6 +0.4 - +0.4 +0.4 +0.2 +0.2 - +0.2 +0.0 - +0'0 +0.0 +-200 +-150 +-100 +-50 +0 +50 +100 +150 +200 +-200 +-150 +-100 +-50 +0 +50 +100 +150 +200 +-200 +-150 +-100 +-50 +0 +50 +100 +150 +200 +G8 Teff = 5480 K +G8 Teff = 4817 K +G8 Teff = 5730 K +1.0 +1.0 +1.0 +0.8 +80 +0.8 +0.6 +0.6 +0.6 +0.4 - +0.4 +0.4 - +0.2 +0.2 - +0.2 +0.0 - +0.0 - +0.0 +-200 +-150 +-100 +-50 +50 +100 +150 +200 +200 +-150 +-100 +-50 +0 +50 +100 +150 +200 +200 +-150 +-100 +-50 +50 +100 +150 +200 +G9 Teff = 5380 K +G9 Teff = 4717 K +G9 Teff = 5630 K +1.0 - +1.0 +1.0 +0.8 +0.8 +0.8 +0.6 +0.6 - +0.6 +0.4 +0.4 +0.4 +0.2 - +0.2 - +0.2 +0.0 - +0'0 +0.0 +200 +-150 +-100 +-50 +50 +100 +150 +200 +-200-150-100 +-50 +0 +50 +100 +150 +200 +-200 +-150-100-50 +50 +100 +150 +200 +K2 Teff = 5100 K +K2 Teff = 4437 K +K2 Teff = 5350 K +1.0 +1.0 +1.0 +0.8 +80 +0.8 +0.6 - +0.6 +0.6 +0.4 - +0.4 +0.4 +0.2 - +0.2 +0.2 +0.0 - +0'0 +0'0 +-200 +150 +100 +50 +0 +50 +100 +150 +200 +-200 +-150 +100 +50 +50 +100 +150 +200 +200 +150 +100 +-50 +0 +50 +100 +150 +200 \ No newline at end of file diff --git a/LNE3T4oBgHgl3EQfAgnY/content/tmp_files/load_file.txt b/LNE3T4oBgHgl3EQfAgnY/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..4527576ea87966fb1b2fd4ad68309c231d502f78 --- /dev/null +++ b/LNE3T4oBgHgl3EQfAgnY/content/tmp_files/load_file.txt @@ -0,0 +1,1546 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf,len=1545 +page_content='Astronomy & Astrophysics manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' SOAP_GPU ©ESO 2023 January 12, 2023 SOAP-GPU: Efficient Spectral Modelling of Stellar Activity Using Graphical Processing Units Yinan Zhao1 and Xavier Dumusque1 Department of Astronomy of the University of Geneva, 51 chemin de Pegasi, 1290 Versoix, Switzerland e-mail: yinan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='zhao@unige.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='ch January 12, 2023 ABSTRACT Context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Stellar activity mitigation is one of the major challenges for the detection of earth-like exoplanets in radial velocity mea- surements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Several promising techniques are now investigating the use of spectral time-series, to differentiate between stellar and planetary perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' In this context, developing a software that can efficiently explore the parameter space of stellar activity at the spectral level is of great importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Aims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' The goal of this paper is to present a new version of the Spot Oscillation And Planet (SOAP) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0 code that can model stellar activity at the spectral level using graphical processing units (GPUs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' We take advantage of the computational power of GPUs to optimise the computationally expensive algorithms behind the original SOAP 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0 code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' For that purpose, we developed GPU kernels that allow to model stellar activity on any given wavelength range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' In addition to the treatment of stellar activity at the spectral level, SOAP-GPU also includes the change of spectral line bisectors from center to limb, and can take as input PHOENIX spectra to model the quiet photosphere, spots and faculae, which allow to simulate stellar activity for a wider space in stellar properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Benchmark calculations show that for the same accuracy, this new code improves the computational speed by a factor of 60 compared with a modified version of SOAP 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0 that generates spectra, when modeling stellar activity on the full visible spectral range with a resolution of R=115’000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Although the code now includes the variation of spectral line bisector with center-to-limb angle, the effect on the derived RVs is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' We also show that it is not possible to fully separate the flux from the convective blueshift effect when modeling spots, due to their lower temperature and thus the appearance of molecular absorption in their spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Rather negligible for the Sun, this degeneracy between the flux and convective blueshift effect become more important when we move to cooler stars, however, this issue does not impact the estimation of the total effect (flux plus convection), and therefore users can trust this output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' The publicly available SOAP-GPU code allows to efficiently model stellar activity at the spectral level, which is essential to test further stellar activity mitigation techniques working at the level of spectral timeseries not affected by other sources of noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Besides a huge gain in performance, SOAP-GPU also includes more physics and is able to model different stars than the Sun, from F to K dwarfs, thanks to the use of the PHOENIX spectral library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' We however note that due to the limited understanding of stellar convection and activity on other stars than the Sun, the more we go away from the solar case, the more the output of the code should be taken with care.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Key words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Methods: data analysis – Techniques: radial velocities – Techniques: spectroscopic - Stars: activity 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Introduction The radial velocity (RV) method has been proven to be one of the most successful method to detect exoplanets since the dis- covery of the first exoplanet orbiting a solar-type star (Mayor & Queloz 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' In order to detect earth-like planets orbiting in the habitable zone of its parent star, a precision of a few dozens of cms−1 must be reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Although the state-of-the-art spec- trographs such as ESPRESSO, and EXPRESS are not far from that precision (50 and 58 cms−1, respectively Pepe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Brewer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 2020), the main limitation to detect Earth-like plan- ets with the RV technique is stellar activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Two major physical processes dominating stellar activity on a time scale of the host star’s rotational period are the flux imbalance due to the temper- ature difference and therefore contrast between active and quiet regions (hereafter flux effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Saar & Donahue 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Du- musque et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Donati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 2017)) and the inhibition of convective blueshift (hereafter CB effect).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' The CB effect is due to the presence of strong local magnetic fields inside active re- gions, which suppress the CB inside those regions and leads to positive RV variations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Cavallini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 1985a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Meunier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Many methods have been proposed to mitigate activity- induced variations using photometric and spectroscopic time se- ries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' In the one-dimensional time series space, many parametric models based on analytic forms or different Gaussian process (GP) frameworks have been developed to model stellar activ- ity using photometry or spectroscopic activity indicators (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Aigrain et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Rajpaul et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Aigrain et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Gilbertson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 2020b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Barragán et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Jointly modeling the data with Keplerians to model planets in addition to a GP to model stellar activity may significantly reduce the stellar activity but may also lead to overfitting when the GP kernel or priors are not wisely set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' This is particularly dangerous when the planetary properties are not constrained from transit observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Due to inherent problems in modeling stellar activity in one- dimensional time series, the community is now shifting toward modeling it in a two-dimensional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Collier Cameron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Article number, page 1 of 17 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='04259v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='SR] 11 Jan 2023 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' SOAP_GPU (2021) calculated the autocorrelation function (ACF) of cross- correlation function (hereafter CCF Baranne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 1996), to iso- late Doppler shift from shape shift variations and applied prin- ciple component analysis (PCA) on the obtained ACFs to model shape changes related to stellar activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' A planet signal of am- plitude ∼ 40 cm/s can be recovered when the algorithm is ap- plied to the HARPS-N solar data (Dumusque et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Col- lier Cameron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Dumusque et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' (2022a) projected CCFs time series onto the Fourier basis func- tions and modelled line variability using different basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Results on simulated data show a 48% reduction in RV rms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' de Beurs et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' (2022) trained a convolutional neural network (CNN) on both simulated CCFs and HARPS-N solar CCFs and were able to significantly reduce stellar activity effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' The idea behind building the CCF is to extract with the best precision the RV information contained in a spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' How- ever, key variations at the spectral level related to stellar activ- ity may be lost when performing the dimensionality reduction imposed by the CCF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Therefore, several methods have been pro- posed to disentangle stellar activities at the spectral level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Davis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' (2017) applied PCA to simulated spectral time series and demonstrated that eigen-vectors are spectral line dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Ra- jpaul et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' (2020) used GP to directly derive RV information from spectral time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' (2017) also applied mul- tivariate GP to model stellar activity on PCA-reduced spectral dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Cretignier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' (2022), based on the knowledge that the impact of stellar activity is line-depth dependant (Cretignier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 2020a), used PCA to model stellar activity in the flux-flux gra- dient space (named the “shell” space) and results on HD10700 (τ Ceti) and HD12861 (α Cen B) indicates the method can successfully remove variations from non-Doppler origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Last by not least, Binnenfeld et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' (2020, 2022) are developing the unit-sphere representation periodogram (USuRPER), to seper- ate Doppler from other RV variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' This technique is based on representing spectra as unit vectors in a multidimensional hy- perspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' The spectral time series used to evaluate the performance of the algorithms developed to mitigate stellar activity at the spec- tral level are either obtained from simulated data or real obser- vations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' The major issue with simulations, is that most of them only model the RV activity effect at the CCF level (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Du- musque et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Herrero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 2016) due to computational inefficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' A few other libraries of simulated spectra affected by stellar activity exist, but generating them takes hours to run, which is not convenient when exploring the parameter space in stellar activity and properties (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Gilbertson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 2020a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Du- musque 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Regarding real observations, solar data obtained by the HARPS-N solar telescope (Collier Cameron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Dumusque et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 2021), HELIOS on HARPS1 and more recently the solar feed of NEID (Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 2022) are the best we can get, in terms of S/N and sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' However, those spectra corresponds for the most part to quiet activity phases of the Sun (end of cy- cle 24 end beginning of cycle 25) and can only used to mitigate stellar activity for star very similar to the Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' When moving to stellar observations, the recent Extreme precision Spectrograph (EXPRES) Stellar Signals Project (ESSP) shared some valuable data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' However, due to the small number of stars and the rather small number of spectra available, it was rather difficult to com- pare different activity mitigation techniques together (Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' As a conclusion of this discussion, it is essential for the community to have access to a code that can simulate efficiently 1 https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='eso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='org/public/announcements/ann18033/ stellar activity at the spectral level, and for a wide range of stellar properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' In this paper, we present a new code, Spot Oscillation And Planet Graphical Process Unit (SOAP-GPU) based on GPU computation that can efficiently model simplified and realistic stellar activity at the spectral level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' In Sect 2, we revisit the ar- chitecture of the SOAP 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0 code it is based on (Dumusque et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 2014) and discuss about its limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' The algorithms behind SOAP-GPU are presented in Sect 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' In Sect 4, we explore the physical parameters of stellar activity and simulation of different cases are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Finally, we draw our conclusion in Sect 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' The SOAP-GPU code is publicly available on Github and Zen- odo2 along with a brief manual and some examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Revisiting SOAP 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0 In this section, we revisit the code Spot Oscillation And Planet (SOAP 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0 Dumusque et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' This code aims at modeling both the flux effect and the CB effect of active regions affecting RV measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Although the public version of the SOAP 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0 code can only simulate stellar activity at the level of the CCFs, modeling the effect at the spectral level follow the same ideas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' In this section, we first discuss the basic algorithms behind SOAP 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0 and demonstrate the limit of the code, in terms of computa- tional efficiency, when we want to model stellar activity at the spectral level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' The stellar disk is initialized with velocity and intensity fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Left: The intensity in each cell is computed depending on a limb dark- ening law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Right: The velocity in each cell is computed considering ro- tational period, stellar inclination and radius of the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' As we can see, iso-velocity lines are not vertical as we implemented differential rota- tion in SOAP-GPU, which was not the case in SOAP 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' The structure of SOAP 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0 SOAP 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0 first computes the “quiet” (without any active region) emission spectrum of the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' To do so, a 2-dimension stellar disk containing N × N cells is initialized (N being the resolu- tion of the disk, the same parameter called “grid” in Boisse et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Velocity and intensity of all disk cells are computed based on the physical configuration of the star (rotational period, stel- lar inclination and radius of the star) and a limb darkening law (as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' In each cell, the quiet photosphere spectrum is injected, weighted by the cell intensity (limb-darkening), and Doppler-shifted to the projected velocity of that cell (rotation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Linear interpolation is applied at this step to project the Doppler- shifted spectrum into the original wavelength grid, to make sure that spectra in different cells are on a common wavelength grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' We note that the public version of SOAP 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0 was using the CCF 2 code available here https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='com/YinanZhao21/SOAP_ GPU and https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='5281/zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='7499461 Article number, page 2 of 17 Light ratio field Velocity field (km/s) 300 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0 300 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='5 250 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='9 250 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='8 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='7 150 150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='5 100 100 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='5 50 50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='4 0 + 0 + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0 0 50 100 150 200 250 300 0 50 100 150 200 250 300Yinan Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' : SOAP-GPU of the high-resolution Kitt Peak Observatory Fourier Transform Spectrograph (FTS) quiet photosphere spectrum (S quiet(λ), Wal- lace et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 1998) as approximation of the quiet Sun to increase computational speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' However, injecting the original spectrum is possible, with the only difference that the dimension of the input is ∼500000, compared to 400 for the CCF, and that we will need to apply a Doppler-shift each time we want to change the veloc- ity of this spectrum, while a simple translation was sufficient in the case of the CCF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' After injecting the quiet photosphere spec- trum in each cells, the integrated quiet solar spectrum is obtained by summing the content of all the cells together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' All those pro- cesses are summarized in the pseudo code below (Algorithm 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Algorithm 1 Quiet spectrum integration 1: for Xlocation = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' N do 2: for Ylocation = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' , N do 3: Shift S quiet(λ) with velocity velX,Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' S quiet(λ) → S quiet(λ ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 4: Do linear interpolation to project the spectrum back to the original wavelength grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' S quiet(λ ′) → S ′ quiet(λ) 5: Weight S ′ quiet(λ) by limb-darkening intensity IX,Y and integrate spectra along disk surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Squiet+ = IX,Y ×S ′ quiet(λ) 6: end for 7: end for The next step consists in initializing the active regions us- ing the following parameters: the number of active regions, their size, their corresponding latitudes and longitudes, their types (ei- ther spot or faculae) and the resolution of the active region con- tour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' An active region spectrum is also needed at this step to model the CB effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' The original SOAP 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0 code uses the CCF of the observed spot spectrum in the visible obtained from the Kitt Peak Observatory FTS (S active(λ) with λ the same as for S quiet(λ) Wallace et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' The spectrum used for faculae re- gions is the same, with the difference that the contrast of such a region follow what is observed in the Sun (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 3 in Me- unier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 2010), thus brighter than the quiet sun and with a center-to-limb brightening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Other groups use synthetic spectra at different temperature to model the quiet photospehere, spots and faculae, and include the effect of CB using results from magneto- hydrodynamical simulations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' the STARSIM 2 code3 Herrero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' We note that injecting observed or synthetic spectra have their advantages and drawbacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Using observed spectra allows to better model the inhibition of convection inside ac- tive regions, but we note that if we use the observed spectra of a spot to model a facula (because an observed spectrum of a facula across the entire visible spectral range does not seem to exist), molecular features will be present in the facula spectrum despite the temperature being higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Using synthetic spectra on the contrary allows to better model the temperature, and there- fore spectral features, of the injected spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' The choice of the spectra will be addressed in the later sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' In order to simulate a spectral time series, we need to calcu- late the disk location of active regions at each timestamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' As shown in Equation 1 and Equation 2 of (Boisse et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 2012), active regions are first put in the center of the disk and their initial configuration is obtained using a rotation matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Next, to get the position of those active regions as a function of time, another rotation matrix is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' At each timestamp, the code evaluates which active regions are visible, and which ones 3 code available here https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='com/rosich/starsim-2 are hidden behind the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' This is performed by the function Localize(lat, long, i, ph), where lat and long are the latitude and longitude of the active region center, i is the inclination angle of the stellar disk and ph is the rotational phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' The output of this function is a binary;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' one if visible, zero otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' If a region is visible, the code proceed to estimate the difference between the quiet solar spectrum and active spectrum at the location of the active regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' The difference for the flux effect, the CB effect and the com- bination of the two (total) in each cell can be calculated using the following equations: ∆S ′ flux(X, Y) = S ′ quiet(X, Y) − Iratio × S ′ quiet(X, Y), (1) ∆S ′ bconv(X, Y) = S ′ quiet(X, Y) − S ′ active(X, Y), (2) ∆S ′ tot(X, Y) = S ′ quiet(X, Y) − Iratio × S ′ active(X, Y), (3) where Iratio is the contrast of the spot or faculae region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' The code then integrates over all the cells covered by active regions to get final difference between the quiet spectrum and active spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' The final spectrum of each effect at each timestamp can be cal- culated by: Sintegrated, final = Sintegrated,quiet − ∆Sintegrated,quiet−active.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' (4) Once a spectrum for each timestamp is obtained, the code then lowers the resolution of the integrated spectrum to match the resolution provided in the configuration file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' The pseudo code that describes how active regions are included, and how the final integrated spectra is obtained is summarized below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' The limitation of SOAP 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0 The structure of SOAP 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0 provides an efficient way to estimate stellar activities on spectroscopic measurement by simulating CCFs at different timestamps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' The major drawback when chang- ing the input from CCFs to spectra is the dimension of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' The dimension of the input CCFs in SOAP 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0 was 400 in veloc- ity space while the input high-resolution spectra we want to use have a dimension of ∼ 500000 in the wavelength domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' From Algorithm 1 and Algorithm 2, we clearly see that the linear in- terpolation is repeatedly called when injecting the spectrum in each cell, which is computationally expensive for an array with dimension of ∼ 500000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' For example, SOAP 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0 takes ∼ 800 seconds to calculate an integrated quiet sun spectrum using a 300×300 disk-grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Another issue is how the code handles multi- ple active regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Each active region is modeled independently, without information from other regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' This algorithm cannot handle the case in which some active regions overlap with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' From real observations, we know that some active regions have complicated configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' For example, most of active re- gions are a combination of a large faculae presenting a small spot in its center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' In this context, a more computationally efficient and generalized algorithm is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Description of SOAP-GPU In the previous section we’ve demonstrated the limitation of SOAP 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0 when modeling stellar activity at the spectral level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Here, we present a new version of SOAP, based on Graphical Processing Unit (GPU) computing, that is much more efficient in term of computational speed, but also that adds some physical complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Article number, page 3 of 17 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' SOAP_GPU Algorithm 2 Active region updates 1: for nregion = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' N do 2: for ttimestep = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' , T do 3: Localize(lat, long, i, ph).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 4: if Localize = 1 then 5: Shift S quiet(λ) with velocity velX,Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' S quiet(λ) → S quiet(λ ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 6: Do linear interpolation to project the spectrum back to the original wavelength space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' S quiet(λ ′) → S ′ quiet(λ) 7: Weight S ′ quiet(λ) by limb-darkening intensity IX,Y 8: Shift S active(λ) with velocity velX,Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' S active(λ) → S active(λ ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 9: Do linear interpolation to project the spec- trum back to the original wavelength space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' S active(λ ′) → S ′ active(λ) 10: Weight S ′ active(λ) by limb-darkening intensity IX,Y 11: Use Equations 1 to 3 to calculate the difference of each effect 12: end if 13: Compute summation of ∆S ′(X, Y) for each effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 14: end for 15: end for 16: for ttimestep = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' , T do 17: for nregion = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' N do 18: Use Equation 4 to update final spectrum at tT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 19: Lower the resolution of final spectrum at tT to match the HARPS-N observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 20: end for 21: end for 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' The basic concept of GPU computing The popularity of artificial intelligence has in recent year sig- nificantly increased due to the programmability of graphic hard- wares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' GPU computing uses graphical card as a co-processor for parallel computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Compared with CPU, GPU solves problems by breaking them into separate tasks and processing them simul- taneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' The basic computational unit that can independently perform simple calculation in a graphic card is called a thread.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' A group of threads that communicate and share memory with each other is called a block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' The new version of SOAP presented here, SOAP-GPU, is written using the Compute Unified Device Architecture (CUDA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' CUDA is a compiler and toolkit for programming NVIDIA GPUs, and is an extension of the C/C++ programming language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' CUDA invokes kernel functions by using the syntax of <<< Nblocks, Nthreads >>>.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' This syntax allows the user to define the thread hierarchy before launching in parallel the same pro- gram function called kernel to many threads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' In order to launch the computation at the level of the GPU, a host function defined in CPU controls the data transfer between CPU and GPU and can execute the kernel function inside the GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Threads in the same block can be accessed as 1D, 2D or 3D structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' In order to perform the thread level calculation, the index of individual thread and block need to be accessed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' The index of each thread in the same block can be expressed as threadIdx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' If the block is launched as the 1D structure, each thread in the same block can be accessed as threadIdx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' The number of the treads used in each 1D block can be obtained as blockDim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Grid is a group of blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' It can be either 1D, 2D or 3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' For the 1D grid, the index of each block in the grid can be expressed as blockIdx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Since the input spectra of quiet sun and active region are both 1D, we used the con- figuration of 1D grid with 1D block and the global index is index = blockIdx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='x ∗ blockDim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='x + threadIdx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Fast linear interpolation with GPU As mentioned in previous sections, the major limitation in SOAP 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0 is the way it handles linear interpolation in each disk cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' A GPU provides thousands of cores which can be implement for linear interpolation for large data array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Considering that both quiet sun and spot spectra are evenly sampled in the wavelength domain, then the input wavelength can be described as: λn = λ0 + nk, (5) where n is the pixel number and k is the step size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' When a Doppler shift is applied, the wavelength array is modified as fol- low: λ ′ n = λn + λn f(β), (6) where f(β) = − � 1 − � (1+β) (1−β) � and β = v/c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' The variable v is the velocity for each cell and c is the speed of light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Since we need to project the shifted spectrum back to the original wavelength space S (λ ′) → S ′(λ), we have to find the index m which satisfies λ ′ n < λm < λ ′ n+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' For the left side, we have: λ ′ n < λm, λ ′ n = λn + λn f(β) = λ0 + nk + λ0 f(β) + nk f(β) < λ0 + mk, so we have: n(1 + f(β)) + f(β)λ0 k < m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' (7) For the right side, we have: m < (n + 1)(1 + f(β)) + f(β)λ0 k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' (8) Once the integer m is known, we can estimate the flux for λm using the spectrum derivative: S ′ m = ∆S n (λ ′ n+1 − λ ′ n) × (λm − λ ′ n) + S n, (9) where S n = S (λn) and ∆S n = S n+1 − S n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Equations 7 to 9 can be parallelised using GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' We launch 1D grid of 1D blocks with <<< Nblocks, Nthreads >>> to per- form the linear interpolation mentioned above and the number of blocks and threads satisfies Diminput_spectrum = Nblocks × Nthreads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' The pseudo code for this part is summarised in Algorithm 3 and the quiet sun spectra integration can be rewritten as Algorithm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Algorithm 3 Fast interpolation with GPU 1: index = blockIdx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='x ∗ blockDim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='x + threadIdx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='x 2: indextarget = ceil(index ∗ (1 + f(β)) + f(β) ∗ λ0/k) 3: S ′ indextarget = ∆S index (λ′ index+1−λ′ index) × (λindextarget − λ ′ index) + S index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Article number, page 4 of 17 Yinan Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' : SOAP-GPU Algorithm 4 Quiet spectrum integration with GPU 1: for Xlocation = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' N do 2: for Ylocation = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' , N do 3: Apply Doppler shift with velocity velX,Y and derive S ′ quiet(λ) using GPU fast interpolation 4: Weight S ′ quiet(λ) by limb-darkening intensity IX,Y and integrate spectra along disk surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Squiet+ = IX,Y ×S ′ quiet(λ) 5: end for 6: end for 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Active region updates As addressed in the previous section, one of the disadvantage of SOAP 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0 is that each active region is modeled independently, which makes the code unable to handle complicated active re- gion configurations: some active regions may overlap with each other;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' spots may be surrounded by facualae regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Here we propose a revised algorithm to update active regions: an empty disk map called In foMap is allocated in the GPU first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' At each timestamp, A list of active regions with their properties is up- loaded and the code calculates the location of active regions pro- jected on the disk map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' If some regions are visible, we update the corresponding pixels with their active region types in the information map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' For example, if a faculae region is visible at (xn, yn), In foMap(xn, yn) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' If there are multiple regions with the same type overlapping with each other, the overlapping region in the information map will remain the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' This will avoid the over-calculation for the overlapping region issue in the SOAP 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0 since each acitve region is calculated independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' This algorithm can also simulate complicated active region con- figurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' For example, a spot surrounded by a large faculae can be simulated by updating the information map with a fac- ulae first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' If the spot region is embedded inside the faculae, the overlapping region in the information map will be updated with the type of the spot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' The pseudo code of this part is summarised in Algorithm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Algorithm 5 Active region updates with GPU 1: for ttimestep = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' , T do 2: for nregion = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' N do 3: Localize(lat, long, i, ph).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 4: if Localize = 1 then 5: Updating InfoMap(xn, yn) = the type of active region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 6: end if 7: end for 8: Inject velocity velX,Y with GPU fast interpolation for the active regions in the information map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' and derive S ′ active(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 9: Weight S ′ active(λ) by limb-darkening intensity IX,Y 10: Use Equations 1 to 3 to calculate the difference of each effect 11: Compute summation of ∆S ′(X, Y) for each effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 12: Use Equation 4 to derive the final spectrum at tT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 13: Lower the resolution of final spectrum at tT to match the HARPS-N observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 14: end for 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Differential rotation In the original SOAP 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0 code, there is no differential rotation implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' In order to better model stellar activity, differential rotation is included when the stellar disk is initialized according to the equation ω = ω0 + ω1 sin2(θ), where ω0 = 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='371◦/day and ω1 = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='587◦/day for the Sun (Borgniet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' To generalise this for other stars, the user can select in the configu- ration of SOAP-GPU a rotation period and a differential rotation rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' ω0 is then equal to 360/PROT and ω1 to DIFF_ROT*PROT (PROT=25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='05 and DIFF_ROT=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='18 for the solar case to repro- duce the above equation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Results Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' The computation speed comparison between SOAP 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0 and SOAP GPU: The integrated quiet sun spectrum is calculated with dif- ferent disk resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' When disk resolution is below 10, SOAP 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0 is faster than SOAP-GPU since the communication between CPU and GPU in SOAP-GPU is time-consuming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' For resolutions above 10, SOAP-GPU is significantly faster than SOAP 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' With a typical res- olution value of 300, the quiet disk spectrum integration in SOAP-GPU is 100 times faster than in SOAP 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Performance and precision comparison with SOAP 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0 We examined the performance of SOAP-GPU in two aspects: computational speed and accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' SOAP-GPU code is executed on a Nvidia RTX-3090 card while we run the modified SOAP 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0 that generates spectra in a MacBook Pro with 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='6 GHz 6-Core Intel Core i7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' We analysed the speed performance of SOAP- GPU by calculating the time it takes to obtain an integrated quiet sun spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' The input quiet sun spectrum has a dimension of 547840, thus the kernel function fast interpolation is launched with <<< Nblocks, Nthreads >>>=<<< 1070, 512 >>>.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' We note that Nthreads is fixed to 512 and Nblocks is an adaptive number based on the dimension of the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' SOAP 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0 is executed with the same simulation configuration on a single CPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' We com- puted the integrated quiet sun spectrum with different disk res- olution and their computational time is show in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' When the disk resolution is very low, smaller than 10, SOAP 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0 is faster than SOAP-GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' This is not surprising since the data transfer between GPU and CPU in SOAP-GPU is the dominating factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' When the disk resolution increases, SOAP-GPU is signif- icantly faster than SOAP 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' When the resolution is above 100, the quiet sun spectrum integration of SOAP-GPU is 100 times faster than SOAP 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0 and both computational curves linearly in- crease in log-log space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Article number, page 5 of 17 103 SOAP 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0 SOAP GPU time (seconds) 102 101 Comuputation 100 10-1 100 101 102 Disk resolution (N)A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' SOAP_GPU Boisse et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' (2012) found no significant change in their re- sults with resolution beyond 300, therefore, we used this disk resolution for the following of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' For the typical disk resolution of 300, a spot at disc center with an area of 1% of the entire disk will be contained in a grid of 34×34 cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Due to the small size of the grid for such a configuration, the fast interpo- lation algorithm (see Algorithm 3) is only able to gain a factor of ∼10 in computation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' If the spot size increases to 9% of the entire disk, the simulation can then gain almost the full speed boost from fast interpolation (100 time faster).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Fast interpolation at the level of the active region modelisation makes therefore sig- nificant improvements in computational speed when considering high-resolution simulations or simulations with large active re- gions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Comparison of the RVs derived from the simulated spectra mod- eled by SOAP2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0 and SOAP-GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' A single equatorial spot with 1% area of the entire disk surface is simulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' It took 1749.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='3 seconds to simulate those spectra with SOAP 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0 while only 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='9 seconds with SOAP-GPU on a Nvidia RTX-3090 card.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' The computation speed is im- proved by a factor of 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' We also examined the accuracy of SOAP-GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' We simu- lated a single equatorial spot with a 1% area of the entire disk surface using a disk resolution of 300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' It took 1749.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='3 seconds to simulate those spectra with SOAP 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0 for 100 timestamps while only 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='9 seconds using SOAP-GPU, which corresponds to a gain of a factor 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' The modeled RVs relative to the flux effect, the CB effect and the total effect are derived from the simulated spectra by cross-correlating them with the same mask originally used in SOAP 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0, and measuring the RV as the mean of a Gaus- sian profile fitted to the obtained CCFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Figure 3 illustrates that the simulated spectra from SOAP-GPU provides the same RVs as the spectra from SOAP 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Exploration of active region properties The dynamics of active regions plays an important role for un- derstanding the stellar activity-induced RVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Most of previous study aimed at investigating these effects with real observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' For example, Meunier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' (2010) derived the stellar activity induced RVs by using Michelson Doppler Imager/Solar and He- liospheric Observatory (MDI/SOHO) magnetograms images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' At the simulation level, Gilbertson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' (2020a) investigated the ef- fect of spot evolution on the long-term and at the spectral level, using a modified version of SOAP 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' However, they only con- sidered spots, and only their decaying phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' In order to illustrate the effects of active region dynamics, we discuss in this section the photometric and RV variations observed when an active re- gion changes in size, when different number of active regions are present and when the active region configuration changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' The size evolution of active region To explore different active region evolution scenarios, we de- veloped and included an evolution module in SOAP-GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' This module can model evolution in three different ways: i) a linear growing phase, ii) a linear decaying phase or iii) a growing and decaying phase modeled by an an asymmetric Gaussian func- tion (Muraközy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Other user-defined functions can be added to this module if desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' We show in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 4 the impact of active region evolution on the light-curve and on the differ- ent RVs derived (flux, CB and total effects).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' For the asymmet- ric Gaussian evolution phase, the maximum size is set to 10000 millionths of solar hemisphere (MSH) equivalent to 1% of the visible hemisphere, the FWHM to 10 days and an asymmetry factor of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' For the growing only, or decaying only evolution phases, the initial size is set to 10000 MSH and the growth or decay rate is set to 400 MSH/day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' We found that both flux and CB effects are sensitive to the evolution of active regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Complex active regions SOAP-GPU also allows users to simulate complex active region configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' From the observational point of view, facluae and spots are not independent from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' The facula distribu- tion is based on the spot distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' This leads to a complex configuration in which spots may overlap faculae (Borgniet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Chapman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' In order to model such a configu- ration, the SOAP-GPU config file allows users to define the dis- tribution of active regions, as a sequence of spots and faculae with given properties (size, initial longitude, initial latitude).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' For example, in order to simulate a spot surrounded by a facula, the user can define the location and the size of the large facula first and then define a smaller spot at the same location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' The region of overlap will be replaced by the spot as mentioned in Algorithm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' A simulation of this case is illustrated in Figure 5, with a 1% spot surrounded by a 9% facula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Since the spot has a higher con- trast than the faculae, the light curve and RVs of the flux effect is dominated by the spot while the RVs of the CB effect is domi- nated by facula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Overall, the CB RV effect induced by the facula dominates all the other contributions, and thus the total RVs is affected mainly by the facula, as it was already demonstrated in several studies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Meunier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Dumusque et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Milbourne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Exploration of spectral properties In this section, we explore the input spectra properties and demonstrate how the derived RV behaves depending on the wavelength domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Chromatic effects of different wavelength coverage To explore the effect induced by different wavelength coverage, we injected into SOAP-GPU only the red or only the blue part of the quiet sun and spot spectra (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' The red and blue parts have the same dimension of 204800, which is different from the full spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' As the fast interpolation kernel function depends on the dimensions of the input spectra, the code automatically configures the kernel with the option <<< 400, 512 >>>.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Article number, page 6 of 17 CPU Tot CPU FIux CPU Bconv 4 GPU Tot GPU FIuX GPUBconv 2 [s/u] RV 0 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0 PhaseYinan Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' : SOAP-GPU Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' SOAP-GPU simulation of different active region evolution curves: A single spot with a latitude of 30◦and longitude of 180◦is simulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Three spot size evolution types are demonstrated: i) a fast growth and slow decay evolution is shown in the first column, ii) a linear decay evolution curve in the second column and iii) a linear growth curve in the third column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' The evolution curves and the simulated light curves are shown in the first two rows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' The RVs of the total effect, the flux effect and the CB effect are present in the rest of the rows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' The simulation of an non-evolving spot (red dashed line ) is also shown in each figure for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' The measured RVs are shown in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' The RVs of the CB effect are different between the blue and red parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' One notable thing is that the RVs of the CB effect simulated from the red inputs goes below zero, while we expect the CB effect to only be positive, as it corresponds to an inhibition of CB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' To confirm that nothing was wrong at the level of the code, we injected for the spotted region the same spectrum as the quiet Sun, but we red-shifted it by 300 m/s to model at first order the inhibition of CB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' In this idealist case, the CB effect does not provide negative values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' After further investigation, those negative values comes from the fact that the spot temperature is lower than the quiet photosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Thus, spectral lines will change in depth, which will induce a flux effect even when not considering the contrast of the active regions when estimating the CB effect (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' In the case of the Sun, this flux effect seen in the CB derived RVs is mainly coming from the red part of the spectrum due to molec- ular absorption that can be seen in the spot spectrum, but not in the quiet photosphere spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' As we will see in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='3, we do not obtain negative values when injecting PHOENIX solar equivalent spectra for the quiet and active Sun instead of the Kitt Peak solar quiet and active atlases, however we still see a slight asymmetry in the derived CB RV effect, pointing toward a small flux effect contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' This is likely because PHOENIX spectra are not able to model all the absorptions coming from molecular bands, and thus the flux effect seen in the CB estimation only is stronger for the real solar spectra than for the synthetic spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' This issue prevent us of fully separating the flux from the CB effect, however, we note that the total effect (flux + CB) should be modeled properly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' We note that this feature was not visible in SOAP 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0, as after computing the CCF for the quiet and actives regions, we were renormalising them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' We note that in SOAP 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0, we used a fixed contrast to model the flux effect of active regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' This contrast was derived by comparing the Planck function of the quiet Sun effective tem- perature and of the spot or facula temperature4, at the average wavelength of the input spectra 5293 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Now that we use spectra 4 the config file in SOAP 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0 allows to give an effective temperature for the quiet photosphere, 5778 K for the Sun, and a temperature difference with respect to this former value for the spot spectrum (663 K as the default value in SOAP 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' For a facula, the temperature is dependent on the center-to-limb angle and was following what is observed on the Sun (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 3 in Meunier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Article number, page 7 of 17 20000 Gauss Decay Growth Non Non Non area 15000 Active region 10000 5000 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='000 Light 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='996 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='994 Non Non Non Gauss Decay Growth 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='5 Non Non Non 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0 Gauss Decay Growth RV (m/s) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0 Tot 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0 Non Non Non 4 Gauss Decay Growth RV (m/s) 2 0 Flux 2 4 6 4 Non Non Non Gauss Decay Growth 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='2 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0 Phase Phase PhaseA&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' SOAP_GPU Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Three different active region configurations are simulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' A sin- gle spot located at the latitude of 30◦and the longitude of 180◦with a fixed size of 1% of the entire solar disk is present in black.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' A single fac- ula with the same coordinates and a fixed size of 9% of the entire solar disk is shown in blue line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' The 1% spot region surrounded with 9% fa- clua region is labeled in red dashed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' The top panel demonstrates the light curves of different configurations and the rest of the panels shows the RVs of the total effect, the flux effect and the CB effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Injecting different size of spectra in SOAP-GPU input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Three different sets of quiet sun and spot spectra, with different wavelength ranges are used as input to SOAP-GPU: The entire spectra with length 547840 is labeled in black.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' The blue and red parts of the spectra with length 204800 are over plotted in blue and red, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' as input, and not CCFs, we implemented a contrast that is wave- length dependant to model the chromatic effects of stellar activ- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' To do so, we introduced a new GPU kernel function called SOAPcontrast <<< Nblocks, Nthreads >>>.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' This new kernel al- lows to perform the wavelength dependent contrast calculation: each wavelength pixel is first accessed by the global index of the kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Next on each thread, it derives the contrast by calculating the ratio of two Planck functions at two different effective tem- peratures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' The absolute value of the contrast in the blue part of the spectrum is higher than in the red part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' This implies that the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' The chromatic effect of RVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' SOAP-GPU is initialized with three different spectra: entire wavelength coverage, and only red and blue spectral parts (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' A single spot with a size of 1%, a latitude of 30◦and a longitude of 180◦is modeled by the code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' The measured RVs with different input spectra are labeled with black, red and blue, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Top: The measured RVs of the total effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Middle: The measured RVs of the flux effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' An offset of 1 ms−1 is added in the red and blue RVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Bottom: The measured RVs of the CB effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' flux effect for the blue part is stronger than in the red part, which can be seen in the middle panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Convection as a function of center-to-limb angle Solar spectral line profiles become asymmetric due to convective motions varying with physical depth inside the solar photosphere (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Dravins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 1981;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Gray 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' This effect also leads to a change in shape of the bisector of spectral lines from disk-center to the limb, as photons are coming from different physical depths (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Cavallini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 1985b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' In order to better model the effect of convection in SOAP-GPU, we derived this effect from very-high spatial and spectral resolution observations of the Sun (Löhner- Böttcher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Stief et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Löhner-Böttcher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' We note that the varying shape of spectral line with center- to-limb µ angle is also modelled in the STARSIM 2 code (Her- rero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 2016) by fitting a fourth-order polynomial function on magneto-hydrodynamic CIFIST 3D models (Ludwig et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' However, this fifth-order polynomial is only valid for line depth as strong as ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='5 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 6 in Herrero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 2016)5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' This was enough to model the shape change of the CCFs in STAR- SIM 2, which does not go deeper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' However, when working at the spectral line level, this polynomial will give completely wrong estimate for the core of deep lines, due to extrapolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' We therefore used the quiet sun observations at different µ angles provided in Löhner-Böttcher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' We first measured the bisectors of all the available iron deep lines at different µ angle in 5 The parameters mentioned here are derived from the published code ( https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='com/rosich/starsim-2) Article number, page 8 of 17 Light curve 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='002 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='998 Spot only Faculae only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='996 Combined Spot only Tot RV (m/s) 20 Faculae only Combined 10 0 Flux RV (m/s) 4 Spot only Faculae only 2 Combined 2 Bconv RV (m/s) Spot only 20 Faculae only Combined 10 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0 Phase1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0 Iflux 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='8 Normalized 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='2 Quiet Quiet_blue 1:8: Quiet_red I flux 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='8 Normalized 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='2 Spot Spot_blue 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0 Spot_red 4000 4500 5000 5500 6000 6500 Wavelength (A)5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0 Full Blue Tot RV (m/s) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='5 Red 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0 4 Full Flux RV (m/s) Blue 2 Red 0 2 4 4 Full Bconv RV (m/s) Blue 2 Red 0 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0 PhaseYinan Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' : SOAP-GPU Löhner-Böttcher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' (2019)6 , and fitted them using polynomial functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' For µ angle smaller then 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='5, we used a straight line to fit the bisectors of the selected deep lines, to prevent strong divergence when extrapolating the fit towards very large depths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' For µ angle larger or equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='5, 3rd order polynomial func- tions are used to capture the curvature of the bisectors around disk center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' The final bisectors, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 8 are obtained by interpolating and extrapolating those bisectors from depth 0 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' To obtain the dependency of the spectral line bisector as a function of µ in an active region, we use the observations pre- sented in Cavallini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' (1985a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' We parameterised the bisec- tors of the FeI at 6301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='5008Å, for the disk center (µ = 1) and different center-to-limb angles (µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='82, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='66 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='44).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' The bottom part of the bisectors, below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='5, is fitted using a straight line, the upper part for which we have data, using a 5th-order polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Rather than extrapolating the fitted polynomial to- wards very shallow depths, which can give unrealistic redshifted values, we decided to use the more redshifted data value of the top bisector for extrapolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' We show in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 8 the obtained active bisectors from depth 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Once we have our model for line bisectors at different µ an- gles, we can use the Python module Convec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='model to apply those bisectors to the original spectra, and thus obtain different spec- tra for different µ angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Each cell in the stellar disk takes the bisector that has the closest µ angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' However, the code first has to remove the original bisector from the spectral lines of the in- put quiet and active Kitt Peak solar spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' To do so, we select the same lines as in Löhner-Böttcher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' (2019) in the input spectra and measure their individual bisectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' To model the av- erage bisector of the lines selected in the quiet spectrum, we use a second-order polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' For the active spectrum, due to the lower effective temperature, the wings of certain lines fitted are blended, which strongly impact the bisector measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' We therefore rejected bisector points that are significantly off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Then, we model the average active bisector of the lines by fitting the regions below and above a depth of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='5 with two different linear models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Fitting a higher-order polynomial for those active bi- sectors was giving unrealistic values when extrapolating to very small or very large depths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' The measured individual bisectors with our models are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' To finally obtain quiet and active spectra with proper bisector shape as a function of µ an- gles, we remove the original bisectors of the quiet and active Kitt peak solar spectra, and then add the bisectors measured for different µ angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' This is done by shifting each point in those spectra depending on their normalised depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' To inject the proper bisectors for different µ angles in our original spectra, we first remove the original bisector, which changes any CB difference between the quiet and active solar spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' We therefore need to impose a shift between the bisec- tors of quiet and active regions in order to properly model the in- hibition of CB inside active regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' We here make two assump- tions: i) the CB is fully inhibited at µ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='2 in the quiet Sun, and ii) it is also fully inhibited for magnetic regions, and this at all µ angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Using the first assumption, we measure for the quiet Sun the maximum shift between the bisector at µ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='85, which is the bisector that is the most blueshifted, and the bisector at µ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' This maximum happens at a depth of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='58 and equals to 375 m/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' This value is extremely similar to the 340 m/s CB value derived from a fit to the data of Liebing et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' (2021) (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 6 We use the following lines: FeI 5250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='2084Å, FeI 5250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='6453Å, FeI 5434.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='5232Å, FeI 5432.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='9470Å, FeI 5576.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0881Å, FeII 6149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='2460Å, FeI 6173.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='3344Å, FeI 6301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='5008Å and FeI 6302.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='4932Å To match the CB relation derived in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='3, we rescale the maximum difference between the µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='85 and µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='2 to be 340 m/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' We note that this rescaling is negligible in the case of the Sun, however, it will be really needed in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='3 when using PHOENIX spectra as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Using the second assumption, we impose that at the same depth of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='58, the difference in veloc- ity between the quiet bisector at µ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='85 and all active bisectors is also 340 m/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' We show the proper shift between the quiet and active bisectors in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' We show the RV impact of considering the µ angle depen- dency on the observed solar spectra in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' As we can see, the impact is not significant when looking at the shape of the signal as a function of phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' This come from the fact that due to limb-darkening and the projection of active regions on the limb, most of the signal comes from larger µ angles (close to disc center).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' The only significant difference is for the amplitude of the CB effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' This is because we forced the CB difference be- tween the quiet and active sun to be 340 m/s, while the CB differ- ence between the quiet and active Kitt peak solar spectra is less than 300 m/s when measuring the average difference between the quiet and active CCF bissectors (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 2 in Dumusque et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' We note that the complexity of modifying the bisectors depending on the center-to-limb angle is not strongly justified when using real solar spectra as input due to the small difference observed in the estimated RVs, however, this step is critical when working with synthetic spectra that does not include the proper bisectors, as described in Setc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' We are conscious that depending on the magnetic field of an active region, the inhibition of the CB will be different and there- fore the bisectors more or less redshifted compared to the quiet Sun, as seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 1 in Cavallini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' (1985a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Also, faculae tend to have weaker magnetic fields than spots and in our case, we model those two active regions with the same bisectors and the same CB inhibition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' It is therefore likely that the CB effect for faculae is slightly overestimated, and this will translate in larger RV amplitudes when modeling the CB effect for faculae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' In summary, in this subsection we present a framework to model the Sun but also other stars (see also next subsection).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Different bisectors at different µ are derived from the quiet pho- totsphere (Löhner-Böttcher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 2019) and facuale (Cavallini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 1985b) and are injected into the input spectra for which we have removed any variation in line bisector from a vertical line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Simulation based on the PHOENIX spectral database The implementation of convective motions described in the pre- ceding section allow us to use synthetic spectra as input, since the effect of convection can be injected using the Convec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='model module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' In order to study stellar activity affecting the data used in RV, a high resolution spectral library is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' For SOAP- GPU, we decided to make it easy for the user to use as input PHOENIX high-resolution spectra (Husser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' We note however that SOAP-GPU can accept other spectral libraries, but it might be a little more difficult for the users to properly setup the inputs since the parameters to remove bisectors of input spec- tra are only optimized for the PHOENIX spectra and the so- lar atlas from the Kitt Peak Observatory FTS (Wallace et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' The PHOENIX library propose a collection of spectra with the wavelength coverage from 500Å to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='5µm with reso- lutions of 500,000 in the optical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' The library covers stellar ef- fective temperature from 2300K to 12000K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Since the spectra in the PHOENIX library are not normalized, which is critical to perform the injection of CB described in the preceding section, Article number, page 9 of 17 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' SOAP_GPU Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Average bisectors of quiet and active solar regions from the disk center (µ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0) to the limb (µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Continuous lines: Fifth-order poly- nomial fit to the quiet sun bisectors of the FeI 5250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='2084Å, FeI 5250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='6453Å, FeI 5434.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='5232Å, FeI 5432.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='9470Å, FeI 5576.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0881Å, FeI 6149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='2460Å, FeI 6173.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='3344Å, FeI 6301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='5008Å and FeI 6302.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='4932Å lines as measured by the Laser Absolute Reference Spectrograph (LARS) at the German Vacuum Tower Telescope (Löhner-Böttcher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Dashed lines: Fit of the bisectors of the FeI 6301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='5008Å spectral line inside a faculae region, as measured by the Fabry-Perot interferometer at the Donati Solar Tower (Cavallini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 1985a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Below a depth of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='5, a linear fit is per- formed, while a fifht-order polynomial is used to model the top part of the bisector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' To prevent unrealistic value when interpolating the polynomial above a normalised flux of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='9 where no measurement exists, we selected the most redshifted part of the top bisector, explaining the vertical values for very shallow depths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' The two vertical lines are shifted by 340 m/s which corresponds to the solar convective blueshift value derived from a fit to the data of Liebing et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' (2021) (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' The active bisectors at different µ angles are all shifted by those 340 m/s at a depth of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='58 as we make the hypothesis that convection is fully suppressed in magnetic regions (see Sect 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='2 for more information).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Bisectors of the FeI lines used in Löhner-Böttcher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' (2019) and fitted model to account for the CB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Left: Bisectors from the quiet Kitt Peak solar spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Right: Bisectors from the active Kitt Peak solar spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' We rejected the bottom part of the 6301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='5008Å bisector because it was significantly off by 2500 m/s due to strong contamination by other weak lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' we used the open source package “Rassine” (Cretignier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' (2020b)) to perform spectral normalization first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' The continuum or spectral energy distribution (SED) of input spectra derived from “Rassine” are denoted as SEDquiet and SEDactive respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' The inhibition of CB when working with PHOENIX spectra can be rewritten as: ∆S ′ bconv(X, Y) = S ′ quiet,n(X, Y) − S ′ active,n(X, Y), (10) where S ′ quiet,n and S ′ active,n are the normalized quiet and active spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Since the contrast between the quiet and active region Article number, page 10 of 17 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0 μ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='20 μ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='8 μ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='40 μ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='50 μ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='60 Normalized flux 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='6 μ= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='70 μ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='80 μ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='85 μ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='4 μ= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='95 μ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='00 Active region μ =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='44 Active region μ =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='2 Active region μ =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='82 Active region μ =l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0 200 0 200 400 Doppler velocity (m/s)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='8 Fit Binned data = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='6 Fel5250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='2084A 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='6 Normalized Normalized Fel 5250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='6453A Fel5434.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='5232A Fel 5432.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='9470 A Fit 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='4 Binned data Fel5576.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0881A Fel 5250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='2084A Fel 6301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='5008A Fel5250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='6453A 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='2 Fel 5434.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='5232A Fel5432.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='9470A Fel5576.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0881A 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0 Fel 6301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='5008 A 200 100 0 100 200 300 300-250-200-150-100 50 0 50 Doppler Velocity (m/s) Doppler Velocity (m/s)Yinan Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' : SOAP-GPU Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Comparison of the RVs derived using two different configu- rations for the input spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' A single equatorial spot with 1% area of the entire disk surface is simulated in both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Red line: RVs derived from simulated spectra with observed quiet sun and spot spectra without µ dependent bisector injection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Dotted line: RVs derived from simulated spectra with µ dependent bisector injection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' The input spectra with µ- angle dependency are generated with the Python module Convec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' The RVs of the flux, CB and total effect don’t change significantly when the µ-dependent CB is introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' is naturally included in the continuum of input PHOENIX spec- tra, the flux effect can be rewritten as: ∆S ′ flux(X, Y) = S ′ quiet,n(X, Y) × SEDquiet − LB(X, Y) × S ′ quiet,n(X, Y) × SEDactive, (11) where LB is the function of limb brightening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' For simulation of spot regions, LB = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0 along the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' For simulation of faculae regions, SOAP 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0 was using ∆T f = 250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='9 − 407.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='7µ + 190.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='9µ2 to model the limb brightening in temperature domain (Meu- nier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Since this equation is only valid for the Sun, we use the empirical equation derived from 3D MHD simula- tions to model other spectral types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 3D MHD simulations mod- elling faculae on the Sun can reproduce extremely well the limb- brightening observed for faculae (Norris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Using sim- ilar simulations, Johnson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' (2021a) model what would be the limb-brightening on other stars (see Figure 3 and Table 1 in Johnson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' (2021a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Using the parametrisation in Johnson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' (2021a), we derived the limb brightening curves of faculae for a G2, K0 and M0 dwarfs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' We then linearly interpolated be- tween the G2 and K0 and K0 and M0 simulations to obtain the limb-brightening dependence for a G8 and G9 dwarf, and a K2 dwarf, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' To obtain the dependence for a F9 dwarf, we linearly extrapolated from the G2 and K0 models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' The derived limb brightening curves from F9 to K2 are shown in Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Finally, the total effect from flux and convection can be de- rived using the following equation: ∆S ′ tot(X, Y) = S ′ quiet,n(X, Y) × SEDquiet − LB(X, Y) × S ′ active,n(X, Y) × SEDactive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' (12) Convection velocity changes with spectral type, and de- creases towards cooler stars than the Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' This is a well known Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Faculae intensity contrast as a function of µ for different spec- tral types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' The limb brightening curves for the G2, K0 and M0 dwarfs are derived from the parametrisation of MHD simulations for 500G fac- ulae, shown in Table 1 in Johnson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' (2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' The limb brightening curves for other spectral types are linearly interpolated by using the two closet limb brightening curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' For spectral types hotter than G2, we ob- tain limb brightening by performing linear extrapolation using the G2 and K0 curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' The limb brightening derived from Meunier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' (2010) is labeled with orange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' effect, that comes out from observations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Meunier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Liebing et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 2021) and magneto hydrodynamic mod- els of stellar phostophere (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Allende Prieto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' As in Liebing et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' (2021), we show in the top panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 12 that the CB is a cubic function of effective temperature in the range 4800 to 6300 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' The parametrisation that we obtain is CBvel = 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='2388 × ((Teff − 4400)/1000)3 + 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='2791.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Once we have this relation to measure the velocity of CB as a function of effective temperature, we simply have to rescale the difference between the quiet Sun bisectors for µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='85 and µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='2 at depth 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='58 to be equal to the value given by our relation, and we also impose that the active bisectors are shifted by the same value, at the same depth (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='2 for justification).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' This allows us to model properly the change of convective velocity as a function of stellar effective temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' We note that as in the case of the Sun, before injecting the proper bisector for the quiet and active regions, we first have to remove the bisector present in the PHOENIX spectra that we use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' This is a necessary step as the PHOENIX spectral library is obtained from 1D atmospheric models and cannot properly reproduce line bisector shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' We show in the appendix, like for Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 9 in the case of the Sun, how the bisector of the original PHOENIX spectra for the quiet stellar region, a faculae and a spot, are fitted before being removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 13, we show the results of a few simulations consid- ering µ dependent input spectra and a single 1% equatorial spot (left panel) or a single 1% facula (right panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' We highlight the RV outputs of SOAP-GPU when using PHOENIX spectra, with the quiet Sun temperature set to Teff = 5778K, the spot temper- ature set to Teff = 5115K, 5015K and 5215K and the facula tem- perature set to Teff = 5928K, 6028K and 6128K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' We also show the result when inputting the Kitt Peak solar quiet and spot or facula spectra (Teff = 5778K and 5115K or 6028K at disk cen- ter, respectively) as modified in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='2, and including (us- ing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 11, 10 and 12) the SED from corresponding PHOENIX spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' As we can see, the amplitude of the RV flux effect in- creases with an increase in temperature difference between the quiet photosphere and the spot or facula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' The CB effect does not change in amplitude with temperature difference and thus the larger amplitude observed for larger difference in temperature Article number, page 11 of 17 6 With μ-conv Tot RV (m/s) Without μ-conv 2 2 4 4 Withμ-conv Flux RV (m/s) 2 Without μ-conv 0 2 4 Withμ-conv Bconv RV (m/s) 3 Without μ-conv 2 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0 Phase0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='25 Spectral type: F9 Spectraltype: G8 Spectral type: G9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='20 Spectral type: K2 G2 Meunier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 2010 Intensity Contrast G2 500G 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='15 K0 500G M0500G 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0 μA&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' SOAP_GPU Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' CB velocity as a function of spectral type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Top: Data from Liebing et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' (2021) and cubic fit to them, giving as relation CBvel = 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='2388 × ((Teff − 4400)/1000)3 + 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='2791.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' We show with coloured stars the CB velocity value for different spectral types that we model in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Bottom: The bisector of the quiet photosphere measured from Löhner-Böttcher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' (2019) at µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='85 (thin lines) and the bisector of an active region measured from Cavallini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' (1985a) at µ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0 (thick lines) for different spectral types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' We impose that the maximum differ- ence between the quiet bisector at µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='85 and µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='2 (not shown here), happening at a depth of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='58 is equal to the CB velocity given by the relation found in the top panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' We also force the difference between the quiet bisector at µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='85 and the active bisectors, independent of their µ angle, to be equal to the same value at a depth of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' between quiet and active regions is solely driven by the change in contrast of the active regions with temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' While for the flux effect, the simulation using PHOENIX spectra or observed solar spectra as input gives the same results, which is not surpris- ing as we use the same SED, this is not the case for the CB ef- fect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' The amplitude derived show a small discrepancy of ∼20%, and the maximum has a slight phase shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' This likely comes from a different flux effect contribution seen in the derived CB RV, due to molecular absorption not perfectly modeled in the PHOENIX spectra compared to solar real observations (see dis- cussion in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' We note that this asymmetry was already something seen in the original SOAP 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0 paper (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 6 in Du- musque et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Something also interesting to note, that we see when using as input both the PHOENIX and solar spectra is the bump in the CB RV effect when the active region crosses the center of the disk (phase=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' This is induced by the fact that CB is maximum at µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='85 and not at disk center, as was shown in Löhner-Böttcher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='14 we show the result of the estimated CB RV effect for an equatorial 1% spot or facula for stars of different tempera- ture (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' spectral type): 6050 K (F9), 5778 K (G2), 5480 K (G8), 5380 K (G9) and 5100 K (K2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' For the spot, we see a positive only effect for the F9 and G2 simulations, which is expected, but for later spectral type, we start to see the emergence of a flux effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' This effect, as already discussed in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='1 comes from the absorption of molecules, that change significantly over a few hundreds of Kelvin for the quiet photosphere at 5480 K and a spot at 4817 K for the G8 simulation for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Also, we see that the more we go towards cooler stars, the more the CB ef- fect show a flux contribution, and this comes from the fact that molecular absorption is not linear with effective temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Al- though molecular absorption prevent us of clearly separating the flux effect of spots from the CB effect like in the case of the F9 or G2 star, the total RV effect including both contributions is still properly estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Therefore, users should be careful when interpreting the estimated RV induced by the inhibition of con- vection for spots on stars with spectral type later than the Sun, however, they can trust the total RV effect estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Regarding the facula, we observe a positive only effect for all spectral type, and thus the CB effect is properly modeled for this type of active regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Limitation of SOAP-GPU when moving away from solar twins On a careful note, we want to warn the user that a lot of the physics included in SOAP-GPU is based on solar observations, like for example the variation of the bisector of the quiet photo- sphere with respect to the center-to-limb angle (Löhner-Böttcher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 2019), or the bisector of an active region extracted from solar observations (Cavallini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 1985a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Therefore, although we try to correct for some effects, like the variation of CB veloc- ity as a function of spectral type, the more we go away from the solar case, the more we should be careful about interpreting the results coming out of SOAP-GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' We note that when modifying the bisector shape of solar or PHEONIX spectra for the disk center, to account for different convection velocity across spectral type, we model the effect of the “third signature” of granulation (Gray 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' The inherent shape of the bisector (known as the “second signature” of granu- lation) and how it varies with µ (Löhner-Böttcher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 2019) is still the one observed for the Sun and it is well known in the liter- ature that the bisector shape of disc-integrated observations, and therefore by analogy at disk center, varies significantly among luminosity class and spectral type (see Figure 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='15 in Gray 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Ba¸stürk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' In those papers, we see that inside the small range from early G’s dwarfs to early K’s dwarfs, the bisec- tor shapes does not change drastically and therefore, although the integrated spectra for those stars won’t be realist in terms of line bisector, the way SOAP-GPU models stellar activity should still be quite realistic as in the end, what counts is the differential between the activity and quiet phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' As we can see in the pre- ceding subsection, besides SOAP-GPU not being able anymore to separate the inhibition of the CB effect from the flux effect for early K’s, the tests we performed show that the code seems to behaves quite well in estimating the total RV effect induced by spots and faculae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' However, outside of this small range from G to K dwarfs, bisectors are completely different, and we warn the users that the present version of SOAP-GPU might give very unrealistic stellar integrated spectra and estimation of stellar ac- tivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' There is perhaps only one exception for dwarfs cooler than early K’s, for which the velocity of convection decreases to level that are very difficult to measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' For those stars, stellar activity is dominated by the flux effect from spot and faculae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Therefore, for such stars, users should ignore the output from the CB effect and only consider the flux effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' To better model stellar activity for stars different than the Sun, we would require disk-resolved spectra for other stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Those could come from 3D MHD simulations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Johnson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 2021b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Dravins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 2021), or thanks to spatially resolved spec- troscopic observation across stellar surfaces thanks to transiting Article number, page 12 of 17 700 Datafrom Liebing etal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='2021 Cubic fit 600 Spectral type: F9 Spectraltype:G2 (Sun) Spectraltype:G8 500 Spectral type: G9 RV (m/s) Spectraltype:K2 400 300 CB 200 100 4800 5000 5200 5400 5600 5800 6000 6200 Teff (K) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0 Spectral type: F9 Spectral type: G2 (Sun) Spectraltype:G8 Spectral type: G9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='8 Spectral type:K2 I flux Normalized 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0 400 200 0 200 400 Doppler velocity (m/s)Yinan Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' : SOAP-GPU Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Simulation of an equatorial 1% spot (left) and 1% facula (right) with different temperatures using PHOENIX spectral library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' The quiet Sun spectrum is extracted from PHOENIX spectral library with log(g) of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='5 and Teff = 5778K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' The effective temperature of the spot spectra are 5015K, 5115K and 5215K, and for the facula spectra 5928K, 6028K and 6128K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' We also show the results of using the observed Kitt Peak solar spectra including the PHOENIX SEDs (using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 11, 10 and 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' The µ dependent CB is included in the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Left: For the spot we see that when the difference in temperature between the spot and the quiet photosphere increases, the RV flux effect becomes larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' We note a rather large discrepancy in the CB effect between the observed solar and PHOENIX input spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' This is likely due to the fact that the observed solar spectra are not well normalized (see Sect 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Right: For the facula, we note the same problem of negative values for the CB effect, which expected as the same badly normalised solar active spectrum is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' We also see that considering the corresponding PHOENIX SED when using the observed solar spectra significantly change the flux contribution (see Sect 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Considering the PHOENIX SED gives results much closer to the PHOENIX simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Simulation of the CB RV effect of an equatorial 1% spot (top) and 1% facula (bottom) for different temperature of the quiet photo- sphere (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' different spectral type) using the PHOENIX spectral library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' The temperature difference between the quiet photosphere ∆Teff for spot and facula is fixed to 663K and 250 K, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Due to the strong absorption effect of molecules in spots on stars cooler than the Sun, we see the appearance of a flux effect in the derivation for the CB effect only (see Sect 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' planets (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Dravins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Both approach are challeng- ing, however could led to a much more realistic modelisation of stellar activity from other stars than our Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' We note that it is rather easy to input different spectra in SOAP-GPU than the ones provided (the Sun and PHEONIX spectra), and there- fore if users have access to disk-resolved spectra with more re- alistic line-bisectors, SOAP-GPU could still be used to obtain efficiently disk-integrated spectra and model the corresponding stellar activity effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' We note that the CB injection part is an individual module in SOAP-GPU that users can easily modify.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Conclusions In this paper, we present a GPU-based improvement to SOAP 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0, named SOAP-GPU, that allows to efficiently model stellar activity at the spectral level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' With the implementation of GPU in- terpolation and summation, benchmark calculations demonstrate that SOAP-GPU improves the computational speed by a factor of 60 when modeling stellar activity on a full visible spectral range at R=115’000 of resolution, while having the same accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Beside the huge gain in speed, SOAP-GPU also provides a more complex modelisation of stellar activity compared to SOAP 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Complex active region scenarios, with regions over- lapping is now handled by the code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' This is mainly useful when modeling active phases of a star like the Sun, with hundreds of active regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' The contrast of the active regions is now wave- length dependant, and therefore change for each wavelength of the modeled spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' The dependence of line bisector with center-to-limb angle µ, following the work of Löhner-Böttcher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' (2019) and Cavallini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' (1985a), is also now accounted for for the Kitt Peak observed quiet and active atlases, but also for PHOENIX spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Although the induced RV effect is rather negligible when using the observed solar spectra as input, includ- ing the framework to change line bisector is crucial to properly model convection when injecting PHOENIX spectra as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' The use of PHOENIX spectra allows us now to model a wide variety of stars with different stellar and active region properties, and allows us as well to better model faculae, as the correspond- ing spectrum now has the proper effective temperature (SOAP 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0 was using the Kitt Peak sunspot atlas to model faculae).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Article number, page 13 of 17 6 SolarspectrawithPhoenixSED Phoenix spectra,spot Teff=5015K 4 Phoenix spectra,spot Teff=5115K : RV (m/s) Phoenixspectra,spotTeff=5215K 2 Tot 0 2 4 Flux RV (m/s) 2 0 2 4 RV (m/s) 3 2 Bconv F 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0 Phase6 Solar spectra with Phoenix SED Phoenixspectra,faculaeTeff=5928K 4 Phoenix spectra,faculae Teff=6028K Tot RV (m/s) Phoenix spectra,faculaeTeff=6128K 2 0 2 4 Flux RV (m/s) 2 0 2 4 4 RV (m/s) 3 2 Bconv 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0 PhaseF9Teff=6050K G2 Teff=5779K Spot RV (m/s) G8Teff=5480K G9Teff=5380K 2 K2Teff=5100K 0 4 Faculae RV (m/s) 3 2 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0 PhaseA&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' SOAP_GPU When modeling the inhibition of CB effect using as input the solar Kitt Peak quiet and active spectral atlases, we noticed that the derived RVs go negative, which is not expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' This comes from molecular absorption that can be seen in the spot spectrum due to lower temperature compared to the quiet Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Even though we do not include the contrast of the active region when modeling the RV CB effect only (see Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 2 and 10), the difference in flux at the level of molecular absorption bands will show up as a flux effect in the estimated CB RVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Positive val- ues will be added to the CB RV effect before the spot crosses the stellar center, and negative values after, therefore creating an asymmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' When modeling other stars than the Sun using PHOENIX spectral library, users should be aware that a lot of physics in- cluded in SOAP-GPU are based on solar observations, and al- though the code tries to correct for known effects like the vari- ation of CB velocity as a function of effective temperature (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='e spectral type, see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='3), the more we go away from the Sun, the more the results should be interpreted with caution (see discussion in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' The modelisation of stellar activity for other stars than the Sun is currently limited by the knowl- edge we have about disk-resolved bisectors for such stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Such information is very challenging to obtain, however, 3D MHD simulations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Johnson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 2021b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Dravins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 2021) and resolved spectroscopic observation of other stars due to plane- tary transits (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Dravins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 2017) could significantly help.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Also, when modeling stars of later spectral type than the Sun, we are not able anymore to separate clearly the inhibition of CB effect from the flux effect due to the strong absorption of molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' However, the output for the total RV effect (flux plus inhibition of CB effects) should be modeled properly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' SOAP- GPU have been tested up to a K2 star (Teff = 5100 K) with spots 663 K cooler and give satisfactory results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Modeling later spec- tral type is challenging mainly due to continuum normalisation of the PHOENIX spectra and injection of spectral line bisector due to line blending, and users should be very careful about the interpretation of the results for such stars with the present code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' There are still some improvements that could be made to bet- ter model the physics at play.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Although the spot and facula spec- trum used when considering PHOENIX spectra as input are of different temperature, and therefore in spectral content, we still associate to those regions the same active bisector as measured for the Sun on a facula (Cavallini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 1985a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Spots are induced by stronger magnetic fields than facula, and thus it is likely that the bisector of spectral lines will be slightly different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Although it is possible to know what is the bisector of a few spectral lines inside a spot at disk center, to our knowledge, no measurement of spot line bisectors for different µ angles are published.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Cav- allini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' (1985a) also show in their Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 1 that depending on the facula observed, the bisector shape changes due likely to dif- ferent magnetic field strength and therefore different level of CB inhibition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' As can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 10, the effect of inducing µ dependant spectral line shape in the quiet and active regions is rather small, and although with more solar data about spots and faculae we could better model the physics at play, results in terms of RV derivation would be rather similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' This likely comes from the fact due to limb-darkening, most of the weight is put on the disk center, where spectral lines does not change significantly in shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' With the performance of SOAP-GPU, it is now possible to model activity at the spectral level for complex stellar surfaces with many active regions and for a long period of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' A so- lar activity simulator, either based on statistical properties of so- lar active regions (similar to Borgniet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 2015) or on the ob- served distribution of those (similar to e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Meunier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 2010) will be published in a forthcoming paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' We encourage any per- son working on techniques to separate the activity effect from planetary signals at the spectral level, to test their framework on SOAP-GPU simulations, where photon-noise, instrumental and telluric systematics are not perturbing the spectral timeseries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' We thank the anonymous referee for the insightful and con- structive comments on this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' We thank Michael Crerignier for his help in normalizing PHOENIX spectra with RASSINE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' We also thank Xiang Gao for the constructive comments on GPU computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' This project has received fund- ing from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement SCORE No 851555).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' This work has been carried out within the framework of the National Centre of Competence in Research PlanetS supported by the Swiss National Sci- ence Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' The authors acknowledge the financial support of the SNSF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' References Aigrain, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=', Parviainen, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=', & Pope, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' S.' metadata={'source': 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Collet, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=', Lo Curto, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=', & Selam, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 2011, A&A, 535, A17 Baranne, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=', Queloz, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=', Mayor, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 1996, A&AS, 119, 373 Barragán, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=', Aigrain, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=', Rajpaul, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=', & Zicher, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=', & Zucker, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 2020, A&A, 642, A146 Boisse, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=', Bonfils, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=', & Santos, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 2012, A&A, 545, 109 Borgniet, S.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=', Blackman, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 2020, AJ, 160, 67 Cavallini, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=', Ceppatelli, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=', & Righini, A.' metadata={'source': 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E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=', Shahaf, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 2021, MNRAS, 505, 1699 Collier Cameron, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=', Mortier, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=', Phillips, D.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 2017, MNRAS, 465, 3343 Dravins, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=', Lindegren, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=', & Nordlund, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 1981, A&A, 96, 345 Dravins, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=', Ludwig, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=', Dahlén, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=', & Pazira, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 2017, A&A, 605, A91 Dravins, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=', Ludwig, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=', & Freytag, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=', Jones, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=', & Stenning, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 2020b, ApJ, 905, 155 Gray, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=', Hinkle, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=', & Livingston, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 1998, An atlas of the spectrum of the solar photosphere from 13,500 to 28,000 cm-1 (3570 to 7405 A) Wallace, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=', Hinkle, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=', & Livingston, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 2005, An atlas of sunspot umbral spectra in the visible from 15,000 to 25,500 cm-1 (3920 to 6664 Å) Zhao, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=', Ford, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=', & Tinney, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 2022a, ApJ, 935, 75 Zhao, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=', Fischer, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=', Ford, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 2022b, AJ, 163, 171 Article number, page 15 of 17 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' SOAP_GPU Appendix A: Line bisectors of PHOENIX spectra As discussed in Sects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='2 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='3, before injecting the µ dependant bisector for solar or PHOENIX spectra to properly model CB and its inhibition close to the limb and in active re- gions, we need to remove any bisector shape already present in the input spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' As PHOENIX spectral library is gener- ated from 1D spectral synthesis, the line bisectors cannot include properly the CB effect and therefore should be close to straight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='1, we show for each simulation of different spectral types the bisector of a few iron lines that are used in Löhner- Böttcher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' For each spectral type simulated, we show the bisectors for the quiet photosphere, but also for simulated spot and faculae, 663 K cooler or 250 K hotter, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' As expected, most of the bisector are close to straight lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' We how- ever fitted the average bisector with a second order polynomial to remove the small curvatures observed before injecting the proper bisectors at different µ angles (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' It is not clear if those curvatures are real effect in the spectral synthesis, or sim- ply due to blends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' The correction performed is small compared to the bisectors that we inject afterward, therefore if only due to blends, this process does not significantly change the outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Article number, page 16 of 17 Yinan Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' : SOAP-GPU Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Bisector of PHOENIX spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' For each input seed spectrum using PHOENIX spectral library, we use five strong iron lines: FeI 5250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='2084Å (green), FeI 5250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='6453Å (cyan), FeI 5434.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='5232Å (purple), FeI 6173.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='3344Å (orange) and FeI 6301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='5008Å (yellow) to measure the average bisector of the input spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Bisector outliers outside a window of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='1Å around each line center are rejected to avoid those points, certainly affected by line blending, to bias our measurement of line bisector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Each line correspond to a different spectral type, and from left to right, we can see the bisector of the spectrum used for the quiet photosphere, a spot region (663 K cooler) and a facula region (250 K hotter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' We average those line bisectors at certain depth (as shown by the red dots) and fit the obtained data with a second order polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' The fitted bisector is used to remove the bisector of input seed spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content=' Article number, page 17 of 17 F9 Teff = 6050 K F9 Teff = 5387K F9 Teff = 6300 K 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfAgnY/content/2301.04259v1.pdf'} +page_content='8 0.' metadata={'source': 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AI Art Another Industrial Revolution in the Making? +Alexis Newton, Kaustubh Dhole +Department of Computer Science +Emory University +{annewto, kdhole}@emory.edu +Abstract +A major shift from skilled to unskilled workers was one of +the many changes caused by the Industrial Revolution, when +the switch to machines contributed to decline in the social +and economic status of artisans, whose skills were dismem- +bered into discrete actions by factory-line workers. We con- +sider what may be an analogous computing technology: the +recent introduction of AI-generated art software. AI art gener- +ators such as Dall-E and Midjourney can create fully rendered +images based solely on a user’s prompt, just at the click of a +button. Some artists fear if the cheaper price and conveyor- +belt speed that comes with AI-produced images is seen as +an improvement to the current system, it may permanently +change the way society values/views art and artists. In this ar- +ticle, we consider the implications that AI art generation in- +troduces through a post-industrial revolution historical lens. +We then reflect on the analogous issues that appear to arise +as a result of the AI art revolution, and we conclude that the +problems raised mirror those of industrialization, giving a vi- +tal glimpse into what may lie ahead. +Introduction +The industrial revolution caused a major shift from skilled to +unskilled labor when machines contributed to a massive lay- +off of artisans in favor of factory-line workers. William Pelz +describes how prior to the industrial revolution people had +been working in much the same way for thousands of years, +producing goods through human labor with some assistance +via animals or water power. However, he says, “All of this +changed with the rise of the machine: tools would no longer +serve people, but rather people would serve machines” (Pelz +2016). Pelz argues that due to this massive change in goods +production, the people became an “appendage” to the ma- +chine, as humans were suddenly at the mercy of machines +using them to make goods. As factory life took hold, com- +pensation was no longer based on paying for skill, but rather +on paying for time. This helped to usher in a different kind of +day-to-day experience for most commoners–that of a time- +work discipline. Maxine Berg (2014) notes the change to a +mechanized factory sector from pre-industrial handicrafts. +Many traditional workers had to transition to this type of +work, or risk being left behind by a quickly transforming +industry, leading to the devaluation and destruction of busi- +nesses that could not compete with the cheaper and faster +production that industrialization had to offer. +Creative AI Across Modalities 2023 (creativeai-ws.github.io), +Thirty-Seventh +AAAI +Conference +on +Artificial +Intelligence +(www.aaai.org), February 7-23, Washington, D.C., USA +With the recent introduction of AI-based art models +(Ho, Jain, and Abbeel 2020; Ramesh et al. 2022; Ruiz et al. +2022), we argue that a shift largely reminiscent of the post- +industrial revolution is unfolding. As AI-based models be- +come more and more common, issues artisans experienced +in the mid 1800s are reemerging, which similarly question +the very existence of artists today. As we are on the precipice +of this revolution, it is imperative for all stakeholders, viz, +policymakers, ethicists, computer scientists and artists to un- +derstand what such a shift would entail in order to manage +the consequences of that shift. +In this article, our aim is to view recent developments in +the art industry due to the introduction of artificial intelli- +gence models through a post-industrial lens. In the follow- +ing sections, we first discuss how technologies influenced +views on independent artisans in the aftermath of the indus- +trial revolution. Next, we discuss the positive and negative +implications of the post-industrial view on AI generated art. +Finally, we consider current issues raised by these models, +and conclude by reflecting on the analogous issues seen in +the industrial revolution. +A Historical Shift +From Individual to Factory Worker +The switch from an individualist working environment to a +factory-centered one would permanently change the way so- +ciety viewed independent artisans, bringing in a new age of +commercialization that was predicated on the swift produc- +tion of machine-made products over man-made ones. This +difference on workforce style in is how the Industrial Rev- +olution led to a change in “people’s relationship to crafts- +manship, time, community and their own role in society as a +whole”. +Such a change in working style was caused that it is hard +to imagine a time where machines were not at the fore- +front of civilization. As the industry transformed resemble +today’s working world, small-scale artisans were pushed out +of people’s minds in favor of production that was faster and +cheaper. The individual craftsman all but disappeared from +the forefront of the business world. By the 1850s, most in- +dependent shoemakers had been replaced by shoe factories, +independent weavers had gone out of business, and women +with hand looms were quickly outstripped by the factories +and machines bringing more people cheaper goods of higher +average quality (Smail 1992). +However, a plethora of small-scale and skill-intensive sec- +tors, like those in the metal trades and textile industries, + +managed to develop alongside the rise of factories (Berg +2014). Parts of the world also still value individually-made +fine arts objects, especially cultures in the east like China +and India. Berg points to the idea of luxury fashion in France +or small-scale building restoration which is popular in Eu- +rope (Berg 2014). Though factorization still prevails, it is +also important to note that the since the early 2000s demand +for “niche” artisans has actually shown an upward trend. +The Alliance for Artisan Enterprise (2012) reports that the +global market for artisan-made products has increased by +more than 8% per year since 2002, and is worth more than +$32 billion. One of the reasons for such a trend has been an +increased willingness to pay a premium for distinctive vis- +`a-vis mass-produced, goods. +Machines Have Politics +Winner (1996) provides an illuminating example of trans- +forming industry in “Do Artifacts Have Politics?” when +he looks at the industrial mechanization of Chicago in the +1880s where the switch to factory production hurt skilled +workers. When pneumatic molding machines were imple- +mented by Cyrus McCormick’s reaper manufacturing plants, +it drove individuals out of business (Winner 1996). In many +other industries, this happened because the factory process +was cheaper, but that was not the case here. In this instance, +McCormick and the National Union of Iron Molders were +at war, so even though the iron molding machines were not +cheaper, they were used to push the previous workers away +from unionization and out of business. Therefore, while the +addition of industrial manufacturing hurt many skilled la- +borers naturally through cheaper replacements, it also hurt +them unnaturally through the furthering of political agendas +(Winner 1996). +Unarguably, Winner concludes that the molding machine +thus has politics in that its technical arrangements have be- +come a form of order. Instead of subscribing to the use of +pneumatic molding machines to speed up or cheapen pro- +duction, these machines expressed human motives in their +use towards achieving authority over others. It is undeniable +that even if artifacts and machines do not have politics, they +do indeed have power. +In his famous essay “The Work of Art in the Age of +Mechanical Reproduction,” Benjamin (1935) discusses the +change in perception of art in the age of mass production. +Before mechanical reproduction, art was unique and valued +for its “aura” – which was derived from its authenticity and +its physical and cultural context. However, with the ability +to mechanically reproduce works of art, the traditional bases +of cultural authority and hierarchy were challenged. As art +moved to being based on politics instead of tied to rituals, +it began to serve as a tool for political activism and resis- +tance, ultimately bringing about social and political change +(Benjamin 1935). +Historically, it seems fair to say that the introduction of +new technologies in the industrial revolution had many im- +pacts on the individual craftsmen. Some of these impacts +might be considered a natural course of action, but it is +imperative as we move forward to acknowledge that other +forms of use can be exacted through the introduction of new +technology–use beyond that of merely what a machine phys- +ically produces. +AI Art Generation +We focus on the recent introduction of AI-generated art soft- +ware to the current art world. AI-based art generators such as +Dall-E (Ramesh et al. 2021, 2022) and Midjourney are rel- +atively new pieces of technology that can now create fully +rendered images based solely on a user’s prompt, often pro- +ducing impressive and intricate results. If the industrial rev- +olution changed the way society viewed artisans and crafts- +men, how might the AI art revolution do the same? We now +examine AI image generators as a computing technology +that has the potential to cause this analogous shift in the art +industry. +The Perception of AI Art +The most general fear associated with AI-generated art is +that it could drastically reduce the amount of jobs available +to working commercial artists today in areas such as illustra- +tion, animation, and graphic design. However, some others +are concerned with the idea that the more “traditional” no- +tion of art may also be modified by AI-generated art. +In “The Culture Industry: Enlightenment as Mass De- +ception,” Adorno and Horkheimer (1944) introduce the term +“culture industry,” and compare technological advancement +of mass media and creation to factory production of goods. +They refer to the “assembly-line character of the culture in- +dustry, the synthetic, planned method of turning out its prod- +ucts” as creating a passive society that is being manipulated +into being satisfied by the products of capitalism, rather than +by way of true psychological needs such as freedom, creativ- +ity and happiness (Adorno and Horkheimer 1944). +In 2018, the art-collective Obvious, produced an art piece +“Edmond de Belamy,” via a generative adversarial network +(GAN) software package. The artwork was printed onto a +canvas and sold at auction for $432, 500, over 43 times +its pre-auction estimated value (Cohen 2018). Adorno and +Horkheimer might argue that artwork created by an AI +is created to be an ideal of art, and that AI artwork is +merely feeding into mass-media culture industry that threat- +ens “high arts.” +Oppositionally, the idea that AI-created art could be +worthwhile in itself as an art piece, as seen in its high price +valuation, points heavily towards James Moor’s prediction +that technology is shifting the questions we ask from “How +well does a computer do such and such an activity?” to +“What is the nature and value of such and such an activity?” +(Moor 1985). In our case, the question shifts from “How +well can computers make art?” to “What is art?”. +What is Art? +The current literature on human attitudes towards AI- +generated art presents some evidence as to what people +might feel of this shift in viewpoint. In two recent studies +(Hong and Curran 2019; Mikalonyt 2022) on attitudes to- +wards artwork produced by humans versus by artificial intel- +ligence, researchers found that while most test subjects felt + +that AI-generated art could be considered “art,” they were +much less inclined to consider the art to be produced by an +“artist”. This is a significant distinction because it implies +that while artwork contains artistic value just by existing, +artwork is not made just by putting pen to paper. +Thus we might push the shift Moor describes even further, +from the question of “What is art?” to “What is an artist?”. +“When judging whether an object falls under the category +of ‘artwork,’ the intent of the creator is seen as more impor- +tant than even the appearance of the object in question” - +Mikalonyt (2022) seem to think, this is the reason that their +participants were not unwilling to consider art created by a +“robot” to be “art,” but were significantly more at odds with +calling a “robot” an “artist.” +In shifting the question from the object to the character- +istic identity, we come to a yet unsolved question about AI +art: If artwork generated by AI can be called “art”, but the +model is arguably not an “artist,” then who is the author of +such a piece? Mikalonyte and Kneer posit that this question +has not yet been solved, suggesting that the lack of answer is +reflected in the fact that autonomously generated AI artwork +has yet to be copyrighted, with proposals to give copyright +to the human designers of the artificial intelligence, as well +as to redefine “authorship” so as to include robots in the def- +inition (Mikalonyt 2022). +Human Art from Ends to Means +Besides raising questions in moral philosophy, such an attri- +bution to the functional aspects of art vis-`a-vis the aesthetics +could mean a lot of hope for traditional artists, especially +those who were dependent on the techniques rather than the +aesthetics of the final product. Artists who would only be +differentiated by techniques might resort to promoting un- +usual techniques which are beyond the current scope of AI +art models, or at least AI art models at present (e.g painting +on paper towels or woodblocks, using the back of the paint- +brush, using fingernails to paint). It wouldn’t be surprising +if artists would resort to differentiating factors relying on +“means” rather than “ends”. Artists would especially want +their work to be intrinsically different - earlier that could be +achieved via both unique propositions of outcomes and of +methodology, but now it would largely be the latter. It won’t +be a surprise to witness more conservative forms being re- +inforced (Browne 2022). Analogous possible trends of in- +creased interest in niche artworks as against mass-produced +ones would actually further promote such resorting to tradi- +tional artistry and differentiating means. +Fast-Paced Computational Creativity +Pelz’s viewpoint that people became appendages to ma- +chines during the industrial revolution clearly has art +analogues today. Over the years, a large section of the +art industry has already resorted to computational meth- +ods (Li, Hashim, and Jacobs 2021; Feldman 2017) after wit- +nessing the benefits of generative art. In generative art, new +concepts, forms, shapes, colors, or patterns are created algo- +rithmically. Artists or programmers first establish some cri- +teria, post-which a computer creates new art forms adhering +to those criteria. Such generative art is considered more aes- +thetically pleasing than functional (Ball 2019). Hence wher- +ever the functional aspects of art would be irrelevant, like +in branding and advertising or the larger design sector, this +transition towards computational art can accelerate people’s +dependence on art designs which can be quickly iterated. +Reduced Dependency on Traditional Artists +Suddenly, a piece of art that may have taken days to produce +by a professional can be done with a handful of suggested +words by almost anyone. It must be noted that if the emer- +gence of this technology follows a similar path to that of +mechanization following the industrial revolution, it could +devalue commercial artists significantly and create massive +job loss in an industry that was previously known to require +a deeply human touch. Everything from storyboarding, con- +cept art, and movie creation to advertising work and social +media would be significantly different, causing a massive +problem for the individual artists who may have trained for +years to perfect their skill sets. +Benefits to Business +Art has previously been far from a process one could au- +tomate, but these AI art generators might be the cause of a +grand shift from skilled to unskilled labor in the free-lance +art world. In the past few months, several online publications +have tested using tools like Dall-E or MidJourney to provide +art to accompany their written content (Warzel 2022). +Access to AI art generation tools could be seen as an im- +provement to the current system for many business owners +where there is a strong demand for commercial art. Being +able to generate content for websites, branding, marketing +and sales in a virtually cost-free manner could help small- +business owners to reach bigger audiences. Online publica- +tions have largely stepped away from free-lance artists any- +ways, with many publications hosting content that was cre- +ated elsewhere (i.e. embedded tweets or stock photography). +Stock photography businesses sell royalty-free pictures +for personal or commercial use, usually paying 15 − 40% +to the creator of the photo for each license (Vincent 2022). +Recently, AI have been breaking into this market, with the +distinct art styles of Dall-E or Midjourney popping up on +stock photo websites (Edwards 2022). Shutterstock, one of +the largest stock photo retailers, announced that along-with +OpenAI, they would be banning artwork from other AI gen- +erators from being uploaded to their site, and it would also +create a “Contributor Fund” to help pay the artists whose +work was used to train the AI software (Vincent 2022). +Art Democratization +Platforms like YouTube, Instagram and TikTok, which have +become hosts of content creation, have been able to at- +tract millions of content creators who make money us- +ing their services by providing them with fame and mon- +etary rewards. This has largely happened with the reduced +technical and social barriers that these arguably demo- +cratic platforms have provided. AI art models could also +tread the same path. Democratization of art would mean + +almost anyone can produce artistic creations, including a +person without limbs or someone with a neurological dis- +order that affects their ability to draw or paint. AI-art +models might be exclusively seen as potential attackers +on the most talented segments of the artistic society, but +they will doubtlessly open up a level playing field for +those who considered art out of reach. Besides, such de- +mocratization would also be reflected in crowd-sourced +efforts (Bigham, Kulkarni, and Lasecki 2017; Kittur et al. +2013, 2019; Dhole et al. 2021; Srivastava et al. 2022) which +would seek contributions to aid in developing large creative +models fairly. +Increased Amount of Plagiarism +The Industrial Revolution was replete with examples of in- +dustrial espionage (Harris 1985; Christopher Klein 2019) +where large businesses often stole the work and ideas of +people who had historically performed it. Governments reg- +ularly encouraged individuals to steal ideas, especially from +abroad, since it hardly required applicants to be inventors, +especially if the invention was abroad. The current AI art +models already have been exposed to the artworks of many +artists without giving them proper attribution or even seek- +ing their permission. Millions of generations of artwork have +already been utilizing these styles. Besides artists’ work +being plagiarized, it is unclear how credit would get di- +vided amongst the artists, model trainers and users writing +prompts. While there have attempts to legally delineate the +complete generation pipeline (Fjeld and Kortz 2017; Kim +2020), the black box nature would make it hard for fair credit +attribution. +Increased Carbon Emissions +The dramatic increase in coal and gas usage, which sky- +rocketed pollution levels across major cities and indus- +trial zones, was an unfavorable side consequence of the +Industrial Revolution. Today, with large groups of peo- +ple expected to move towards careers of computational +art, it would be inevitable that training these AI-art mod- +els would also be performed more frequently, via re- +searchers as well as artists raising concerns of carbon +emissions. Strubell, Ganesh, and McCallum (2020)’s lifecy- +cle assessment of training popular large language models re- +vealed that a typical training process took nearly five times +the lifetime carbon emissions of the average American car. +Besides, the largeness of these models also necessitates GPU +usage during inference time. +Furthering Political Agendas +Winner (1996) has been helpful for giving convincing argu- +ments that artifacts and machines in general can have biases. +The 9ft clearance levels of Long Island bridges were a de- +sign decision made by urban planner Robert Moses in order +to restrict buses filled with low-income people and racial mi- +norities from accessing parkways. As a result of the biases +implicit in these designs, people were given limited access +to parts of their own city (Winner 1996). +Benjamin (1935) had argued that as art’s authenticity di- +minishes due to the ease of mechanization, it begins to be +based on politics rather than on rituals. Just like the Long +Island bridges, political agendas intertwined with design +have had crucial consequences. Therefore, it wouldn’t be a +surprise if biases mirrored in the AI art generation of to- +day (Bansal et al. 2022) were exploited to further political +agendas. Hassine and Neeman (2019) revealed that AI gen- +erated art skews mostly white, both in depiction and in rep- +resentation. Unless age, sex or race is specified, prompts to +the system have built in biases towards young white men +(Srinivasan and Uchino 2020). This unconscious bias could +have a manifestation in the real world, just as Robert Moses’ +designs manifested for racial minorities in New York City. +Questions of Concern +Finally, we consider the issues to pubic welfare and society +that AI-art generation introduces through its effects on the +artists of today. We also consider the scope of involvement +that computer scientists and AI have had in creating or con- +tributing to these issues. +Is the threatened change in the status of artists +characterized by the primary and essential +involvement of AI models? +Industrial “sweatshops” mass-producing art for commercial +consumption have been a constant long before the computer +became an element in the equation. From the comic strips of +the 1880s - 1960s to the comic books of the 1930s - present +day, to the mass produced landscape art created for furniture +stores in Asian factories since the mid-1950s, art has been +commoditized long before the computer (Hersch 2021). In +these assembly lines, one person would sketch the outlines +of image, another would pencil in details of the people, an- +other would ink those images, yet another would draw in the +backgrounds, and a final hand would color the image. Pro- +duced by an assembly line usually called a “studio,” the art +would be signed either by an arbitrarily selected worker or +even by a completely fictitious artist (Hersch 2021). More to +the point — this was art created by “factory workers” who +acted the same role in production as today’s graphics pro- +grams do (Campbell 2022). Therefore, one could also argue +that AI models might not be essential to the problem. How- +ever, what distinctively stands out with the usage of AI art +models, is the rapid pace of artwork creation and prolifera- +tion, unlike what was witnessed before the arrival of com- +puters or the internet. +Does the threatened change in the status of artists +occur because of exploiting some unique property +of AI models? +The primary difference created by technology is mass ac- +cess. To staff a “studio” with bit-work artists requires a sub- +stantial investment in infrastructure, equipment, and labor +costs. Such programs as are available today are much more +economical and widely available to individuals — from hob- +byists to serious artists to mercenary corporations — than +at any time in our history. So it is legitimate to argue that +AI models have uniquely driven the scale and creativity of + +the problem far more broadly than earlier technology could +have. +Could this issue have even arisen without the +involvement of AI models? +The answer depends heavily on the question of whether AI- +art generation programs are considered as tools or entities. +Motion-picture technology created entirely new art forms. +One could create art with motion across space and time in +a way that entirely changed how our culture thought about +visual art. But the cameras themselves have never — for all +their technological sophistication — been more than tools. +Cameras do not set out with purpose to make movies, and +AI art generation programs do not set out independently to +create images. Both must be employed by users. +Conclusion +Perhaps what AI-art generation software is doing is forc- +ing our society to confront a much larger issue about artists. +Instead of threatening the status of artists, advances in com- +puter technology require us to confront the idea that the view +of artists has already changed, and has been changing. +While on the surface AI-generated art seems unique, +many of the issues that it is raising concerning society’s +views of art and artists are merely more complex callbacks +to the mechanization of artisan’s projects in the industrial +revolution. Specifically, the art generated by AI-generation +algorithms creates problems that mirror those of industrial- +ization. In the same way that the artisan was pushed out, so +too is the artist today. In the same way intellectual property +was compromised during the 1800s, legal loopholes may en- +sure many artists are not duly credited without proper AI +regulation. +Our study of the industrial revolution analogues serves as +a warning to look backwards at the past treatment of creators +and consumers of industry in the wake of newly introduced +technologies. We pose that this may be an important step to +take before experiencing the consequences of the technical +revolutions that unfold before us today. +But these analogies should not be taken as a discourage- +ment against developing large models or to undermine the +efforts of the field of AI in general. We should actively +strive to improve technical parameters of these models, by +accounting for the possibility of potential damage early on, +as these models have and already display tremendous poten- +tial for business as well as for democratization. +Limitations +How AI art generation tools will affect artists is an extremely +subjective and multifaceted subject, and forecasting it pre- +cisely will not be easy. We have provided comparisons based +on events that occurred post the industrial revolution. How- +ever, we think that empirical evidence would be helpful in +better understanding many of the issues raised. Our objec- +tive was to present as thorough and comprehensive an anal- +ysis as possible by considering the technical, political and +industrial implications of art. Our section on “What is Art?” +is quite limited due to the rich history concerning this ques- +tion from a philosophical point of view, but we still feel it +was important to include. Nonetheless, we believe that our +work will serve as a crucial gateway for both the engineering +and humanities disciplines to facilitate dialogue and advance +debate about the impact of AI art tools. +Acknowledgments +We thank Dr. Kristin Williams and Dr. Steve Newton for +their crucial thoughts and feedback in numerous drafts. We +also thank the anonymous reviewers and meta reviewer for +their invaluable suggestions. +References +Adorno, T.; and Horkheimer, M. 1944. +Dialectic +of +Enlightenment. +Philosophy +Archive +@ +marx- +ists.org. +Translated +by +Andy +Blunden +(1998), +https://www.marxists.org/reference/archive/adorno/1944/culture-industry.htm. +Ball, +M. +2019. +How +to +Start +Creative +Coding. +https://www.arts.ac.uk/study-at-ual/short-courses/stories/how-to-start-creative-coding. +Bansal, H.; Yin, D.; Monajatipoor, M.; and Chang, K.-W. +2022. 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Shutterstock will start selling AI-generated +stock imagery with help from OpenAI. +Warzel, C. 2022. I Went Viral in the Bad Way. +Winner. +1996. +Do +artifacts +have +politics? +https://link.springer.com/article/10.1007/BF02583549#citeas. +Accessed: 2022-10-015. + diff --git a/LNE4T4oBgHgl3EQfiQ3g/content/tmp_files/load_file.txt b/LNE4T4oBgHgl3EQfiQ3g/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f13f624a101ce26925466c9164d67539eb9cd230 --- /dev/null +++ b/LNE4T4oBgHgl3EQfiQ3g/content/tmp_files/load_file.txt @@ -0,0 +1,529 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf,len=528 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content='05133v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content='AI] 12 Jan 2023 Is AI Art Another Industrial Revolution in the Making?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' Alexis Newton, Kaustubh Dhole Department of Computer Science Emory University {annewto, kdhole}@emory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content='edu Abstract A major shift from skilled to unskilled workers was one of the many changes caused by the Industrial Revolution, when the switch to machines contributed to decline in the social and economic status of artisans, whose skills were dismem- bered into discrete actions by factory-line workers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' We con- sider what may be an analogous computing technology: the recent introduction of AI-generated art software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' AI art gener- ators such as Dall-E and Midjourney can create fully rendered images based solely on a user’s prompt, just at the click of a button.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' Some artists fear if the cheaper price and conveyor- belt speed that comes with AI-produced images is seen as an improvement to the current system, it may permanently change the way society values/views art and artists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' In this ar- ticle, we consider the implications that AI art generation in- troduces through a post-industrial revolution historical lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' We then reflect on the analogous issues that appear to arise as a result of the AI art revolution, and we conclude that the problems raised mirror those of industrialization, giving a vi- tal glimpse into what may lie ahead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' Introduction The industrial revolution caused a major shift from skilled to unskilled labor when machines contributed to a massive lay- off of artisans in favor of factory-line workers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' William Pelz describes how prior to the industrial revolution people had been working in much the same way for thousands of years, producing goods through human labor with some assistance via animals or water power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' However, he says, “All of this changed with the rise of the machine: tools would no longer serve people, but rather people would serve machines” (Pelz 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' Pelz argues that due to this massive change in goods production, the people became an “appendage” to the ma- chine, as humans were suddenly at the mercy of machines using them to make goods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' As factory life took hold, com- pensation was no longer based on paying for skill, but rather on paying for time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' This helped to usher in a different kind of day-to-day experience for most commoners–that of a time- work discipline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' Maxine Berg (2014) notes the change to a mechanized factory sector from pre-industrial handicrafts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' Many traditional workers had to transition to this type of work, or risk being left behind by a quickly transforming industry, leading to the devaluation and destruction of busi- nesses that could not compete with the cheaper and faster production that industrialization had to offer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' Creative AI Across Modalities 2023 (creativeai-ws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content='io), Thirty-Seventh AAAI Conference on Artificial Intelligence (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content='aaai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content='org), February 7-23, Washington, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=', USA With the recent introduction of AI-based art models (Ho, Jain, and Abbeel 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' Ramesh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' Ruiz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' 2022), we argue that a shift largely reminiscent of the post- industrial revolution is unfolding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' As AI-based models be- come more and more common, issues artisans experienced in the mid 1800s are reemerging, which similarly question the very existence of artists today.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' As we are on the precipice of this revolution, it is imperative for all stakeholders, viz, policymakers, ethicists, computer scientists and artists to un- derstand what such a shift would entail in order to manage the consequences of that shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' In this article, our aim is to view recent developments in the art industry due to the introduction of artificial intelli- gence models through a post-industrial lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' In the follow- ing sections, we first discuss how technologies influenced views on independent artisans in the aftermath of the indus- trial revolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' Next, we discuss the positive and negative implications of the post-industrial view on AI generated art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' Finally, we consider current issues raised by these models, and conclude by reflecting on the analogous issues seen in the industrial revolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' A Historical Shift From Individual to Factory Worker The switch from an individualist working environment to a factory-centered one would permanently change the way so- ciety viewed independent artisans, bringing in a new age of commercialization that was predicated on the swift produc- tion of machine-made products over man-made ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' This difference on workforce style in is how the Industrial Rev- olution led to a change in “people’s relationship to crafts- manship, time, community and their own role in society as a whole”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' Such a change in working style was caused that it is hard to imagine a time where machines were not at the fore- front of civilization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' As the industry transformed resemble today’s working world, small-scale artisans were pushed out of people’s minds in favor of production that was faster and cheaper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' The individual craftsman all but disappeared from the forefront of the business world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' By the 1850s, most in- dependent shoemakers had been replaced by shoe factories, independent weavers had gone out of business, and women with hand looms were quickly outstripped by the factories and machines bringing more people cheaper goods of higher average quality (Smail 1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' However, a plethora of small-scale and skill-intensive sec- tors, like those in the metal trades and textile industries, managed to develop alongside the rise of factories (Berg 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' Parts of the world also still value individually-made fine arts objects, especially cultures in the east like China and India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' Berg points to the idea of luxury fashion in France or small-scale building restoration which is popular in Eu- rope (Berg 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' Though factorization still prevails, it is also important to note that the since the early 2000s demand for “niche” artisans has actually shown an upward trend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' The Alliance for Artisan Enterprise (2012) reports that the global market for artisan-made products has increased by more than 8% per year since 2002, and is worth more than $32 billion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' One of the reasons for such a trend has been an increased willingness to pay a premium for distinctive vis- `a-vis mass-produced, goods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' Machines Have Politics Winner (1996) provides an illuminating example of trans- forming industry in “Do Artifacts Have Politics?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' when he looks at the industrial mechanization of Chicago in the 1880s where the switch to factory production hurt skilled workers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' When pneumatic molding machines were imple- mented by Cyrus McCormick’s reaper manufacturing plants, it drove individuals out of business (Winner 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' In many other industries, this happened because the factory process was cheaper, but that was not the case here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' In this instance, McCormick and the National Union of Iron Molders were at war, so even though the iron molding machines were not cheaper, they were used to push the previous workers away from unionization and out of business.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' Therefore, while the addition of industrial manufacturing hurt many skilled la- borers naturally through cheaper replacements, it also hurt them unnaturally through the furthering of political agendas (Winner 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' Unarguably, Winner concludes that the molding machine thus has politics in that its technical arrangements have be- come a form of order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' Instead of subscribing to the use of pneumatic molding machines to speed up or cheapen pro- duction, these machines expressed human motives in their use towards achieving authority over others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' It is undeniable that even if artifacts and machines do not have politics, they do indeed have power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' In his famous essay “The Work of Art in the Age of Mechanical Reproduction,” Benjamin (1935) discusses the change in perception of art in the age of mass production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' Before mechanical reproduction, art was unique and valued for its “aura” – which was derived from its authenticity and its physical and cultural context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' However, with the ability to mechanically reproduce works of art, the traditional bases of cultural authority and hierarchy were challenged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' As art moved to being based on politics instead of tied to rituals, it began to serve as a tool for political activism and resis- tance, ultimately bringing about social and political change (Benjamin 1935).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' Historically, it seems fair to say that the introduction of new technologies in the industrial revolution had many im- pacts on the individual craftsmen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' Some of these impacts might be considered a natural course of action, but it is imperative as we move forward to acknowledge that other forms of use can be exacted through the introduction of new technology–use beyond that of merely what a machine phys- ically produces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' AI Art Generation We focus on the recent introduction of AI-generated art soft- ware to the current art world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' AI-based art generators such as Dall-E (Ramesh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' 2021, 2022) and Midjourney are rel- atively new pieces of technology that can now create fully rendered images based solely on a user’s prompt, often pro- ducing impressive and intricate results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' If the industrial rev- olution changed the way society viewed artisans and crafts- men, how might the AI art revolution do the same?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' We now examine AI image generators as a computing technology that has the potential to cause this analogous shift in the art industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' The Perception of AI Art The most general fear associated with AI-generated art is that it could drastically reduce the amount of jobs available to working commercial artists today in areas such as illustra- tion, animation, and graphic design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' However, some others are concerned with the idea that the more “traditional” no- tion of art may also be modified by AI-generated art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' In “The Culture Industry: Enlightenment as Mass De- ception,” Adorno and Horkheimer (1944) introduce the term “culture industry,” and compare technological advancement of mass media and creation to factory production of goods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' They refer to the “assembly-line character of the culture in- dustry, the synthetic, planned method of turning out its prod- ucts” as creating a passive society that is being manipulated into being satisfied by the products of capitalism, rather than by way of true psychological needs such as freedom, creativ- ity and happiness (Adorno and Horkheimer 1944).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' In 2018, the art-collective Obvious, produced an art piece “Edmond de Belamy,” via a generative adversarial network (GAN) software package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' The artwork was printed onto a canvas and sold at auction for $432, 500, over 43 times its pre-auction estimated value (Cohen 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' Adorno and Horkheimer might argue that artwork created by an AI is created to be an ideal of art, and that AI artwork is merely feeding into mass-media culture industry that threat- ens “high arts.” Oppositionally, the idea that AI-created art could be worthwhile in itself as an art piece, as seen in its high price valuation, points heavily towards James Moor’s prediction that technology is shifting the questions we ask from “How well does a computer do such and such an activity?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' to “What is the nature and value of such and such an activity?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' (Moor 1985).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' In our case, the question shifts from “How well can computers make art?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' to “What is art?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' What is Art?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' The current literature on human attitudes towards AI- generated art presents some evidence as to what people might feel of this shift in viewpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' In two recent studies (Hong and Curran 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' Mikalonyt 2022) on attitudes to- wards artwork produced by humans versus by artificial intel- ligence, researchers found that while most test subjects felt that AI-generated art could be considered “art,” they were much less inclined to consider the art to be produced by an “artist”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' This is a significant distinction because it implies that while artwork contains artistic value just by existing, artwork is not made just by putting pen to paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' Thus we might push the shift Moor describes even further, from the question of “What is art?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' to “What is an artist?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' “When judging whether an object falls under the category of ‘artwork,’' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' the intent of the creator is seen as more impor- tant than even the appearance of the object in question” - Mikalonyt (2022) seem to think,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' this is the reason that their participants were not unwilling to consider art created by a “robot” to be “art,”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' but were significantly more at odds with calling a “robot” an “artist.”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' In shifting the question from the object to the character- istic identity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' we come to a yet unsolved question about AI art: If artwork generated by AI can be called “art”,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' but the model is arguably not an “artist,”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' then who is the author of such a piece?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' Mikalonyte and Kneer posit that this question has not yet been solved, suggesting that the lack of answer is reflected in the fact that autonomously generated AI artwork has yet to be copyrighted, with proposals to give copyright to the human designers of the artificial intelligence, as well as to redefine “authorship” so as to include robots in the def- inition (Mikalonyt 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' Human Art from Ends to Means Besides raising questions in moral philosophy, such an attri- bution to the functional aspects of art vis-`a-vis the aesthetics could mean a lot of hope for traditional artists, especially those who were dependent on the techniques rather than the aesthetics of the final product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' Artists who would only be differentiated by techniques might resort to promoting un- usual techniques which are beyond the current scope of AI art models, or at least AI art models at present (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content='g painting on paper towels or woodblocks, using the back of the paint- brush, using fingernails to paint).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' It wouldn’t be surprising if artists would resort to differentiating factors relying on “means” rather than “ends”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' Artists would especially want their work to be intrinsically different - earlier that could be achieved via both unique propositions of outcomes and of methodology, but now it would largely be the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' It won’t be a surprise to witness more conservative forms being re- inforced (Browne 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' Analogous possible trends of in- creased interest in niche artworks as against mass-produced ones would actually further promote such resorting to tradi- tional artistry and differentiating means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' Fast-Paced Computational Creativity Pelz’s viewpoint that people became appendages to ma- chines during the industrial revolution clearly has art analogues today.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' Over the years, a large section of the art industry has already resorted to computational meth- ods (Li, Hashim, and Jacobs 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' Feldman 2017) after wit- nessing the benefits of generative art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' In generative art, new concepts, forms, shapes, colors, or patterns are created algo- rithmically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' Artists or programmers first establish some cri- teria, post-which a computer creates new art forms adhering to those criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' Such generative art is considered more aes- thetically pleasing than functional (Ball 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' Hence wher- ever the functional aspects of art would be irrelevant, like in branding and advertising or the larger design sector, this transition towards computational art can accelerate people’s dependence on art designs which can be quickly iterated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' Reduced Dependency on Traditional Artists Suddenly, a piece of art that may have taken days to produce by a professional can be done with a handful of suggested words by almost anyone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' It must be noted that if the emer- gence of this technology follows a similar path to that of mechanization following the industrial revolution, it could devalue commercial artists significantly and create massive job loss in an industry that was previously known to require a deeply human touch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' Everything from storyboarding, con- cept art, and movie creation to advertising work and social media would be significantly different, causing a massive problem for the individual artists who may have trained for years to perfect their skill sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' Benefits to Business Art has previously been far from a process one could au- tomate, but these AI art generators might be the cause of a grand shift from skilled to unskilled labor in the free-lance art world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' In the past few months, several online publications have tested using tools like Dall-E or MidJourney to provide art to accompany their written content (Warzel 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' Access to AI art generation tools could be seen as an im- provement to the current system for many business owners where there is a strong demand for commercial art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' Being able to generate content for websites, branding, marketing and sales in a virtually cost-free manner could help small- business owners to reach bigger audiences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' Online publica- tions have largely stepped away from free-lance artists any- ways, with many publications hosting content that was cre- ated elsewhere (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' embedded tweets or stock photography).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' Stock photography businesses sell royalty-free pictures for personal or commercial use, usually paying 15 − 40% to the creator of the photo for each license (Vincent 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' Recently, AI have been breaking into this market, with the distinct art styles of Dall-E or Midjourney popping up on stock photo websites (Edwards 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' Shutterstock, one of the largest stock photo retailers, announced that along-with OpenAI, they would be banning artwork from other AI gen- erators from being uploaded to their site, and it would also create a “Contributor Fund” to help pay the artists whose work was used to train the AI software (Vincent 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' Art Democratization Platforms like YouTube, Instagram and TikTok, which have become hosts of content creation, have been able to at- tract millions of content creators who make money us- ing their services by providing them with fame and mon- etary rewards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' This has largely happened with the reduced technical and social barriers that these arguably demo- cratic platforms have provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' AI art models could also tread the same path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' Democratization of art would mean almost anyone can produce artistic creations, including a person without limbs or someone with a neurological dis- order that affects their ability to draw or paint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' AI-art models might be exclusively seen as potential attackers on the most talented segments of the artistic society, but they will doubtlessly open up a level playing field for those who considered art out of reach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' Besides, such de- mocratization would also be reflected in crowd-sourced efforts (Bigham, Kulkarni, and Lasecki 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' Kittur et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' 2013, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' Dhole et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' Srivastava et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' 2022) which would seek contributions to aid in developing large creative models fairly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' Increased Amount of Plagiarism The Industrial Revolution was replete with examples of in- dustrial espionage (Harris 1985;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' Christopher Klein 2019) where large businesses often stole the work and ideas of people who had historically performed it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' Governments reg- ularly encouraged individuals to steal ideas, especially from abroad, since it hardly required applicants to be inventors, especially if the invention was abroad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' The current AI art models already have been exposed to the artworks of many artists without giving them proper attribution or even seek- ing their permission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' Millions of generations of artwork have already been utilizing these styles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' Besides artists’ work being plagiarized, it is unclear how credit would get di- vided amongst the artists, model trainers and users writing prompts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' While there have attempts to legally delineate the complete generation pipeline (Fjeld and Kortz 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' Kim 2020), the black box nature would make it hard for fair credit attribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' Increased Carbon Emissions The dramatic increase in coal and gas usage, which sky- rocketed pollution levels across major cities and indus- trial zones, was an unfavorable side consequence of the Industrial Revolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' Today, with large groups of peo- ple expected to move towards careers of computational art, it would be inevitable that training these AI-art mod- els would also be performed more frequently, via re- searchers as well as artists raising concerns of carbon emissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' Strubell, Ganesh, and McCallum (2020)’s lifecy- cle assessment of training popular large language models re- vealed that a typical training process took nearly five times the lifetime carbon emissions of the average American car.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' Besides, the largeness of these models also necessitates GPU usage during inference time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' Furthering Political Agendas Winner (1996) has been helpful for giving convincing argu- ments that artifacts and machines in general can have biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' The 9ft clearance levels of Long Island bridges were a de- sign decision made by urban planner Robert Moses in order to restrict buses filled with low-income people and racial mi- norities from accessing parkways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' As a result of the biases implicit in these designs, people were given limited access to parts of their own city (Winner 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' Benjamin (1935) had argued that as art’s authenticity di- minishes due to the ease of mechanization, it begins to be based on politics rather than on rituals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' Just like the Long Island bridges, political agendas intertwined with design have had crucial consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' Therefore, it wouldn’t be a surprise if biases mirrored in the AI art generation of to- day (Bansal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' 2022) were exploited to further political agendas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' Hassine and Neeman (2019) revealed that AI gen- erated art skews mostly white, both in depiction and in rep- resentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' Unless age, sex or race is specified, prompts to the system have built in biases towards young white men (Srinivasan and Uchino 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' This unconscious bias could have a manifestation in the real world, just as Robert Moses’ designs manifested for racial minorities in New York City.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' Questions of Concern Finally, we consider the issues to pubic welfare and society that AI-art generation introduces through its effects on the artists of today.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' We also consider the scope of involvement that computer scientists and AI have had in creating or con- tributing to these issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' Is the threatened change in the status of artists characterized by the primary and essential involvement of AI models?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' Industrial “sweatshops” mass-producing art for commercial consumption have been a constant long before the computer became an element in the equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' From the comic strips of the 1880s - 1960s to the comic books of the 1930s - present day, to the mass produced landscape art created for furniture stores in Asian factories since the mid-1950s, art has been commoditized long before the computer (Hersch 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' In these assembly lines, one person would sketch the outlines of image, another would pencil in details of the people, an- other would ink those images, yet another would draw in the backgrounds, and a final hand would color the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' Pro- duced by an assembly line usually called a “studio,” the art would be signed either by an arbitrarily selected worker or even by a completely fictitious artist (Hersch 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' More to the point — this was art created by “factory workers” who acted the same role in production as today’s graphics pro- grams do (Campbell 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' Therefore, one could also argue that AI models might not be essential to the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' How- ever, what distinctively stands out with the usage of AI art models, is the rapid pace of artwork creation and prolifera- tion, unlike what was witnessed before the arrival of com- puters or the internet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' Does the threatened change in the status of artists occur because of exploiting some unique property of AI models?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' The primary difference created by technology is mass ac- cess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' To staff a “studio” with bit-work artists requires a sub- stantial investment in infrastructure, equipment, and labor costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' Such programs as are available today are much more economical and widely available to individuals — from hob- byists to serious artists to mercenary corporations — than at any time in our history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' So it is legitimate to argue that AI models have uniquely driven the scale and creativity of the problem far more broadly than earlier technology could have.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' Could this issue have even arisen without the involvement of AI models?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' The answer depends heavily on the question of whether AI- art generation programs are considered as tools or entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' Motion-picture technology created entirely new art forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' One could create art with motion across space and time in a way that entirely changed how our culture thought about visual art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' But the cameras themselves have never — for all their technological sophistication — been more than tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' Cameras do not set out with purpose to make movies, and AI art generation programs do not set out independently to create images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' Both must be employed by users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' Conclusion Perhaps what AI-art generation software is doing is forc- ing our society to confront a much larger issue about artists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' Instead of threatening the status of artists, advances in com- puter technology require us to confront the idea that the view of artists has already changed, and has been changing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' While on the surface AI-generated art seems unique, many of the issues that it is raising concerning society’s views of art and artists are merely more complex callbacks to the mechanization of artisan’s projects in the industrial revolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' Specifically, the art generated by AI-generation algorithms creates problems that mirror those of industrial- ization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' In the same way that the artisan was pushed out, so too is the artist today.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' In the same way intellectual property was compromised during the 1800s, legal loopholes may en- sure many artists are not duly credited without proper AI regulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' Our study of the industrial revolution analogues serves as a warning to look backwards at the past treatment of creators and consumers of industry in the wake of newly introduced technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' We pose that this may be an important step to take before experiencing the consequences of the technical revolutions that unfold before us today.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' But these analogies should not be taken as a discourage- ment against developing large models or to undermine the efforts of the field of AI in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' We should actively strive to improve technical parameters of these models, by accounting for the possibility of potential damage early on, as these models have and already display tremendous poten- tial for business as well as for democratization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' Limitations How AI art generation tools will affect artists is an extremely subjective and multifaceted subject, and forecasting it pre- cisely will not be easy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' We have provided comparisons based on events that occurred post the industrial revolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' How- ever, we think that empirical evidence would be helpful in better understanding many of the issues raised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' Our objec- tive was to present as thorough and comprehensive an anal- ysis as possible by considering the technical, political and industrial implications of art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' Our section on “What is Art?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' is quite limited due to the rich history concerning this ques- tion from a philosophical point of view, but we still feel it was important to include.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' Nonetheless, we believe that our work will serve as a crucial gateway for both the engineering and humanities disciplines to facilitate dialogue and advance debate about the impact of AI art tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' Acknowledgments We thank Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' Kristin Williams and Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' Steve Newton for their crucial thoughts and feedback in numerous drafts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' We also thank the anonymous reviewers and meta reviewer for their invaluable suggestions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content='org/wp-content/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' Accessed: 2022-11-04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' Vincent, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' Shutterstock will start selling AI-generated stock imagery with help from OpenAI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' Warzel, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' I Went Viral in the Bad Way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' Winner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' Do artifacts have politics?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' https://link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content='springer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content='com/article/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content='1007/BF02583549#citeas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} +page_content=' Accessed: 2022-10-015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfiQ3g/content/2301.05133v1.pdf'} diff --git a/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf b/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..6f581f8f1712b101c41ff27de9ccdb9d4c5e9e9b --- /dev/null +++ b/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid 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Kojima,1 N. Katayama,1, ∗ Y. Matsuda,1 M. Shiomi,1 R. Ishii,2 and H. Sawa1 +1Department of Applied Physics, Nagoya University, Nagoya 464-8603, Japan +2The Institute for Solid State Physics, Tokyo University, Tokyo 277-8581, Japan +(Dated: January 11, 2023) +Vanadium atoms in layered LiVO2 form in-plane periodic vanadium trimers at low temperatures, +but the trimers appear randomly in the stacking direction because there are many trimer configura- +tions with comparable lattice energy. We detailed an original modeling scheme to represent glassy +states with a completely disordered trimer configuration in the stacking structure. Through PDF +analysis using this model, we show that the synthesis method can yield two types of low-temperature +stacking structures: a completely disordered stacking structure and a short-range order in the stack- +ing structure. The phase transition temperature of the former sample is about 15 K lower than that +of the latter. We discuss that this is due to the strong trimer frustration that appears in the sam- +ple without short-range order, which suppresses the phase transition temperature, similar to the +frustration effect in conventional spin systems. +In transition metal compounds with orbital degrees of +freedom, transition metal ions often spontaneously as- +semble to form “molecules” in solids at low tempera- +tures [1–13]. Since electrons are trapped and localized in +bonding orbitals during molecular formation, molecular +formation generally appears as a drastic first-order tran- +sition to a nonmagnetic insulating state with a large en- +tropy change [3, 10, 14, 15]. Because of these properties, +molecular formation has attracted considerable attention +not only from the perspective of the fundamental physics +of strongly correlated electrons [16, 17] and low dimen- +sionality [18], but also from an applied perspective, such +as sensors [19] and phase-change materials [7, 14, 20– +22]. Molecular formation in solids significantly alters the +surrounding crystalline field, supporting the emergence +of long-range configurations with the most stable lattice +energies at low temperatures. +If there are many pat- +terns of molecular arrangement with similar lattice en- +ergies and these patterns are not uniquely determined, +what structures and physical properties can arise from +these ground-state frustrations? +Layered LiVO2 with a two-dimensional triangular lat- +tice may provide a fascinating playground for studying +such an issue [12]. As shown in Figure 1(a), LiVO2 has a +stacking structure with three periodic layers, which un- +dergoes a nonmagnetic-paramagnetic transition at about +500 K upon heating [14]. The low-temperature nonmag- +netism is due to the long-range trimerization of vanadium +on in-plane triangular lattice. On the other hand, how- +ever, as shown in Figure 1(b), there are three degrees of +freedom in the arrangement of the trimer for adjacent +VO2 layers, and the energies of the lattice structures of +these three patterns (i-iii) are equivalent. As a result, +the ordered structure should not be uniquely determined. +Such a trimer frustration state may seem analogous to +∗ Corresponding +author.; +katayama.naoyuki.m5@f.mail.nagoya- +u.ac.jp +spin frustration [23–29], but is purely a frustrated state +of structural origin. Therefore, it is an exciting research +challenge to explore the new electronic phases and phys- +ical properties that emerge from “trimer frustration,” as +in conventional spin systems with geometric frustration. +However, this requires techniques for accurately model- +ing the structural state of LiVO2 and evaluating it with +experimental methods. +Here, we report on the modeling of the random trimer +arrangement in the stacking direction, and the PDF anal- +ysis of LiVO2 based on this model. +Our analysis re- +veals important differences in the local structure of the +low-temperature phase of LiVO2 depending on the syn- +thesis method. +Samples synthesized by a combination +of solid- and solution-reaction methods show completely +disordered glassy trimer arrangement, while those syn- +thesized by the solid-phase reaction method show short- +range order in the trimer arrangement in the stacking +direction. DSC measurements of both samples revealed +that the entropy change associated with the phase tran- +sition is maximized as the Li/V ratio approaches 1.0. On +the other hand, regardless of the Li/V ratio, the trimer- +ization temperature of the sample with a completely dis- +ordered trimer arrangement in the stacking direction was +decreased by nearly 15 K compared to the short-range or- +dered sample. We discuss that these results indicate that +strong trimer frustration suppresses the low-temperature +phase in samples with a completely disordered trimer ar- +rangement in the stacking direction, similar to the sup- +pression of antiferromagnetic order in spin systems with +strong magnetic frustration. +We have grown two types of samples, named “as- +grown samples” and “solution samples,” depending on +the synthesis method. +“As-grown samples” were syn- +thesized by solid-state reaction. +Li2CO3 (99.9%) and +V2O3 (99.99%) were mixed in the ratio of Li/V ∼ 1.00, +placed on an alumina boat, and sintered at 625 ℃ for +24 hours with H2/Ar=5% gas flowing. +The obtained +samples were regrinded and sintered at 750 ℃ for 12 +arXiv:2301.03833v1 [cond-mat.str-el] 10 Jan 2023 + +2 +hours with H2/Ar=5% gas flowing. The “solution sam- +ple” was obtained by immersing it in a large excess of +0.2 M n-BuLi/Hexane solution for 24 hours under an +Ar atmosphere in a glove box after the solid phase re- +action so that the Li content was almost 1.0. +The Li +content of these samples was evaluated by ICP measure- +ment. In the following, these samples used in the exper- +iments are labeled “as-grown(0.97)”, “as-grown(0.96)”, +“as-grown(1.01)”, and “solution(1.01)” according to the +Li content estimated by ICP measurement. +ICP measurement was performed using a SPECTRO +ARCOS MV130 (Hitachi High-Tech). The obtained sam- +ples were subjected to DSC measurements using a 204 F1 +Phoenix (Netzsch). The temperature rise and fall rates +were 10 K/min. +Synchrotron X-ray diffraction experi- +ments were performed at BL5S2 of the Aichi SR using a +quadruple PILATUS 100 K detector at an E = 20 keV +X-ray energy. Diffraction experiments to obtain the pair +distribution function (PDF) were carried out at BL04B2 +of SPring-8. The experiments were performed using E += 61 keV X-rays, and a combination of four CdTe and +three Ge point detectors was used. Rietveld and Le Bail +analysis were performed by Rietan-FP [30]. PDF conver- +sion was performed using a dedicated package [31]. After +corrections, the PDF was obtained by Fourier transform +with 0.2 < Q (˚A) < 25.5 and ∆Q = 0.01 (˚A). The sim- +ulations of the PDF were performed using PDFgui [32]. +VESTA was used to draw the crystal structure [33]. +Synchrotron X-ray diffraction results revealed that all +the samples used in this study are almost single-phase. +Details of the Rietveld analysis are summarized in Sup- +plemental Information [34]. Figure 1(c) shows a part of +the powder X-ray diffraction image at 300 K. For the so- +lution(1.01) sample, a sawtooth 1/3 1/3 0 superlattice +peak appears. The superlattice peaks are due to the 1/3 +1/3 0 superlattice peaks appearing as diffuse streaks in +the l direction as shown in Figure 1(d), indicating that +the trimer is periodically aligned in the plane and ran- +domly aligned in the stacking direction. This has been +reported previously [12]. On the other hand, as shown in +Figure 1(c), the as-grown samples maintain the sawtooth +superlattice peak but generate a sharper peak at 1/3 1/3 +0 than the solution(1.01) sample. This suggests that the +diffuse streak condenses at 1/3 1/3 0, as shown in Fig- +ure 1(e), and short-range order develops in the trimer +arrangement pattern along the stacking direction in the +as-grown sample. +The development of short-range order in as-grown +samples can be confirmed by pair distribution function +(PDF) analysis. Figure 2(a) is a magnified view of the +PDF data in the low and high r regions. +The three +peaks occurring in the low r region indicate the nearest- +neighbor V-O distance, intra-trimer V-V distance, and +inter-trimer V-V distance as shown in Figure 2(b). The +PDF data of the solution and as-grown samples are sim- +ilar in this r region, indicating that they have similar +in-plane structures. On the other hand, the magnified +view of the high r region shows that the PDF patterns +FIG. 1. +(a) Horizontal view of the crystal structure of layered +LiVO2. (b) Three energetically equivalent trimer patterns in +relation to adjacent VO2 layers. +(c) Part of X-ray powder +diffraction data. The highlighted area corresponds to 1/3 1/3 +0. Compared to the solution sample, a sharp peak is clearly +generated in this area in the as-grown samples. +The inset +of the graph shows data normalized so that the background +and peak tops (both indicated by dashed lines) are aligned +to clarify the shape of the superlattice peaks. (d) Schematic +of superlattice reflections of a solution sample with uniform +streaks in the c∗ direction. (e) Schematic of superlattice re- +flection of as-grown sample about to condense to 1/3 1/3 0. +of both samples are very different. The PDF spectrum +of the solution (1.01) sample shows a “glassy” pattern of +broadening peaks. This reflects the lack of order in the +arrangement of the trimer in the stacking direction. On +the other hand, in the as-grown sample, the peak shapes +are sharper than in the solution sample, the peaks are +clearly separated from each other, and even the peaks +with weaker intensities are clearly recognizable. +To investigate short-range ordering, PDF simulation +patterns in the trimer glassy state were created and fitted +to the PDF data. Since three trimer patterns appear in +each VO2 layer, 3n trimer ordered structures appear per +n VO2 layers. The glass pattern simulations were created +by calculating all of the PDF patterns produced by the 3n +possible patterns and adding them together with a weight +of 1/3n [12]. As shown in Figure 2(c), the simulation data +are in good agreement with the experimental PDF data +in the range 1.5 < r(˚A) < 40, indicating the appearance +of a trimer glassy state in solution(1.01). On the other + +Solution +(1.01) +As-grown +(0.97) +As-grown +(0.96)3 +FIG. 2. +(a) PDF data at 1.5 < r(˚A) < 3.5 and 30 < r(˚A) < +40. (b) Distance between atoms in a VO2 plane. (c,d) G(r) +pattern of (c) solution(1.01) and (d) as-grown(0.96) fitted by +glass simulation data. (e) Confidence R(r) of the refinement +obtained in various r regions. (f) G(r) of solution(1.01) and +as-grown(0.96) and their difference ∆G(r). +hand, as shown in Figure 2(d), for the as-grown samples, +a large residual appears above 15 ˚A, which corresponds +to the thickness of three layers, and the residual expands +as r increases. This result can be understood as follows. +First, the reduced G(r) at 1.5 < r(˚A) < 4.0 contains +mainly the component corresponding to the in-plane in- +teratomic distance. The reduced G(r) at 4.0 < r(˚A) < 15 +contains information on the distances between the atoms +in the nearest and next nearest VO2 layers. If there is +a short-range order in the trimer arrangement along the +stacking direction, large residuals are likely to appear at +4.0 < r(˚A) < 15. +However, as shown in Figure 1(b), +the relationships between the trimer in a VO2 layer and +the three trimer patterns in the nearest and next-nearest +VO2 layers are all equivalent, resulting in the same PDF +pattern regardless of which trimer pattern appears. In +other words, the PDF pattern at 4.0 < r(˚A) < 15 is con- +stant regardless of the presence or absence of short-range +order. +When the correlation length of the short-range +order is longer than 15 ˚A, large residuals appear in the +region r(˚A) > 15. This occurs in the fitting of as-grown +samples. +In order to quantitatively investigate the agreement +between the simulated data and the experimentally ob- +tained PDF data, we defined the following evaluation +function R(r), +R(r) = +��r +r′=r−5 {G(r′)exp. − s · G(r′)calc.}2 +�r +r′=r−5 G(r′)2exp. +. +(1) +where G(r)exp. is the experimental value and G(r)calc. is +the simulated reduced PDF data of the trimer glass pat- +tern. Details on how to obtain G(r)calc. are described +in Supplemental Information [34]. s is a dimensionless +scale factor to normalize G(r)exp. and G(r)calc.. +The +value of s that minimizes R(5) was defined as the sample- +specific scale factor and used to calculate R(r) in various +r ranges. +Equation (1) is based on the so-called box- +car refinement concept and is useful for estimating the +correlation length of the short-range order in as-grown +samples. The results are shown in Figure 2(e), where the +R(r) values for the solution(1.01) sample remain low in all +r regions analyzed, while the R(r) values of the as-grown +samples tend to increase uniformly at r(˚A) ≥ 15. In suf- +ficiently large r(˚A) regions, well beyond the correlation +length of the short-range order, there is no correlation +between distant atoms. Therefore, in sufficiently large +r(˚A) regions, the reduced PDF data for the solution and +as-grown samples should again agree well. This can be +roughly inferred from the r dependence of the difference +between the PDF data of the two samples. As shown in +Figure 2(f), the residuals clearly decrease in the region +above 100 ˚A, which roughly corresponds to the correla- +tion length. +The cause of the difference between the solution and +as-grown samples is not clear but is speculated as follows. +In the as-grown sample, the lattice energy degeneracy +due to the numerous stacking patterns is thought to be +lifted, and short-range order is realized in the trimer ar- +rangement in the stacking direction. Since there is no sig- +nificant difference in the basic lattice structure between +the solution and as-grown samples, the interlayer Li ion +sites are likely responsible. Since the short-range order +also appears in as-grown(1.01), the lack of Li ions is not +the origin of the short-range order. +A possible expla- +nation is the disorder of the Li ion sites. As shown in +Figure 3(a), almost all Li ions are ordered into octahe- +dral sites between VO2 layers, but there are also tetrahe- +dral sites where extra Li ions can enter as shown in Fig- +ure 3(b). If the Li ions fully occupy the octahedral site, +the Coulomb potential of the Li ions on the VO2 layer +is uniform and the lattice energy degeneracy associated +with trimerization is preserved, so strong frustration is +expected, as shown in Figure 3(c). However, if deficien- +cies or other disturbances exist at the octahedral site, the + +Intra-trimer +Solutign(l1.01) +-2.6A +As-grown 0.96) +Inter-trime4 +FIG. 3. +(a) Octahedral site with majority Li ions. (b) Tetra- +hedral site with excess Li ions. (c) Relationship between Li ion +and three possible trimer configuration. Li ions are located +between the upper and lower VO2 layers. (d) Relationship +to the trimer when Li is deficient. (e,f) Relation between Li +ions at the tetrahedral sites and (e) upper and (f) lower VO2 +layers. +random potential should lift the lattice energy degener- +acy and produce a stable ordered structure, as shown in +Figure 3(d). Interestingly, the random insertion of Li ions +into the tetrahedral site does not lift the lattice energy +degeneracy because it gives equal random potentials for +the three trimer patterns that appear in the neighboring +VO2 layers as shown in Figures 3(e) and (f). Therefore, +we speculate that the solution reaction with n-BuLi may +have an annealing effect that encourages Li ions to move +between the layers to fully occupy the low-potential oc- +tahedral sites, in addition to the effect of adjusting the +Li ion content. How does the presence or absence of such +short-range order affect electronic properties? +To explore the effect of such short-range ordering on +the physical properties, DSC measurements were per- +formed, and as shown in Figures 4(a) and (b), the entropy +change associated with the phase transition was always +larger in the solution sample than in the as-grown sam- +ple. This is understood to be due to the optimization of +the electronic state of V in the solution sample as a result +of controlling the Li content with n-BuLi. On the other +hand, in contrast to the trend of the entropy change, the +phase transition temperature of the solution sample is ∼ +15 K lower than that of the as-grown sample. This re- +sult is clearly inconsistent with the entropy change data +suggesting stabilization of the trimer structure. +These results seem to indicate that the phase transi- +tion temperature is suppressed in samples without the +FIG. 4. +(a) (upper) Entropy change measured by DSC on +heating process, and (lower) the phase transition tempera- +tures. The Li/V ratio is determined by ICP measurements. +(b) DSC data for as-grown and solution samples. +short-range ordering of the trimer arrangement in the +stacking direction compared to samples with short-range +ordering. This is reminiscent of frustration effects in spin +systems. The present trimer frustration state in LiVO2 +is a unique state formed by the coupling of electrons and +lattice degrees of freedom, and is a consequence of pure +lattice ordering. Nevertheless, it is similar to spin sys- +tems in that the presence of frustration suppresses the +phase transition, and weakening the frustration induces +an ordered state and increases the transition tempera- +ture. This seems to indicate that frustration effects sim- +ilar to those in spin systems can be realized in lattice +systems. +In spin-frustrated systems, the strength of frustration +is quantified by the absolute value of the ratio of the +Weiss temperature to the N´eel temperature (frustration +factor). In the present trimer frustration, the spin gap +estimated from NMR measurements may be an indica- +tor of the strength of the frustration. +From previous +NMR measurements on LiVO2, the spin gap in the low- +temperature phase is estimated to be ∆ ∼ 3400 K [35] +(1600 K [36]), which is much larger than the phase tran- +sition temperature of LiVO2, Tc ∼ 500 K. This seems to +indicate that the trimer transition temperature in LiVO2 +is strongly suppressed. +One might attribute the energy differences to the pres- +ence of local orbital degeneracy lifted state (ODL) that +develops prior at high temperatures, as recent PDF stud- +ies of the local structure have found in many systems that +form orbital molecules at low temperatures [11, 13, 37– +42]. This may be the case for LiVS2, an analog of LiVO2. +LiVS2 has a different stacking structure than LiVO2 and +no trimer frustration [12], but a trimer transition occurs +at 314 K with a gap of ∆ ∼ 1900 K [10, 43, 44]. Above +the phase transition temperature, PDF analysis shows +that a zigzag chain-like short-range order develops, sug- +gesting that orbital degeneracy is already locally lifted +at high temperatures [11]. However, this is not the case +for LiVO2. Our previous PDF studies on LiVO2 have + +(a) +(b) +(c) +Lower +(e) +Excess Li +layer +equivalent +Upper +layer +Possible trimers : +Possible trimers : +degenerate +degenerate +(d) +Li deficiency +(f) +Excess Li +Possible trimers : +Possible trimers : +non-degenerate +degenerateSolution +As-grown(0.96) +Solution +As-grown +(1.01)5 +shown that no such short-range order develops above the +phase transition temperature in LiVO2 [12]. The above +indicates that LiVO2 and LiVS2 are not similar and each +has unique physics for trimer formation. +It should be noted that we were able to address such +a physics of trimer frustration because of our success in +modeling the trimer glassy state of LiVO2 and identi- +fying its structure by PDF analysis. +The existence of +vanadium trimer formation in LiVO2 was pointed out +more than half a century ago based on the lattice sym- +metry of the low-temperature phase [45]. +Subsequent +studies have confirmed the V-V distance splitting asso- +ciated with trimer formation by EXAFS [46] and PDF +analysis [47], electron diffraction analysis [14], and the +NMR measurement using a single crystalline sample [48]. +All of these results support the in-plane appearance of +the trimer, but the identification of the crystal structure +containing the trimer had not been successful for more +than half a century. +This is because the trimer disor- +der in the stacking direction, intrinsic to LiVO2, has not +been properly modeled. Coupled with glassy state mod- +eling, PDF analysis was essential in the present results +to reveal that LiVO2 is the playground where the new +physics of trimer frustration emerges. This achievement +can never be revealed by conventional average structure +analysis, and may point to a new direction in structural +analysis. +Finally, we point out the importance of this trimer +frustration in terms of applications. The latent heat cal- +culated from the entropy change of LiVO2 (∆H ∼ 326 +Jcc−1) is equivalent to that of H2O (∆H ∼ 306 Jcc−1) +and is promising as a phase change material (PCM) prod- +uct [7, 14, 20–22]. If the phase transition temperature can +be manipulated by controlling trimer frustration, it could +be a PCM material that can be used at various temper- +atures. Such studies are beyond the scope of this study, +but they clearly demonstrate the importance of both the +fundamental and applied aspects of trimer frustration. +ACKNOWLEDGMENTS +All authors thank Dr. +K. Ohara and S. Hashimoto +for fruitful discussion. The work leading to these results +has received funding from the Grant in Aid for Scientific +Research (Nos. JP17K17793, JP20H02604, JP21K18599, +JP21J21236). This work was carried out under the Visit- +ing Researcher’s Program of the Institute for Solid State +Physics, the University of Tokyo, and the Collaborative +Research Projects of Laboratory for Materials and Struc- +tures, Institute of Innovative Research, Tokyo Institute of +Technology. PXRD experiments were conducted at the +BL5S2 of Aichi Synchrotron Radiation Center, Aichi Sci- +ence and Technology Foundation, Aichi, Japan (Propos- +als No. 202002076, No. 202104111, No. 2021L3002 and +No. 202105170), and at the BL04B2 of SPring-8, Hyogo, +Japan (Proposals No. 2018B1128, No. 2019A1218, No. +2021A1112 and No. 2021B1119). +[1] P. G. Radaelli, Y. Horibe, M. J. Gutmann, H. Ishibashi, +C. H. Chen, R. M. Ibberson, Y. Koyama, Y. S. Hor, +V. Kiryukhin, and S. W. 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B 87, 075135 (2013). + diff --git a/R9E2T4oBgHgl3EQfWgcY/content/tmp_files/load_file.txt b/R9E2T4oBgHgl3EQfWgcY/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a887133d7c078321594f74c8242f82eae34b29c4 --- /dev/null +++ b/R9E2T4oBgHgl3EQfWgcY/content/tmp_files/load_file.txt @@ -0,0 +1,799 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf,len=798 +page_content='Short-range order and increased transition temperature in LiVO2 with weakened trimer frustration K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Kojima,1 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Katayama,1, ∗ Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Matsuda,1 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Shiomi,1 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Ishii,2 and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Sawa1 1Department of Applied Physics, Nagoya University, Nagoya 464-8603, Japan 2The Institute for Solid State Physics, Tokyo University, Tokyo 277-8581, Japan (Dated: January 11, 2023) Vanadium atoms in layered LiVO2 form in-plane periodic vanadium trimers at low temperatures, but the trimers appear randomly in the stacking direction because there are many trimer configura- tions with comparable lattice energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' We detailed an original modeling scheme to represent glassy states with a completely disordered trimer configuration in the stacking structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Through PDF analysis using this model, we show that the synthesis method can yield two types of low-temperature stacking structures: a completely disordered stacking structure and a short-range order in the stack- ing structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' The phase transition temperature of the former sample is about 15 K lower than that of the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' We discuss that this is due to the strong trimer frustration that appears in the sam- ple without short-range order, which suppresses the phase transition temperature, similar to the frustration effect in conventional spin systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' In transition metal compounds with orbital degrees of freedom, transition metal ions often spontaneously as- semble to form “molecules” in solids at low tempera- tures [1–13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Since electrons are trapped and localized in bonding orbitals during molecular formation, molecular formation generally appears as a drastic first-order tran- sition to a nonmagnetic insulating state with a large en- tropy change [3, 10, 14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Because of these properties, molecular formation has attracted considerable attention not only from the perspective of the fundamental physics of strongly correlated electrons [16, 17] and low dimen- sionality [18], but also from an applied perspective, such as sensors [19] and phase-change materials [7, 14, 20– 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Molecular formation in solids significantly alters the surrounding crystalline field, supporting the emergence of long-range configurations with the most stable lattice energies at low temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' If there are many pat- terns of molecular arrangement with similar lattice en- ergies and these patterns are not uniquely determined, what structures and physical properties can arise from these ground-state frustrations?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Layered LiVO2 with a two-dimensional triangular lat- tice may provide a fascinating playground for studying such an issue [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' As shown in Figure 1(a), LiVO2 has a stacking structure with three periodic layers, which un- dergoes a nonmagnetic-paramagnetic transition at about 500 K upon heating [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' The low-temperature nonmag- netism is due to the long-range trimerization of vanadium on in-plane triangular lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' On the other hand, how- ever, as shown in Figure 1(b), there are three degrees of freedom in the arrangement of the trimer for adjacent VO2 layers, and the energies of the lattice structures of these three patterns (i-iii) are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' As a result, the ordered structure should not be uniquely determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Such a trimer frustration state may seem analogous to ∗ Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' katayama.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content='naoyuki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content='m5@f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content='mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content='nagoya- u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content='jp spin frustration [23–29], but is purely a frustrated state of structural origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Therefore, it is an exciting research challenge to explore the new electronic phases and phys- ical properties that emerge from “trimer frustration,” as in conventional spin systems with geometric frustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' However, this requires techniques for accurately model- ing the structural state of LiVO2 and evaluating it with experimental methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Here, we report on the modeling of the random trimer arrangement in the stacking direction, and the PDF anal- ysis of LiVO2 based on this model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Our analysis re- veals important differences in the local structure of the low-temperature phase of LiVO2 depending on the syn- thesis method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Samples synthesized by a combination of solid- and solution-reaction methods show completely disordered glassy trimer arrangement, while those syn- thesized by the solid-phase reaction method show short- range order in the trimer arrangement in the stacking direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' DSC measurements of both samples revealed that the entropy change associated with the phase tran- sition is maximized as the Li/V ratio approaches 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' On the other hand, regardless of the Li/V ratio, the trimer- ization temperature of the sample with a completely dis- ordered trimer arrangement in the stacking direction was decreased by nearly 15 K compared to the short-range or- dered sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' We discuss that these results indicate that strong trimer frustration suppresses the low-temperature phase in samples with a completely disordered trimer ar- rangement in the stacking direction, similar to the sup- pression of antiferromagnetic order in spin systems with strong magnetic frustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' We have grown two types of samples, named “as- grown samples” and “solution samples,” depending on the synthesis method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' “As-grown samples” were syn- thesized by solid-state reaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Li2CO3 (99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content='9%) and V2O3 (99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content='99%) were mixed in the ratio of Li/V ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content='00, placed on an alumina boat, and sintered at 625 ℃ for 24 hours with H2/Ar=5% gas flowing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' The obtained samples were regrinded and sintered at 750 ℃ for 12 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content='03833v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content='str-el] 10 Jan 2023 2 hours with H2/Ar=5% gas flowing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' The “solution sam- ple” was obtained by immersing it in a large excess of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content='2 M n-BuLi/Hexane solution for 24 hours under an Ar atmosphere in a glove box after the solid phase re- action so that the Li content was almost 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' The Li content of these samples was evaluated by ICP measure- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' In the following, these samples used in the exper- iments are labeled “as-grown(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content='97)”, “as-grown(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content='96)”, “as-grown(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content='01)”, and “solution(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content='01)” according to the Li content estimated by ICP measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' ICP measurement was performed using a SPECTRO ARCOS MV130 (Hitachi High-Tech).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' The obtained sam- ples were subjected to DSC measurements using a 204 F1 Phoenix (Netzsch).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' The temperature rise and fall rates were 10 K/min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Synchrotron X-ray diffraction experi- ments were performed at BL5S2 of the Aichi SR using a quadruple PILATUS 100 K detector at an E = 20 keV X-ray energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Diffraction experiments to obtain the pair distribution function (PDF) were carried out at BL04B2 of SPring-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' The experiments were performed using E = 61 keV X-rays, and a combination of four CdTe and three Ge point detectors was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Rietveld and Le Bail analysis were performed by Rietan-FP [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' PDF conver- sion was performed using a dedicated package [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' After corrections, the PDF was obtained by Fourier transform with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content='2 < Q (˚A) < 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content='5 and ∆Q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content='01 (˚A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' The sim- ulations of the PDF were performed using PDFgui [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' VESTA was used to draw the crystal structure [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Synchrotron X-ray diffraction results revealed that all the samples used in this study are almost single-phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Details of the Rietveld analysis are summarized in Sup- plemental Information [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Figure 1(c) shows a part of the powder X-ray diffraction image at 300 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' For the so- lution(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content='01) sample, a sawtooth 1/3 1/3 0 superlattice peak appears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' The superlattice peaks are due to the 1/3 1/3 0 superlattice peaks appearing as diffuse streaks in the l direction as shown in Figure 1(d), indicating that the trimer is periodically aligned in the plane and ran- domly aligned in the stacking direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' This has been reported previously [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' On the other hand, as shown in Figure 1(c), the as-grown samples maintain the sawtooth superlattice peak but generate a sharper peak at 1/3 1/3 0 than the solution(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content='01) sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' This suggests that the diffuse streak condenses at 1/3 1/3 0, as shown in Fig- ure 1(e), and short-range order develops in the trimer arrangement pattern along the stacking direction in the as-grown sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' The development of short-range order in as-grown samples can be confirmed by pair distribution function (PDF) analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Figure 2(a) is a magnified view of the PDF data in the low and high r regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' The three peaks occurring in the low r region indicate the nearest- neighbor V-O distance, intra-trimer V-V distance, and inter-trimer V-V distance as shown in Figure 2(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' The PDF data of the solution and as-grown samples are sim- ilar in this r region, indicating that they have similar in-plane structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' On the other hand, the magnified view of the high r region shows that the PDF patterns FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' (a) Horizontal view of the crystal structure of layered LiVO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' (b) Three energetically equivalent trimer patterns in relation to adjacent VO2 layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' (c) Part of X-ray powder diffraction data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' The highlighted area corresponds to 1/3 1/3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Compared to the solution sample, a sharp peak is clearly generated in this area in the as-grown samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' The inset of the graph shows data normalized so that the background and peak tops (both indicated by dashed lines) are aligned to clarify the shape of the superlattice peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' (d) Schematic of superlattice reflections of a solution sample with uniform streaks in the c∗ direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' (e) Schematic of superlattice re- flection of as-grown sample about to condense to 1/3 1/3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' of both samples are very different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' The PDF spectrum of the solution (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content='01) sample shows a “glassy” pattern of broadening peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' This reflects the lack of order in the arrangement of the trimer in the stacking direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' On the other hand, in the as-grown sample, the peak shapes are sharper than in the solution sample, the peaks are clearly separated from each other, and even the peaks with weaker intensities are clearly recognizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' To investigate short-range ordering, PDF simulation patterns in the trimer glassy state were created and fitted to the PDF data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Since three trimer patterns appear in each VO2 layer, 3n trimer ordered structures appear per n VO2 layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' The glass pattern simulations were created by calculating all of the PDF patterns produced by the 3n possible patterns and adding them together with a weight of 1/3n [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' As shown in Figure 2(c), the simulation data are in good agreement with the experimental PDF data in the range 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content='5 < r(˚A) < 40, indicating the appearance of a trimer glassy state in solution(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content='01).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' On the other Solution (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content='01) As-grown (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content='97) As-grown (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content='96)3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' (a) PDF data at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content='5 < r(˚A) < 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content='5 and 30 < r(˚A) < 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' (b) Distance between atoms in a VO2 plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' (c,d) G(r) pattern of (c) solution(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content='01) and (d) as-grown(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content='96) fitted by glass simulation data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' (e) Confidence R(r) of the refinement obtained in various r regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' (f) G(r) of solution(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content='01) and as-grown(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content='96) and their difference ∆G(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' hand, as shown in Figure 2(d), for the as-grown samples, a large residual appears above 15 ˚A, which corresponds to the thickness of three layers, and the residual expands as r increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' This result can be understood as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' First, the reduced G(r) at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content='5 < r(˚A) < 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content='0 contains mainly the component corresponding to the in-plane in- teratomic distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' The reduced G(r) at 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content='0 < r(˚A) < 15 contains information on the distances between the atoms in the nearest and next nearest VO2 layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' If there is a short-range order in the trimer arrangement along the stacking direction, large residuals are likely to appear at 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content='0 < r(˚A) < 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' However, as shown in Figure 1(b), the relationships between the trimer in a VO2 layer and the three trimer patterns in the nearest and next-nearest VO2 layers are all equivalent, resulting in the same PDF pattern regardless of which trimer pattern appears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' In other words, the PDF pattern at 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content='0 < r(˚A) < 15 is con- stant regardless of the presence or absence of short-range order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' When the correlation length of the short-range order is longer than 15 ˚A, large residuals appear in the region r(˚A) > 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' This occurs in the fitting of as-grown samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' In order to quantitatively investigate the agreement between the simulated data and the experimentally ob- tained PDF data, we defined the following evaluation function R(r), R(r) = ��r r′=r−5 {G(r′)exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' − s · G(r′)calc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' }2 �r r′=r−5 G(r′)2exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' (1) where G(r)exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' is the experimental value and G(r)calc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' is the simulated reduced PDF data of the trimer glass pat- tern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Details on how to obtain G(r)calc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' are described in Supplemental Information [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' s is a dimensionless scale factor to normalize G(r)exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' and G(r)calc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content='. The value of s that minimizes R(5) was defined as the sample- specific scale factor and used to calculate R(r) in various r ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Equation (1) is based on the so-called box- car refinement concept and is useful for estimating the correlation length of the short-range order in as-grown samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' The results are shown in Figure 2(e), where the R(r) values for the solution(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content='01) sample remain low in all r regions analyzed, while the R(r) values of the as-grown samples tend to increase uniformly at r(˚A) ≥ 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' In suf- ficiently large r(˚A) regions, well beyond the correlation length of the short-range order, there is no correlation between distant atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Therefore, in sufficiently large r(˚A) regions, the reduced PDF data for the solution and as-grown samples should again agree well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' This can be roughly inferred from the r dependence of the difference between the PDF data of the two samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' As shown in Figure 2(f), the residuals clearly decrease in the region above 100 ˚A, which roughly corresponds to the correla- tion length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' The cause of the difference between the solution and as-grown samples is not clear but is speculated as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' In the as-grown sample, the lattice energy degeneracy due to the numerous stacking patterns is thought to be lifted, and short-range order is realized in the trimer ar- rangement in the stacking direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Since there is no sig- nificant difference in the basic lattice structure between the solution and as-grown samples, the interlayer Li ion sites are likely responsible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Since the short-range order also appears in as-grown(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content='01), the lack of Li ions is not the origin of the short-range order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' A possible expla- nation is the disorder of the Li ion sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' As shown in Figure 3(a), almost all Li ions are ordered into octahe- dral sites between VO2 layers, but there are also tetrahe- dral sites where extra Li ions can enter as shown in Fig- ure 3(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' If the Li ions fully occupy the octahedral site, the Coulomb potential of the Li ions on the VO2 layer is uniform and the lattice energy degeneracy associated with trimerization is preserved, so strong frustration is expected, as shown in Figure 3(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' However, if deficien- cies or other disturbances exist at the octahedral site, the Intra-trimer Solutign(l1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content='01) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content='6A As-grown 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content='96) Inter-trime4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' (a) Octahedral site with majority Li ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' (b) Tetra- hedral site with excess Li ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' (c) Relationship between Li ion and three possible trimer configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Li ions are located between the upper and lower VO2 layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' (d) Relationship to the trimer when Li is deficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' (e,f) Relation between Li ions at the tetrahedral sites and (e) upper and (f) lower VO2 layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' random potential should lift the lattice energy degener- acy and produce a stable ordered structure, as shown in Figure 3(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Interestingly, the random insertion of Li ions into the tetrahedral site does not lift the lattice energy degeneracy because it gives equal random potentials for the three trimer patterns that appear in the neighboring VO2 layers as shown in Figures 3(e) and (f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Therefore, we speculate that the solution reaction with n-BuLi may have an annealing effect that encourages Li ions to move between the layers to fully occupy the low-potential oc- tahedral sites, in addition to the effect of adjusting the Li ion content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' How does the presence or absence of such short-range order affect electronic properties?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' To explore the effect of such short-range ordering on the physical properties, DSC measurements were per- formed, and as shown in Figures 4(a) and (b), the entropy change associated with the phase transition was always larger in the solution sample than in the as-grown sam- ple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' This is understood to be due to the optimization of the electronic state of V in the solution sample as a result of controlling the Li content with n-BuLi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' On the other hand, in contrast to the trend of the entropy change, the phase transition temperature of the solution sample is ∼ 15 K lower than that of the as-grown sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' This re- sult is clearly inconsistent with the entropy change data suggesting stabilization of the trimer structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' These results seem to indicate that the phase transi- tion temperature is suppressed in samples without the FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' (a) (upper) Entropy change measured by DSC on heating process, and (lower) the phase transition tempera- tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' The Li/V ratio is determined by ICP measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' (b) DSC data for as-grown and solution samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' short-range ordering of the trimer arrangement in the stacking direction compared to samples with short-range ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' This is reminiscent of frustration effects in spin systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' The present trimer frustration state in LiVO2 is a unique state formed by the coupling of electrons and lattice degrees of freedom, and is a consequence of pure lattice ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Nevertheless, it is similar to spin sys- tems in that the presence of frustration suppresses the phase transition, and weakening the frustration induces an ordered state and increases the transition tempera- ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' This seems to indicate that frustration effects sim- ilar to those in spin systems can be realized in lattice systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' In spin-frustrated systems, the strength of frustration is quantified by the absolute value of the ratio of the Weiss temperature to the N´eel temperature (frustration factor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' In the present trimer frustration, the spin gap estimated from NMR measurements may be an indica- tor of the strength of the frustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' From previous NMR measurements on LiVO2, the spin gap in the low- temperature phase is estimated to be ∆ ∼ 3400 K [35] (1600 K [36]), which is much larger than the phase tran- sition temperature of LiVO2, Tc ∼ 500 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' This seems to indicate that the trimer transition temperature in LiVO2 is strongly suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' One might attribute the energy differences to the pres- ence of local orbital degeneracy lifted state (ODL) that develops prior at high temperatures, as recent PDF stud- ies of the local structure have found in many systems that form orbital molecules at low temperatures [11, 13, 37– 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' This may be the case for LiVS2, an analog of LiVO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' LiVS2 has a different stacking structure than LiVO2 and no trimer frustration [12], but a trimer transition occurs at 314 K with a gap of ∆ ∼ 1900 K [10, 43, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Above the phase transition temperature, PDF analysis shows that a zigzag chain-like short-range order develops, sug- gesting that orbital degeneracy is already locally lifted at high temperatures [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' However, this is not the case for LiVO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Our previous PDF studies on LiVO2 have (a) (b) (c) Lower (e) Excess Li layer equivalent Upper layer Possible trimers : Possible trimers : degenerate degenerate (d) Li deficiency (f) Excess Li Possible trimers : Possible trimers : non-degenerate degenerateSolution As-grown(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content='96) Solution As-grown (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content='01)5 shown that no such short-range order develops above the phase transition temperature in LiVO2 [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' The above indicates that LiVO2 and LiVS2 are not similar and each has unique physics for trimer formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' It should be noted that we were able to address such a physics of trimer frustration because of our success in modeling the trimer glassy state of LiVO2 and identi- fying its structure by PDF analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' The existence of vanadium trimer formation in LiVO2 was pointed out more than half a century ago based on the lattice sym- metry of the low-temperature phase [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Subsequent studies have confirmed the V-V distance splitting asso- ciated with trimer formation by EXAFS [46] and PDF analysis [47], electron diffraction analysis [14], and the NMR measurement using a single crystalline sample [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' All of these results support the in-plane appearance of the trimer, but the identification of the crystal structure containing the trimer had not been successful for more than half a century.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' This is because the trimer disor- der in the stacking direction, intrinsic to LiVO2, has not been properly modeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Coupled with glassy state mod- eling, PDF analysis was essential in the present results to reveal that LiVO2 is the playground where the new physics of trimer frustration emerges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' This achievement can never be revealed by conventional average structure analysis, and may point to a new direction in structural analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Finally, we point out the importance of this trimer frustration in terms of applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' The latent heat cal- culated from the entropy change of LiVO2 (∆H ∼ 326 Jcc−1) is equivalent to that of H2O (∆H ∼ 306 Jcc−1) and is promising as a phase change material (PCM) prod- uct [7, 14, 20–22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' If the phase transition temperature can be manipulated by controlling trimer frustration, it could be a PCM material that can be used at various temper- atures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Such studies are beyond the scope of this study, but they clearly demonstrate the importance of both the fundamental and applied aspects of trimer frustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' ACKNOWLEDGMENTS All authors thank Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Ohara and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Hashimoto for fruitful discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' The work leading to these results has received funding from the Grant in Aid for Scientific Research (Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' JP17K17793, JP20H02604, JP21K18599, JP21J21236).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' This work was carried out under the Visit- ing Researcher’s Program of the Institute for Solid State Physics, the University of Tokyo, and the Collaborative Research Projects of Laboratory for Materials and Struc- tures, Institute of Innovative Research, Tokyo Institute of Technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Hoshikawa, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Ishigaki, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Kobayashi, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Ohta, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Sawa, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' B 98, 081104(R) (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' [8] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Okamoto, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Amano, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Katayama, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Sawa, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Niki, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Mitoka, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Harima, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Hasegawa, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Ogita, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Tanaka, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Takigawa, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Yokoyama, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Takehana, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Imanaka, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Nakamura, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Kishida, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Takenaka, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' 11, 3144 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' [9] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Miura, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Yasui, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Sato, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Igawa, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Kakurai, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Jpn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' 76, 033705 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' [10] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Katayama, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Uchida, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Hashizume, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Niitaka, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Matsuno, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Matsumura, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Nishihata, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Mizuki, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Takeshita, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Gauzzi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Nohara, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Takagi, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' 103, 146405 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' [11] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Katayama, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Kojima, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Yamaguchi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Hattori, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Tamura, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Ohara, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Kobayashi, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Sugimoto, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Ohta, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Saitoh, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Sawa, npj Quantum Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' 6, 16 (2021).' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' B 100, 235120 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' [13] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Browne, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Kimber, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Attfield, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' 1, 052003(R) (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' [14] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Tian, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Chisholm, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Khalifah, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Jin, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Sales, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Nagler, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Mandrus, Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Bull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' 39, 1319 (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Jpn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' 60, 2550 (1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' [37] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Kimber, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Mazin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Shen, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Jeschke, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfWgcY/content/2301.03833v1.pdf'} +page_content=' Streltsov, D.' 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a/RNFKT4oBgHgl3EQfiS5U/content/tmp_files/2301.11841v1.pdf.txt b/RNFKT4oBgHgl3EQfiS5U/content/tmp_files/2301.11841v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..b23761425456c9a8f6e939ae36e5da8db0dbca97 --- /dev/null +++ b/RNFKT4oBgHgl3EQfiS5U/content/tmp_files/2301.11841v1.pdf.txt @@ -0,0 +1,984 @@ +PhysGraph: Physics-Based Integration Using Graph Neural Networks +OSHRI HALIMI, Technion – Israel Institute of Technology and Meta Reality Labs Research +EGOR LARIONOV, Meta Reality Labs Research +ZOHAR BARZELAY, Meta Reality Labs Research +PHILIPP HERHOLZ, Meta Reality Labs Research +TUUR STUYCK, Meta Reality Labs Research +PhysGraph +Patch-Based +Training Data +Material +Properties +Collision +Geometry +Force +Module +Fig. 1. PhysGraph leverages a recurrent graph neural network in order to minimize user-provided energy potentials in a topology and force independent way. +We demonstrate our method on enhancing coarse resolution cloth geometry with physics-based high resolution detail. The figure shows the coarse geometry +on the left and the enhanced output on the right. Our method is capable of handling body and self collisions as well as the elastic potentials of cloth. By +explicitly separating the force modeling and the integration process, we obtain high generalization power. The method is trained on local patches with only +forces resulting from the garment elastic potential. We show several examples to demonstrate that our method allows to modify material properties and +collision geometry during inference, all leveraging a single model trained on only a limited number of potentials. +Physics-based simulation of mesh based domains remains a challenging task. +State-of-the-art techniques can produce realistic results but require expert +knowledge. A major bottleneck in many approaches is the step of integrating +a potential energy in order to compute velocities or displacements. Recently, +learning based method for physics-based simulation have sparked interest +with graph based approaches being a promising research direction. One of +the challenges for these methods is to generate models that are mesh inde- +pendent and generalize to different material properties. Moreover, the model +should also be able to react to unforeseen external forces like ubiquitous +collisions. Our contribution is based on a simple observation: evaluating +forces is computationally relatively cheap for traditional simulation meth- +ods and can be computed in parallel in contrast to their integration. If we +learn how a system reacts to forces in general, irrespective of their origin, +we can learn an integrator that can predict state changes due to the total +forces with high generalization power. We effectively factor out the physical +model behind resulting forces by relying on an opaque force module. We +demonstrate that this idea leads to a learnable module that can be trained +on basic internal forces of small mesh patches and generalizes to different +mesh typologies, resolutions, material parameters and unseen forces like +Authors’ addresses: Oshri Halimi, Technion – Israel Institute of Technology and Meta +Reality Labs Research; Egor Larionov, Meta Reality Labs Research; Zohar Barzelay, +Meta Reality Labs Research; Philipp Herholz, Meta Reality Labs Research; Tuur Stuyck, +Meta Reality Labs Research. +collisions at inference time. Our proposed paradigm is general and can be +used to model a variety of physical phenomena. We focus our exposition +on the detail enhancement of coarse clothing geometry which has many +applications including computer games, virtual reality and virtual try-on. +CCS Concepts: • Computing methodologies → Physical simulation; +Neural networks. +Additional Key Words and Phrases: Cloth simulation, neural network simu- +lation, graph neural networks +1 +INTRODUCTION +Physics-based simulation has made significant advances in the last +decades and it is now possible to re-create highly realistic physical +phenomena using computer models. Current simulation techniques +provide us with a method that generalizes to novel settings. How- +ever, these simulations are difficult to compute and can require +extensive manual interventions in order to obtain the desired re- +sults. On the other hand, with the proliferation of machine learning +and data-driven techniques, there has been an increased interest +in recreating physical phenomena using neural networks [Allen +arXiv:2301.11841v1 [cs.GR] 27 Jan 2023 + +Patch-Based + Training Data +PhysGraph +Force +Module +Material +Collision +Properties +Geometry2 +• +Halimi, O. et al +et al. 2022a,b; Pfaff et al. 2020; Sanchez-Gonzalez et al. 2020]. How- +ever, it is hard to generalize these neural models to account for all +the variety that can occur as it might not be fully covered in the +dataset or overfitting might occur. This motivates us to leverage +the strengths of both approaches to design a neural model that uses +physics-based information to produce a generalizable and widely +applicable method. In this paper, we present a solution that improves +on research in this direction. +We focus on the specific application of modeling garment defor- +mations. The ability to model garments is crucial for telepresence, +games, virtual try-on and other applications. Cloth state prediction +using data-driven approaches is a long standing and notoriously +difficult problem, due to the high variability in garment shape, de- +formation, and discontinuities caused by frequent collisions against +the body and within the cloth itself. Additionally, it remains cum- +bersome to obtain the required training data [Halimi et al. 2022]. +Despite this, many advances have been made [Bertiche et al. 2020, +2022; Santesteban et al. 2019, 2022a, 2021], which allow us to produce +garment configurations based on the skeleton pose and body shape +as input. Unfortunately these methods rely on networks trained +on specific garments, and thus do not generalize well. Oftentimes, +they are limited to modelling tight fitting clothing as it leverages a +skinning model with respect to the body skeleton. To address these +limitations, others have presented approaches for the animation +of loose clothing with neural networks [Zhang et al. 2021b], or by +leveraging real-time physics-based cloth simulation [Stuyck 2018] +with a learned neural rendering pass to obtain realistic looking +clothing that generalizes to new motion and body shapes [Xiang +et al. 2022]. Despite recent progress, many limitations still remain. +Methods are often limited to fixed underlying body skeletons, fixed +topologies, and material properties and do not handle collisions +gracefully. +In an effort to obtain better generalization, these observations +motivate us to explore the potential of combining machine learning +and physics-based techniques further and exploiting knowledge +about the physical system directly, instead of learning the relation- +ship implicitly. The core idea of our proposed method is to factor +out the force specific components from the integration process. +Force computations are computationally relatively more efficient +for physics-based simulation methods and can be computed in par- +allel in contrast to their integration. Leveraging this design, we can +then learn the integration procedure. This approach fundamentally +prevents overfitting and has the ability to generalize to novel forces. +We implement this design by using a message passing graph neu- +ral network that can integrate physics-based forces provided by +an external force module on arbitrary topologies. The approach is +agnostic to the specific material models and we demonstrate that it +generalizes to unseen forces during inference. Thanks to the design +of the method, we are able to model different garment categories, +both tight and loose, with self and body collisions in a topology +independent way that does not require an underlying skinned body +mesh. +In summary, our main contributions are: +• A novel neural architecture with split responsibilities for force +modeling and force integration, which allows for integrating +physics-based forces in a topology invariant fashion resulting +in a method that generalizes to unseen settings. +• The method is able to resolve collisions with arbitrary ge- +ometries using either triangle mesh or signed distance field +(SDF) representations and, for the first time using a neural +approach, is able to resolve self collisions between multiple +garments. +• The resulting simulation pipeline retains the controllable, +physics-based, well understood material models, providing +the user with meaningful and intuitive control parameters +(i.e. material properties). +• The integrator module is trained in an unsupervised way, +which allows for efficient learning without the need for ex- +pensive and hard to obtain training data. +2 +RELATED WORK +We provide an overview of relevant work related to the modeling of +physical phenomena using neural networks, cloth detail enhance- +ment and neural methods for generating garment deformations. +2.1 +Neural Networks for Modeling Physical Phenomena +Neural networks have been successfully used to model granular +material and fluids [Li et al. 2019; Scarselli et al. 2008] using particle +based approaches and graph neural networks [Battaglia et al. 2018]. +Several other methods focus on the modeling of fluid dynamics +[Thuerey et al. 2020; Um et al. 2018]. Pfaff et al. [2020] introduced a +mesh-based method for the simulation of several phenomena using +graph neural networks with several follow up works [Fortunato +et al. 2022; Sanchez-Gonzalez et al. 2020]. +2.2 +Cloth Geometry Enhancement +Enhancing details on cloth geometry is a long-standing research +problem with early work by Cutler et al. [2005] who presented a pro- +cedural wrinkling model capable of adding art-directed wrinkling in +a production setting. Bergou et al. [2007] proposed a method where +constrained Lagrangian mechanics are used to add physically-based +details to animated thin shells. Müller and Chentanez [2010] pro- +posed a simple and fast method to add wrinkles to dynamic meshes +by attaching a higher resolution wrinkle mesh to the coarse base +mesh. Kavan et al. [2011] presented a method where enhancement +is achieved by learning linear upsampling operators for physically- +based cloth simulations. Rémillard and Kry [2013] proposed a method +to add wrinkling to composite objects consisting of a soft interior +and harder skin. Since then, many follow-up works have been pre- +sented. Rohmer et al. [2010] leverage the stretch tensor computed on +the coarse animation to add temporally coherent wrinkles. Similarly, +Gillette et al. [2015] present a method to add dynamic wrinkling to +coarse animated cloth using a two stage stretch tensor estimation +process. Cloth details can be enhanced using tension field theory +to model coarse geometry after which the amplitude and phase +of the fine wrinkling is added [Chen et al. 2021]. Recently, Wang +[2021] explored specialized techniques for cloth simulation on the +GPU using grid-aligned meshes to gain an edge at reconstructing +fine wrinkles at submillimeter levels. Other methods rely on neural +networks operating on 3D geometry [Liu et al. 2020; Zhang et al. + +PhysGraph: Physics-Based Integration Using Graph Neural Networks +• +3 +2021a]. Lahner et al. [2018] present a data-driven approach to en- +hance detail encoded in a normal map texture. An image-to-image +neural network is trained to enhance detail in image space. +2.3 +Neural Garment Deformations +Learning-based methods aim at predicting a garment’s draping over +a given body mesh. Several methods rely on the SMPL [Loper et al. +2015] parametric body shape model, along with its rigging and skin- +ning functions. In practice, this means that such methods are able +to cast the ML-draping problem as that of predicting corrective +garment deformations. These per-vertex deformations are added to +the garment’s rest-pose vertex position, and are then skinned. Initial +methods train a garment prediction model by regressing ground- +truth vertex positions, calculated using high-fidelity physics simula- +tion data [Patel et al. 2020]; while relying on fixed skinning weights, +transferred from the SMPL weights. The fixed skinning weights limit +the garment vertices to move based on their rest pose location. This +assumption is alleviated by computing post-deformation skinning +weights, utilizing a prediction network [Santesteban et al. 2021]. +Per-pose predictions do not take into account the dynamic nature of +garment deformations. Therefore, Santesteban et al. [2019] utilize a +recurrent model whose predictions depend on past poses too. Such +models require per-garment training, necessitating multiple ground- +truth simulations. Bertiche et al. [2021] alleviates this requirement, +by training in an unsupervised setup. The loss is cast as a set of +physical potentials to minimize: stretching, bending, gravity, and +body-cloth collision. These losses are differentiable, and therefore +can be back-propagated to optimize the network’s weights. San- +testeban et al. [2022a] and Bertiche et al. [2022] both add an inertia +loss to address temporal consistency. The input of Santesteban et al. +[2022a] further takes not only the parameterized body-pose, but +also its shape. However, the network is still specialized per-garment. +De Luigi et al. [2022] alleviates this requirement by predicting a +latent code for any given garment. It thus generalizes over body +shape, body pose, and garment type. Reliance on body-based skin- +ning amounts to limiting the garment to move in correspondence to +the underlying body. However, for loose-garment this is not always +the case. To tackle this, Pan et al. [2022] creates for a garment a +new set of joints and corresponding per-vertex skinning weights, +based on ground-truth simulations on a variety of motions. Drap- +ing prediction then amounts to predicting the joints translation +and rotation parameters. The above methods rely on a paramet- +ric body representation. This limits their applicability to draping +multiple garments layers or stylistic (non-human) avatars. Zhang +et al. [2022] addresses this by representing the underlying body as +a set of sampled points, while Li et al. [2022] separately encodes the +input body and garment meshes using graph-convolution networks. +This separate encoding does not take into account body-garment +interactions. To alleviate this shortcoming, Grigorev et al. [2022] +adds body-garment graph edges, and uses hierarchical message- +passing. Body-garment collisions can also be solved by learning a +collision-free generative deformation space [Santesteban et al. 2021]. +ULNeF [Santesteban et al. 2022b] generalizes to multiple garment by +predicting corrective terms to the garments’ implicit representation; +but is limited to running on human shapes in canonical pose. +The above ML-based garment draping methods achieve impres- +sive results. However, they are all limited in generalizing to arbitrary- +posed bodies with arbitrary layers of interwoven garments and +clothing items (such as a tucked-in shirt, layered with a suit, tie +and a pocket handkerchief as in Fig. 1). Our method allows, for the +first time, to achieve realistic draping of complex topologies, in a +self-supervised manner (see full comparison in Table 2). +3 +METHOD +To evaluate, PhysGraph, we focus on quasi-static simulation of cloth +using elastic energy and contact penalty potential minimization. +Given an energy potential, a classical method would iterate over con- +figurations following the negative energy gradients (forces) until it +finds the optimal point. Here, we decouple the force module responsi- +ble for directly differentiating energy potentials, and the integration +module, which integrates the resulting forces into a displacement +vector for all vertex positions. Given these complementary respon- +sibilities of the modules, our method provides great flexibility to the +system being modeled. The method is agnostic to both the type of +physical forces modeled by the force module and, the mesh connec- +tivity at inference time, allowing it to generalize to different types +of potentials even after the network is trained. The central building +block of our architecture is a recurrent mesh graph network. We +initialize the integration module with an upsampled version of the +physical simulation given a coarse mesh. This way, the large scale +dynamic behavior of the system is captured in the coarse mesh while +our architecture creates finer scale details that are governed by the +static equilibrium of forces. These coarse meshes can be obtained +using classical mesh based simulation or other methods like artist +models and linear blend skinning output. +Our key contribution is an algorithm that performs several itera- +tions of force computation and integration to find an approximate +minimizer of the potential energy provided by the force module +during inference. To this end, the integrator leverages a graph net- +work architecture [Pfaff et al. 2020] which we use in a recurrent +fashion. The subsequent sections will introduce the force module, +the integration module and the graph network architecture. +3.1 +Force module +We credit the generalizability of our method to the separation of +the force formulation and integration. The force generation module +outputs forces based on user-specified potentials given the current +nodal configuration where forces are accumulated at the nodal level, +enabling parallelization of the force computation. The conservative +force potential Φ, is responsible for the forces F = −∇XΦ acting +on the system. Different potentials can be used to model different +physical phenomena. +3.2 +Mesh-Based Graph Networks +The simulation mesh can be interpreted as a graph 𝐺 = (𝑉, 𝐸) onto +which we encode vertex and edge features consisting of 128 values +each. The graph network operates in three phases: Encode, Process +and Decode. +Encode. Vectors of per edge ˜𝑒𝑖 and per vertex features ˜𝑣𝑖 form the +input to the graph network. By applying two multilayer perceptrons + +4 +• +Halimi, O. et al +MLP +MLP +MLP +, +MLP +Decode +Encode +Process Edge +Process Vertex +MLP +Force Module +Integration Module ++ +Fig. 2. Our approach consists of three components. The force module (Sec- +tion 3.1) which evaluates per-node forces for a mesh configuration. The +integration module (Section 3.3) constructs initial feature vectors ˜𝑣𝑖 and +˜𝑒𝑖 based on force and configuration information from previous iterations, +passes these quantities to a graph neural network (Section 3.2) and re- +trieves new displacements 𝐷𝑘. The graph network iterates between an edge +processing step that distributes vertex information to edges and an edge +processing step which distributes edge information to adjacent vertices. +These distribution steps are iterated 𝑀 times. The integration module iter- +ates 𝐾 times and finally returns an estimate for the quasi-static equilibrium +state X𝐾 . +(MLP), one to each vertex feature and one to each edge feature, we +obtain initial features 𝑣0 +𝑖 and 𝑒0 +𝑖 . +Process. We use a fixed amount of 𝑀 = 10 message passing itera- +tions. Each iteration 𝑗 consists of two steps computing new edge +and vertex features. First, 𝑒 𝑗 +𝑖 is computed by passing information +from vertices to edges. In the second step, information is passed +from edges to adjacent vertices to build 𝑣 𝑗 +𝑖 , see Figure 2. Each step +uses a single MLP with weights that are shared for all iterations and +between all vertices and edges, respectively. +Decode. A final MLP is used to decode the final vertex features +𝑣𝑘 into the final displacement vectors. +The set of learnable parameters 𝜃 are the weights of the five MLPs +used during the three phases. +3.3 +Integration Module +The physical configuration of the mesh is given by its embedding +which can be represented as a matrix of stacked position vectors +X ∈ R𝑛×3 where 𝑛 is the number of vertices in the mesh and X𝑖 +represents the position of the 𝑖−th vertex. We assume that the mesh +has a rest configuration X ∈ R𝑛×3, which by definition, experi- +ences no internal forces. The goal of the integration module is +to find an approximation to the static equilibrium configuration +X∗ = argminX Φ(X). Starting with initial vertex positions X𝑘 with +𝑘 = 0, we produce new positions X𝑘+1 by using the pre-trained +graph network. Throughout the simulation, the graph network has +access to the mesh connectivity corresponding to the set of graph +edges 𝐸. For each configuration, we can query the force module to +obtain corresponding forces F𝑘 = −∇XΦ(X𝑘). The inputs to the +graph network are per vertex and edge features. For each vertex +𝑖 we construct the feature vector ˜𝑣𝑖 by concatenating first order +differences of the last 𝐻 configurations and the corresponding force +vectors +˜𝑣𝑖 = +� +X𝑘−1 +𝑖 +− X𝑘 +𝑖 , · · · , X𝑘−𝐻 +𝑖 +− X𝑘−𝐻+1 +𝑖 +, F𝑘 +𝑖 , · · · , F𝑘−𝐻+1 +𝑖 +� +. +(1) +The edge features ˜𝑒𝑖 for the edge connecting vertex 𝑟 and 𝑠 are +comprised of position differences for the current configuration and +the rest state as well as their lengths +˜𝑒𝑖 = +� +X𝑘𝑟 − X𝑘𝑠 , ∥X𝑘𝑟 − X𝑘𝑠 ∥, X𝑟 − X𝑠, ∥X𝑟 − X𝑠 ∥ +� +, +(2) +where ∥ · ∥ is the Euclidean norm. The network outputs displace- +ments D𝑘 that define the next state via X𝑘+1 = X𝑘 + D𝑘. After +𝐾 = 5 iterations we obtain an approximation of the equilibrium +state X𝐾. +3.4 +Training phase +We train the integration module with a dataset of small physical +systems. These act as an input to the recurrent model, consisting +of 𝐾 recurrent force calculation and integration blocks sharing the +same network parameters of the trainable integration module. The +training is supervised by requiring the minimization of the potential, +summed over all the intermediate states. +4 +GARMENT DETAIL ENHANCEMENT +Here, we apply PhysGraph to the cloth upsampling problem. A +given coarse resolution cloth mesh is first subdivided. Then nodal +forces are computed using the force module and integrated into +nodal displacements by the integration module. This is repeated +iteratively over multiple steps, which allows local forces to prop- +agate throughout the rest of the mesh. We show that the method +remains effective regardless of whether the coarse mesh is obtained +through low resolution cloth simulation or other methods such as +artist animation or through procedural or skinning approaches. In +this section, we define the potentials used to model fabric elasticity +and contact. +4.1 +Garment Potentials +To demonstrate cloth modelling, we use springs to model stretching, +dihedral angle penalty to model bending and penetration penalty to +model contact, although the force +module can be any material model +that produces forces. The potentials +are weighted based on element area, +which we define to be the area of a +barycentric subdivision (light gray areas in the inset). This way the + +PhysGraph: Physics-Based Integration Using Graph Neural Networks +• +5 +total area of all edge and vertex weights, respectively, sum up to the +total surface area 𝐴. +Stretching is modelled using edge aligned springs with net elas- +tic potential +Φ𝑠 = 𝑘𝑠 +2𝐴 +∑︁ +𝑒 ∈𝐸 +𝑎𝑒 (𝑙(𝑒) − 𝑙0(𝑒))2, +(3) +where 𝑘𝑠 is the spring stiffness, 𝑙(𝑒) = ∥X𝑖 − X𝑗 ∥ is the edge length +for an edge 𝑒 = (𝑖, 𝑗) and 𝑙0(𝑒) = ∥X𝑖 − X𝑗 ∥ is its rest-length. +The bending potential is defined by the dihedral angles 𝜃𝑑 formed +between the normal vectors to the triangles in each dihedral element +Φ𝑏 = 𝑘𝑏 +2𝐴 +∑︁ +𝑒 ∈𝐷 +𝑎𝑒𝜃2 +𝑑 +(4) +where 𝐷 is the set of interior edges corresponding to dihedral ele- +ments. The gravitational potential is defined by +Φ𝑔 = − 𝑔 +𝐴 +∑︁ +𝑣∈𝑉 +𝑎𝑣𝑚𝑣𝑧𝑣, +where 𝑚𝑣 = 𝜌𝑎𝑣, +(5) +and 𝑧𝑣 is the coordinate along the gravity axis and 𝜌 the mass den- +sity. The external contact potential is modeled using the signed- +distance-function 𝑆𝐷𝐹 (·), which measures the signed distance (neg- +ative inside, positive outside) to the surface of a set of colliders +in the system. The contact potential 𝜙𝑣 = − min(𝑆𝐷𝐹 (X𝑣), 0) is +accumulated over all potentially violating vertices 𝑣 with +Φ𝑒𝑐 = 𝑘𝑒𝑐 +𝐴𝑒𝑐 +∑︁ +𝑣∈𝑉 +𝑎𝑣𝜙𝑣, +where 𝐴𝑒𝑐 = +∑︁ +𝑣∈𝑉 +𝑎𝑣(1 − 𝛿0(𝜙𝑣)), +(6) +and 𝑘𝑒𝑐 is the external contact penalty stiffness. The zero-set in- +dicator function 𝛿0 evaluates to 1 for 0 and to 0 otherwise. Fi- +nally, the self collision potential is modeled by radial compres- +sion springs with rest-length 𝑅 around each vertex. A compres- +sion spring between vertices 𝑢, 𝑣 ∈ 𝑉 , modelled by the potential +𝜓𝑢,𝑣 = max(𝑅 − ∥X𝑢 − X𝑣∥, 0), exerts a force in the outward radial +direction when compressed, which happens when distinct vertices +become closer than 𝑅 apart. The total energy is defined by +Φ𝑠𝑐 = 𝑘𝑠𝑐 +𝐴𝑠𝑐 +∑︁ +𝑢,𝑣∈𝑉 +𝑢∉N𝑑 (𝑣) +(𝑎𝑣 + 𝑎𝑢)𝜓2 +𝑢,𝑣, +𝐴𝑠𝑐 = +∑︁ +𝑢,𝑣∈𝑉 +𝑢∉N𝑑 (𝑣) +(𝑎𝑣 + 𝑎𝑢)(1 − 𝛿0(𝜓𝑢,𝑣)) +(7) +where 𝑘𝑠𝑐 is the self-collision penalty stiffness, and we consider only +interactions of vertices which are not neighbors on the mesh within +some d-ring of 𝑣 denoted N𝑑 (𝑣). +4.2 +Training +The method is trained using patches sampled from a dynamically +simulated t-shirt on a moving human body, example patches are +shown in Figure 1. The patches are subdivided using a self-similarity +subdivision scheme, increasing the mesh resolution by a factor of +16 and, by linearly interpolating the coordinates. We use 𝐾 = 5 +recurrent blocks in our experiments. We stress that we only include +forces resulting from the stretch and bending potentials Φ = Φ𝑠 +Φ𝑏 +at training time. To account for the fact that the patch is a sub- +system of the larger full-cloth system and prevent the flattening of +the patches in the absence of the rest of the cloth-system, we used +fixed boundary conditions while training. +Fig. 3. From left to right: coarsely simulated input, inferring with stretch +and bending forces only, stretch and bending and body collisions and finally, +including all forces on the right. The model was trained with elastic energy +forces only and generalizes to include collisions. +Fig. 4. In contrast +to methods that +rely on skinning +based techniques, +our method nat- +urally +handles +loose clothing. Left +shows +the +low +resolution +dress +and right shows +our +enhanced +result. +5 +RESULTS +Due to the design of our method, we are able to train our inte- +grator network using only the internal cloth potentials and their +resulting forces. At inference time, the trained network is capable +of ingesting numerous forces from a variety of sources to produce +plausible results. We showcase this by applying our trained integra- +tion module to a variety of novel forces. We show results with forces +such as gravity, body collision and self collision. We demonstrate +the effectiveness of PhysGraph for several variations of the cloth +enhancement application with complete multi-layered garment ex- +amples shown in Figure 10. All results were generated using a single +trained network which was only exposed to elastic forces at training +time. Note that the method is not limited to these specific examples. +5.1 +Coarse Simulation Enhancement +We show results for the cloth geometry enhancement in Figure 3 +which shows the resulting geometries after integrating different +force potentials ranging from simply including elastic energy poten- +tials, which were included at training time, to a full model with body +and self-collisions, which includes several forces not seen during +training. Our method is garment independent and works for loose +clothing as can be seen in Figure 4. +5.2 +Linear Blend Skinning Enhancement +To demonstrate generalization to the model input, we show that the +coarse geometry does not need to be obtained from a simulation +and lower cost methods can be used. We showcase the efficacy of + +6 +• +Halimi, O. et al +Fig. 5. We demonstrate that our method is capable of enhancing geometry +obtained from linear blend skinning (left), the middle shows the enhanced +mesh without self collisions and the rightmost shows the predicted gar- +ment incorporating all forces. Note the realistic wrinkling added by our +method while preserving the overall shape and remaining collision free. The +enhancement model includes both body and self collision forces. Note that +for the jacket, the method is even capable of removing the skinning artifacts +near the armpit. +Fig. 6. We demonstrate the ability of enhancing garment geometry using +different materials at inference time. This example shows the apparent +visual difference of varying bend and stretch stiffnesses allowing the user to +generate a potentially expensive coarse sequence once and adjust materials +afterwards. Our method produces plausible results where higher bending +stiffness correctly corresponds to bigger folds. +the method on garments posed using linear blend skinning which +is known to have several issues which distorts the mesh in non- +physical ways. Nevertheless, Figure 5 shows that our method is +capable of producing visually pleasing high resolution meshes where +self-collisions are resolved. +5.3 +Material Generalization +Due to the force module, we are able to modify the material proper- +ties at inference time. Figure 6 shows different material settings for +a pair of pants which are all generated from the same coarse input +geometry. +5.4 +Collision Geometry Generalization +Our method generalizes to different collision geometries as shown in +Figure 7. We show rich cloth interaction with a rigid block, pushing +the cloth through the center of a torus. These colliding geometries +are completely novel with respect to those at training time, yet, the +method produces correct results. +Fig. 7. We show that our method generalizes to different collision geometries +during inference. From left to right, we show the original low resolution +input, a mesh obtained using Loop subdivision [1987], our method, and a +high resolution simulation. Note how our method adds detail compared +to a subdivision which simply smooths. Our method maintains the overall +shape of the low resolution but adds detail, preserving artistic intent. The +high resolution simulation is very distinct from the low resolution making +it difficult to obtain desired results when iterating on lower resolutions. +0 +10 +20 +30 +40 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +iteration +potential +adam 10−2 +adam 10−3 +adam 10−4 +gd 10−1 +gd 100 +gd 101 +PhysGraph +Fig. 8. We demonstrate that our method converges faster and to a lower po- +tential than several gradient-descent optimizers, with varying base learning +rates. For each baseline optimizer, we show increased learning rates until +they divergence. +5.5 +Self Collisions +Interactions of fabric with itself are ubiquitous for cloth simulations +as garments are often layered and display complex interactions. +Therefore, it is essential to model them appropriately. We demon- +strate the importance of including self collisions in Figures 1, 3 and 10, +where we show that our model is capable of including these forces, +providing clean, intersection free meshes as shown on the right. +5.6 +Convergence Analysis +We highlight the effectiveness of PhysGraph by comparing conver- +gence with respect to two baseline optimizers: Adam and gradient +decent. We showcase various learning rates in Figure 8 for which +they still converge. Note how our learned integrator is the most +effective. +5.7 +Comparisons To Related Work +We compare PhysGraph (ours) to two recent ML-approaches for gar- +ment draping. Both SSCH [Santesteban et al. 2021] and SNUG [San- +testeban et al. 2022a] are skinning-based approaches, trained with +ground-truth simulations and self-supervision, respectively. We use + +kb = 1 +kb = 10 +kb = 100 +kb = 100 +ks = 1e4 +ks = 1e4 +ks = 1e3 +ks = 1e4PhysGraph: Physics-Based Integration Using Graph Neural Networks +• +7 +their publicly released t-shirt models for comparison, and emphasize +that both models were trained for this specific garment. In compar- +ison, our method has not seen this garment during training. Our +method takes the skinned mesh as input, and predicts a refined +draping. Figure 9 shows a qualitative comparison and Table 1 pro- +vides quantitative comparison of the system potentials for each +method. While both SNUG and SSCH are limited to a fixed topology +and resolution since both models are specialized for this specific +garment, we are still able to generate detail at several resolutions +without having trained on this topology. Furthermore, our method +provides qualitatively and quantitatively better results with a more +general method which has not been optimized for this particular +setting. Table 2 provides a functional comparison which additionally +includes ULNeF [Santesteban et al. 2022b], Hood [Grigorev et al. +2022] and MeshGraphNet (MGN) [Pfaff et al. 2020]. +SNUG +SSCH +PhysGraph +Low Resolution +PhysGraph +High Resolution +Fig. 9. Comparison to skinning-based ML-draping methods of three differ- +ent frames of the CMU-07-02-poses sequence from Mahmood et al. [2019]. +From left to right: SNUG [Santesteban et al. 2022a]; SSCH [Santesteban et al. +2021]; PhysGraph low resolution prediction; PhysGraph high resolution +refinement. Note how SSCH and SNUG show similar wrinkling regardless +of the pose, whereas PhysGraph is capable of adding fine detail in a more +physically plausible way at several resolutions where detail varies with pose. +Also note that SNUG results in intersections with the body as seen in the +bottom row. +6 +DISCUSSION, LIMITATIONS, AND FUTURE WORK +We propose a novel method for the integration of physics-based +forces using a graph neural network. Our method demonstrates +excellent generalization capabilities and we show a successful appli- +cation to the detail enhancement of coarse cloth geometry for both +tight and loose clothing. Our method is capable of modeling gar- +ment interactions with itself and the body and other collision objects +and we are the first to support collisions with multiple garments +simultaneously using their geometry directly without needing an +SDF or other representation which introduce several limitations for +Total Potential [erg] ↓ +Body Collision(%) ↓ +Skinning +243.2094 +4.5088 +SSCH +106.2101 +0.8257 +SNUG +91.3766 +0.7818 +Ours +74.4635 +0.4813 +Table 1. Quantitative comparison with state-of-the-art methods. ↓ means a +lower value is better. We report the potential of the cloth in [erg] units, using +a mass-spring model under gravity, using 𝑘𝑠 = 1𝑒4 erg/cm2, 𝑘𝑏 = 10 erg, +and 𝜌 = 0.0187 gr/cm2. To obtain the rest edge lengths, we use SNUG’s +rest mesh. The collisions with the body are reported as the percentage of +cloth vertices admitting negative values when used as a query to the body +SDF, which is defined with respect to the body with a collision margin of 2 +mm, similar to SNUG. All the compared potentials are calculated over the +same mesh topology taken from SNUG’s shirt. +Topology +Invariant +Pose +Invariant +Force +Agnostic +Body- +Garment +Collisions +Garment +self +Collisions +Multi +Garment +Collisions +Unsupervised +SSCH +✗ +✓ +✗ +✓ +✗ +✗ +✗ +SNUG +✗ +✓ +✗ +✗ +✗ +✗ +✓ +ULNeF +✓ +✗ +✗ +✗ +✓ +✓ +✗ +MGN +✓ +✓ +✗ +✓ +✓ +✗ +✗ +Hood +✓ +✓ +✗ +✓ +✓ +✗ +✓ +PhysGraph +✓ +✓ +✓ +✓ +✓ +✓ +✓ +Table 2. Summary of Related Work. Our work achieves all desirable features. +modeling layered clothing. Our implementation is not optimized +to enhance cloth geometry at real-time rates. However, currently +the bulk of the computation is within the force module, which is +implemented in Python. We believe that given the engineering re- +sources, the method has the potential to run at interactive rates due +to the parallel nature of the force module. Additionally, there are +active research efforts on improving message-passing architectures +efficiency [Rahman et al. 2021; Xie et al. 2022] which have already +demonstrated the ability to accelerate the computation by two or- +ders of magnitudes. Like most other methods, our current model is +unable to resolve pre-existing self-intersections. However, future +work could include untangling forces [Baraff et al. 2003] as part of +the force module. +In Figure 11, we show how our method performs on a simu- +lated t-shirt sequence, where our generated detail is already mostly +temporally coherent with the exception of areas with clustered col- +lisions. This is encouraging given that the method operates per +frame independently without exploiting temporal information. 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Graph. 41, 6, Article +222 (nov 2022), 15 pages. https://doi.org/10.1145/3550454.3555456 +Zhiqiang Xie, Minjie Wang, Zihao Ye, Zheng Zhang, and Rui Fan. 2022. Graphiler: Opti- +mizing Graph Neural Networks with Message Passing Data Flow Graph. Proceedings +of Machine Learning and Systems 4 (2022), 515–528. +Meng Zhang, Duygu Ceylan, and Niloy J Mitra. 2022. Motion guided deep dynamic 3d +garments. ACM Transactions on Graphics (TOG) 41, 6 (2022), 1–12. +Meng Zhang, Tuanfeng Wang, Duygu Ceylan, and Niloy J Mitra. 2021a. Deep detail +enhancement for any garment. In Computer Graphics Forum, Vol. 40. Wiley Online +Library, 399–411. +Meng Zhang, Tuanfeng Y Wang, Duygu Ceylan, and Niloy J Mitra. 2021b. Dynamic +neural garments. ACM Transactions on Graphics (TOG) 40, 6 (2021), 1–15. + +PhysGraph: Physics-Based Integration Using Graph Neural Networks +• +9 +Fig. 10. We show several more examples of the cloth enhancement process using PhysGraph to demonstrate that our method scales to complicated multi-layer +outfits. We demonstrate a tucked in shirt with belt and an outfit consisting of a hoodie, shirt and pants. Note how our model is capable of resolving collisions +with small geometric features such as the belt loops and pockets. + +10 +• +Halimi, O. et al +Fig. 11. We show several consecutive frames of an enhanced cloth sequence. Despite operating per frame, our method shows mostly temporally consistent +results with the exception of collision heavy regions. Please refer to the supplemental video for the full result. + +33333333 +9 +8 +16 +Coarse +Enhanced +Coarse +Enhanced +ndno +Input +Input + ndno \ No newline at end of file diff --git a/RNFKT4oBgHgl3EQfiS5U/content/tmp_files/load_file.txt b/RNFKT4oBgHgl3EQfiS5U/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c85925ec55c08967c7399b2d6bcfc7fd2959ffa8 --- /dev/null +++ b/RNFKT4oBgHgl3EQfiS5U/content/tmp_files/load_file.txt @@ -0,0 +1,627 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf,len=626 +page_content='PhysGraph: Physics-Based Integration Using Graph Neural Networks OSHRI HALIMI, Technion – Israel Institute of Technology and Meta Reality Labs Research EGOR LARIONOV, Meta Reality Labs Research ZOHAR BARZELAY, Meta Reality Labs Research PHILIPP HERHOLZ, Meta Reality Labs Research TUUR STUYCK, Meta Reality Labs Research PhysGraph Patch-Based Training Data Material Properties Collision Geometry Force Module Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' PhysGraph leverages a recurrent graph neural network in order to minimize user-provided energy potentials in a topology and force independent way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' We demonstrate our method on enhancing coarse resolution cloth geometry with physics-based high resolution detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' The figure shows the coarse geometry on the left and the enhanced output on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Our method is capable of handling body and self collisions as well as the elastic potentials of cloth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' By explicitly separating the force modeling and the integration process, we obtain high generalization power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' The method is trained on local patches with only forces resulting from the garment elastic potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' We show several examples to demonstrate that our method allows to modify material properties and collision geometry during inference, all leveraging a single model trained on only a limited number of potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Physics-based simulation of mesh based domains remains a challenging task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' State-of-the-art techniques can produce realistic results but require expert knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' A major bottleneck in many approaches is the step of integrating a potential energy in order to compute velocities or displacements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Recently, learning based method for physics-based simulation have sparked interest with graph based approaches being a promising research direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' One of the challenges for these methods is to generate models that are mesh inde- pendent and generalize to different material properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Moreover, the model should also be able to react to unforeseen external forces like ubiquitous collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Our contribution is based on a simple observation: evaluating forces is computationally relatively cheap for traditional simulation meth- ods and can be computed in parallel in contrast to their integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' If we learn how a system reacts to forces in general, irrespective of their origin, we can learn an integrator that can predict state changes due to the total forces with high generalization power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' We effectively factor out the physical model behind resulting forces by relying on an opaque force module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' We demonstrate that this idea leads to a learnable module that can be trained on basic internal forces of small mesh patches and generalizes to different mesh typologies, resolutions, material parameters and unseen forces like Authors’ addresses: Oshri Halimi, Technion – Israel Institute of Technology and Meta Reality Labs Research;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Egor Larionov, Meta Reality Labs Research;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Zohar Barzelay, Meta Reality Labs Research;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Philipp Herholz, Meta Reality Labs Research;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Tuur Stuyck, Meta Reality Labs Research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' collisions at inference time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Our proposed paradigm is general and can be used to model a variety of physical phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' We focus our exposition on the detail enhancement of coarse clothing geometry which has many applications including computer games, virtual reality and virtual try-on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' CCS Concepts: • Computing methodologies → Physical simulation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Additional Key Words and Phrases: Cloth simulation, neural network simu- lation, graph neural networks 1 INTRODUCTION Physics-based simulation has made significant advances in the last decades and it is now possible to re-create highly realistic physical phenomena using computer models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Current simulation techniques provide us with a method that generalizes to novel settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' How- ever, these simulations are difficult to compute and can require extensive manual interventions in order to obtain the desired re- sults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' On the other hand, with the proliferation of machine learning and data-driven techniques, there has been an increased interest in recreating physical phenomena using neural networks [Allen arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content='11841v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content='GR] 27 Jan 2023 Patch-Based Training Data PhysGraph Force Module Material Collision Properties Geometry2 Halimi, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' et al et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' 2022a,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Pfaff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Sanchez-Gonzalez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' How- ever, it is hard to generalize these neural models to account for all the variety that can occur as it might not be fully covered in the dataset or overfitting might occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' This motivates us to leverage the strengths of both approaches to design a neural model that uses physics-based information to produce a generalizable and widely applicable method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' In this paper, we present a solution that improves on research in this direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' We focus on the specific application of modeling garment defor- mations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' The ability to model garments is crucial for telepresence, games, virtual try-on and other applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Cloth state prediction using data-driven approaches is a long standing and notoriously difficult problem, due to the high variability in garment shape, de- formation, and discontinuities caused by frequent collisions against the body and within the cloth itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Additionally, it remains cum- bersome to obtain the required training data [Halimi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Despite this, many advances have been made [Bertiche et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' 2020, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Santesteban et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' 2019, 2022a, 2021], which allow us to produce garment configurations based on the skeleton pose and body shape as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Unfortunately these methods rely on networks trained on specific garments, and thus do not generalize well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Oftentimes, they are limited to modelling tight fitting clothing as it leverages a skinning model with respect to the body skeleton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' To address these limitations, others have presented approaches for the animation of loose clothing with neural networks [Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' 2021b], or by leveraging real-time physics-based cloth simulation [Stuyck 2018] with a learned neural rendering pass to obtain realistic looking clothing that generalizes to new motion and body shapes [Xiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Despite recent progress, many limitations still remain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Methods are often limited to fixed underlying body skeletons, fixed topologies, and material properties and do not handle collisions gracefully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' In an effort to obtain better generalization, these observations motivate us to explore the potential of combining machine learning and physics-based techniques further and exploiting knowledge about the physical system directly, instead of learning the relation- ship implicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' The core idea of our proposed method is to factor out the force specific components from the integration process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Force computations are computationally relatively more efficient for physics-based simulation methods and can be computed in par- allel in contrast to their integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Leveraging this design, we can then learn the integration procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' This approach fundamentally prevents overfitting and has the ability to generalize to novel forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' We implement this design by using a message passing graph neu- ral network that can integrate physics-based forces provided by an external force module on arbitrary topologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' The approach is agnostic to the specific material models and we demonstrate that it generalizes to unseen forces during inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Thanks to the design of the method, we are able to model different garment categories, both tight and loose, with self and body collisions in a topology independent way that does not require an underlying skinned body mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' In summary, our main contributions are: A novel neural architecture with split responsibilities for force modeling and force integration, which allows for integrating physics-based forces in a topology invariant fashion resulting in a method that generalizes to unseen settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' The method is able to resolve collisions with arbitrary ge- ometries using either triangle mesh or signed distance field (SDF) representations and, for the first time using a neural approach, is able to resolve self collisions between multiple garments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' The resulting simulation pipeline retains the controllable, physics-based, well understood material models, providing the user with meaningful and intuitive control parameters (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' material properties).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' The integrator module is trained in an unsupervised way, which allows for efficient learning without the need for ex- pensive and hard to obtain training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' 2 RELATED WORK We provide an overview of relevant work related to the modeling of physical phenomena using neural networks, cloth detail enhance- ment and neural methods for generating garment deformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content='1 Neural Networks for Modeling Physical Phenomena Neural networks have been successfully used to model granular material and fluids [Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Scarselli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' 2008] using particle based approaches and graph neural networks [Battaglia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' 2018].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Several other methods focus on the modeling of fluid dynamics [Thuerey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Um et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' 2018].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Pfaff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' [2020] introduced a mesh-based method for the simulation of several phenomena using graph neural networks with several follow up works [Fortunato et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Sanchez-Gonzalez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content='2 Cloth Geometry Enhancement Enhancing details on cloth geometry is a long-standing research problem with early work by Cutler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' [2005] who presented a pro- cedural wrinkling model capable of adding art-directed wrinkling in a production setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Bergou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' [2007] proposed a method where constrained Lagrangian mechanics are used to add physically-based details to animated thin shells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Müller and Chentanez [2010] pro- posed a simple and fast method to add wrinkles to dynamic meshes by attaching a higher resolution wrinkle mesh to the coarse base mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Kavan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' [2011] presented a method where enhancement is achieved by learning linear upsampling operators for physically- based cloth simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Rémillard and Kry [2013] proposed a method to add wrinkling to composite objects consisting of a soft interior and harder skin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Since then, many follow-up works have been pre- sented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Rohmer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' [2010] leverage the stretch tensor computed on the coarse animation to add temporally coherent wrinkles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Similarly, Gillette et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' [2015] present a method to add dynamic wrinkling to coarse animated cloth using a two stage stretch tensor estimation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Cloth details can be enhanced using tension field theory to model coarse geometry after which the amplitude and phase of the fine wrinkling is added [Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Recently, Wang [2021] explored specialized techniques for cloth simulation on the GPU using grid-aligned meshes to gain an edge at reconstructing fine wrinkles at submillimeter levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Other methods rely on neural networks operating on 3D geometry [Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' PhysGraph: Physics-Based Integration Using Graph Neural Networks 3 2021a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Lahner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' [2018] present a data-driven approach to en- hance detail encoded in a normal map texture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' An image-to-image neural network is trained to enhance detail in image space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content='3 Neural Garment Deformations Learning-based methods aim at predicting a garment’s draping over a given body mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Several methods rely on the SMPL [Loper et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' 2015] parametric body shape model, along with its rigging and skin- ning functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' In practice, this means that such methods are able to cast the ML-draping problem as that of predicting corrective garment deformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' These per-vertex deformations are added to the garment’s rest-pose vertex position, and are then skinned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Initial methods train a garment prediction model by regressing ground- truth vertex positions, calculated using high-fidelity physics simula- tion data [Patel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' 2020];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' while relying on fixed skinning weights, transferred from the SMPL weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' The fixed skinning weights limit the garment vertices to move based on their rest pose location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' This assumption is alleviated by computing post-deformation skinning weights, utilizing a prediction network [Santesteban et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Per-pose predictions do not take into account the dynamic nature of garment deformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Therefore, Santesteban et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' [2019] utilize a recurrent model whose predictions depend on past poses too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Such models require per-garment training, necessitating multiple ground- truth simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Bertiche et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' [2021] alleviates this requirement, by training in an unsupervised setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' The loss is cast as a set of physical potentials to minimize: stretching, bending, gravity, and body-cloth collision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' These losses are differentiable, and therefore can be back-propagated to optimize the network’s weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' San- testeban et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' [2022a] and Bertiche et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' [2022] both add an inertia loss to address temporal consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' The input of Santesteban et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' [2022a] further takes not only the parameterized body-pose, but also its shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' However, the network is still specialized per-garment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' De Luigi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' [2022] alleviates this requirement by predicting a latent code for any given garment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' It thus generalizes over body shape, body pose, and garment type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Reliance on body-based skin- ning amounts to limiting the garment to move in correspondence to the underlying body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' However, for loose-garment this is not always the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' To tackle this, Pan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' [2022] creates for a garment a new set of joints and corresponding per-vertex skinning weights, based on ground-truth simulations on a variety of motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Drap- ing prediction then amounts to predicting the joints translation and rotation parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' The above methods rely on a paramet- ric body representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' This limits their applicability to draping multiple garments layers or stylistic (non-human) avatars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' [2022] addresses this by representing the underlying body as a set of sampled points, while Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' [2022] separately encodes the input body and garment meshes using graph-convolution networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' This separate encoding does not take into account body-garment interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' To alleviate this shortcoming, Grigorev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' [2022] adds body-garment graph edges, and uses hierarchical message- passing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Body-garment collisions can also be solved by learning a collision-free generative deformation space [Santesteban et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' ULNeF [Santesteban et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' 2022b] generalizes to multiple garment by predicting corrective terms to the garments’ implicit representation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' but is limited to running on human shapes in canonical pose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' The above ML-based garment draping methods achieve impres- sive results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' However, they are all limited in generalizing to arbitrary- posed bodies with arbitrary layers of interwoven garments and clothing items (such as a tucked-in shirt, layered with a suit, tie and a pocket handkerchief as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Our method allows, for the first time, to achieve realistic draping of complex topologies, in a self-supervised manner (see full comparison in Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' 3 METHOD To evaluate, PhysGraph, we focus on quasi-static simulation of cloth using elastic energy and contact penalty potential minimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Given an energy potential, a classical method would iterate over con- figurations following the negative energy gradients (forces) until it finds the optimal point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Here, we decouple the force module responsi- ble for directly differentiating energy potentials, and the integration module, which integrates the resulting forces into a displacement vector for all vertex positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Given these complementary respon- sibilities of the modules, our method provides great flexibility to the system being modeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' The method is agnostic to both the type of physical forces modeled by the force module and, the mesh connec- tivity at inference time, allowing it to generalize to different types of potentials even after the network is trained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' The central building block of our architecture is a recurrent mesh graph network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' We initialize the integration module with an upsampled version of the physical simulation given a coarse mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' This way, the large scale dynamic behavior of the system is captured in the coarse mesh while our architecture creates finer scale details that are governed by the static equilibrium of forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' These coarse meshes can be obtained using classical mesh based simulation or other methods like artist models and linear blend skinning output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Our key contribution is an algorithm that performs several itera- tions of force computation and integration to find an approximate minimizer of the potential energy provided by the force module during inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' To this end, the integrator leverages a graph net- work architecture [Pfaff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' 2020] which we use in a recurrent fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' The subsequent sections will introduce the force module, the integration module and the graph network architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content='1 Force module We credit the generalizability of our method to the separation of the force formulation and integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' The force generation module outputs forces based on user-specified potentials given the current nodal configuration where forces are accumulated at the nodal level, enabling parallelization of the force computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' The conservative force potential Φ, is responsible for the forces F = −∇XΦ acting on the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Different potentials can be used to model different physical phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content='2 Mesh-Based Graph Networks The simulation mesh can be interpreted as a graph 𝐺 = (𝑉, 𝐸) onto which we encode vertex and edge features consisting of 128 values each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' The graph network operates in three phases: Encode, Process and Decode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Encode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Vectors of per edge ˜𝑒𝑖 and per vertex features ˜𝑣𝑖 form the input to the graph network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' By applying two multilayer perceptrons 4 Halimi, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' et al MLP MLP MLP , MLP Decode Encode Process Edge Process Vertex MLP Force Module Integration Module + Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Our approach consists of three components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' The force module (Sec- tion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content='1) which evaluates per-node forces for a mesh configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' The integration module (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content='3) constructs initial feature vectors ˜𝑣𝑖 and ˜𝑒𝑖 based on force and configuration information from previous iterations, passes these quantities to a graph neural network (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content='2) and re- trieves new displacements 𝐷𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' The graph network iterates between an edge processing step that distributes vertex information to edges and an edge processing step which distributes edge information to adjacent vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' These distribution steps are iterated 𝑀 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' The integration module iter- ates 𝐾 times and finally returns an estimate for the quasi-static equilibrium state X𝐾 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' (MLP), one to each vertex feature and one to each edge feature, we obtain initial features 𝑣0 𝑖 and 𝑒0 𝑖 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' We use a fixed amount of 𝑀 = 10 message passing itera- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Each iteration 𝑗 consists of two steps computing new edge and vertex features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' First, 𝑒 𝑗 𝑖 is computed by passing information from vertices to edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' In the second step, information is passed from edges to adjacent vertices to build 𝑣 𝑗 𝑖 , see Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Each step uses a single MLP with weights that are shared for all iterations and between all vertices and edges, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Decode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' A final MLP is used to decode the final vertex features 𝑣𝑘 into the final displacement vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' The set of learnable parameters 𝜃 are the weights of the five MLPs used during the three phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content='3 Integration Module The physical configuration of the mesh is given by its embedding which can be represented as a matrix of stacked position vectors X ∈ R𝑛×3 where 𝑛 is the number of vertices in the mesh and X𝑖 represents the position of the 𝑖−th vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' We assume that the mesh has a rest configuration X ∈ R𝑛×3, which by definition, experi- ences no internal forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' The goal of the integration module is to find an approximation to the static equilibrium configuration X∗ = argminX Φ(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Starting with initial vertex positions X𝑘 with 𝑘 = 0, we produce new positions X𝑘+1 by using the pre-trained graph network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Throughout the simulation, the graph network has access to the mesh connectivity corresponding to the set of graph edges 𝐸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' For each configuration, we can query the force module to obtain corresponding forces F𝑘 = −∇XΦ(X𝑘).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' The inputs to the graph network are per vertex and edge features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' For each vertex 𝑖 we construct the feature vector ˜𝑣𝑖 by concatenating first order differences of the last 𝐻 configurations and the corresponding force vectors ˜𝑣𝑖 = � X𝑘−1 𝑖 − X𝑘 𝑖 , · · · , X𝑘−𝐻 𝑖 − X𝑘−𝐻+1 𝑖 , F𝑘 𝑖 , · · · , F𝑘−𝐻+1 𝑖 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' (1) The edge features ˜𝑒𝑖 for the edge connecting vertex 𝑟 and 𝑠 are comprised of position differences for the current configuration and the rest state as well as their lengths ˜𝑒𝑖 = � X𝑘𝑟 − X𝑘𝑠 , ∥X𝑘𝑟 − X𝑘𝑠 ∥, X𝑟 − X𝑠, ∥X𝑟 − X𝑠 ∥ � , (2) where ∥ · ∥ is the Euclidean norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' The network outputs displace- ments D𝑘 that define the next state via X𝑘+1 = X𝑘 + D𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' After 𝐾 = 5 iterations we obtain an approximation of the equilibrium state X𝐾.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content='4 Training phase We train the integration module with a dataset of small physical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' These act as an input to the recurrent model, consisting of 𝐾 recurrent force calculation and integration blocks sharing the same network parameters of the trainable integration module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' The training is supervised by requiring the minimization of the potential, summed over all the intermediate states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' 4 GARMENT DETAIL ENHANCEMENT Here, we apply PhysGraph to the cloth upsampling problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' A given coarse resolution cloth mesh is first subdivided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Then nodal forces are computed using the force module and integrated into nodal displacements by the integration module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' This is repeated iteratively over multiple steps, which allows local forces to prop- agate throughout the rest of the mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' We show that the method remains effective regardless of whether the coarse mesh is obtained through low resolution cloth simulation or other methods such as artist animation or through procedural or skinning approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' In this section, we define the potentials used to model fabric elasticity and contact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content='1 Garment Potentials To demonstrate cloth modelling, we use springs to model stretching, dihedral angle penalty to model bending and penetration penalty to model contact, although the force module can be any material model that produces forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' The potentials are weighted based on element area, which we define to be the area of a barycentric subdivision (light gray areas in the inset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' This way the PhysGraph: Physics-Based Integration Using Graph Neural Networks 5 total area of all edge and vertex weights, respectively, sum up to the total surface area 𝐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Stretching is modelled using edge aligned springs with net elas- tic potential Φ𝑠 = 𝑘𝑠 2𝐴 ∑︁ 𝑒 ∈𝐸 𝑎𝑒 (𝑙(𝑒) − 𝑙0(𝑒))2, (3) where 𝑘𝑠 is the spring stiffness, 𝑙(𝑒) = ∥X𝑖 − X𝑗 ∥ is the edge length for an edge 𝑒 = (𝑖, 𝑗) and 𝑙0(𝑒) = ∥X𝑖 − X𝑗 ∥ is its rest-length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' The bending potential is defined by the dihedral angles 𝜃𝑑 formed between the normal vectors to the triangles in each dihedral element Φ𝑏 = 𝑘𝑏 2𝐴 ∑︁ 𝑒 ∈𝐷 𝑎𝑒𝜃2 𝑑 (4) where 𝐷 is the set of interior edges corresponding to dihedral ele- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' The gravitational potential is defined by Φ𝑔 = − 𝑔 𝐴 ∑︁ 𝑣∈𝑉 𝑎𝑣𝑚𝑣𝑧𝑣, where 𝑚𝑣 = 𝜌𝑎𝑣, (5) and 𝑧𝑣 is the coordinate along the gravity axis and 𝜌 the mass den- sity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' The external contact potential is modeled using the signed- distance-function 𝑆𝐷𝐹 (·), which measures the signed distance (neg- ative inside, positive outside) to the surface of a set of colliders in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' The contact potential 𝜙𝑣 = − min(𝑆𝐷𝐹 (X𝑣), 0) is accumulated over all potentially violating vertices 𝑣 with Φ𝑒𝑐 = 𝑘𝑒𝑐 𝐴𝑒𝑐 ∑︁ 𝑣∈𝑉 𝑎𝑣𝜙𝑣, where 𝐴𝑒𝑐 = ∑︁ 𝑣∈𝑉 𝑎𝑣(1 − 𝛿0(𝜙𝑣)), (6) and 𝑘𝑒𝑐 is the external contact penalty stiffness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' The zero-set in- dicator function 𝛿0 evaluates to 1 for 0 and to 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Fi- nally, the self collision potential is modeled by radial compres- sion springs with rest-length 𝑅 around each vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' A compres- sion spring between vertices 𝑢, 𝑣 ∈ 𝑉 , modelled by the potential 𝜓𝑢,𝑣 = max(𝑅 − ∥X𝑢 − X𝑣∥, 0), exerts a force in the outward radial direction when compressed, which happens when distinct vertices become closer than 𝑅 apart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' The total energy is defined by Φ𝑠𝑐 = 𝑘𝑠𝑐 𝐴𝑠𝑐 ∑︁ 𝑢,𝑣∈𝑉 𝑢∉N𝑑 (𝑣) (𝑎𝑣 + 𝑎𝑢)𝜓2 𝑢,𝑣, 𝐴𝑠𝑐 = ∑︁ 𝑢,𝑣∈𝑉 𝑢∉N𝑑 (𝑣) (𝑎𝑣 + 𝑎𝑢)(1 − 𝛿0(𝜓𝑢,𝑣)) (7) where 𝑘𝑠𝑐 is the self-collision penalty stiffness, and we consider only interactions of vertices which are not neighbors on the mesh within some d-ring of 𝑣 denoted N𝑑 (𝑣).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content='2 Training The method is trained using patches sampled from a dynamically simulated t-shirt on a moving human body, example patches are shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' The patches are subdivided using a self-similarity subdivision scheme, increasing the mesh resolution by a factor of 16 and, by linearly interpolating the coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' We use 𝐾 = 5 recurrent blocks in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' We stress that we only include forces resulting from the stretch and bending potentials Φ = Φ𝑠 +Φ𝑏 at training time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' To account for the fact that the patch is a sub- system of the larger full-cloth system and prevent the flattening of the patches in the absence of the rest of the cloth-system, we used fixed boundary conditions while training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' From left to right: coarsely simulated input, inferring with stretch and bending forces only, stretch and bending and body collisions and finally, including all forces on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' The model was trained with elastic energy forces only and generalizes to include collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' In contrast to methods that rely on skinning based techniques, our method nat- urally handles loose clothing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Left shows the low resolution dress and right shows our enhanced result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' 5 RESULTS Due to the design of our method, we are able to train our inte- grator network using only the internal cloth potentials and their resulting forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' At inference time, the trained network is capable of ingesting numerous forces from a variety of sources to produce plausible results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' We showcase this by applying our trained integra- tion module to a variety of novel forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' We show results with forces such as gravity, body collision and self collision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' We demonstrate the effectiveness of PhysGraph for several variations of the cloth enhancement application with complete multi-layered garment ex- amples shown in Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' All results were generated using a single trained network which was only exposed to elastic forces at training time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Note that the method is not limited to these specific examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content='1 Coarse Simulation Enhancement We show results for the cloth geometry enhancement in Figure 3 which shows the resulting geometries after integrating different force potentials ranging from simply including elastic energy poten- tials, which were included at training time, to a full model with body and self-collisions, which includes several forces not seen during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Our method is garment independent and works for loose clothing as can be seen in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content='2 Linear Blend Skinning Enhancement To demonstrate generalization to the model input, we show that the coarse geometry does not need to be obtained from a simulation and lower cost methods can be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' We showcase the efficacy of 6 Halimi, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' et al Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' We demonstrate that our method is capable of enhancing geometry obtained from linear blend skinning (left), the middle shows the enhanced mesh without self collisions and the rightmost shows the predicted gar- ment incorporating all forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Note the realistic wrinkling added by our method while preserving the overall shape and remaining collision free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' The enhancement model includes both body and self collision forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Note that for the jacket, the method is even capable of removing the skinning artifacts near the armpit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' We demonstrate the ability of enhancing garment geometry using different materials at inference time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' This example shows the apparent visual difference of varying bend and stretch stiffnesses allowing the user to generate a potentially expensive coarse sequence once and adjust materials afterwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Our method produces plausible results where higher bending stiffness correctly corresponds to bigger folds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' the method on garments posed using linear blend skinning which is known to have several issues which distorts the mesh in non- physical ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Nevertheless, Figure 5 shows that our method is capable of producing visually pleasing high resolution meshes where self-collisions are resolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content='3 Material Generalization Due to the force module, we are able to modify the material proper- ties at inference time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Figure 6 shows different material settings for a pair of pants which are all generated from the same coarse input geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content='4 Collision Geometry Generalization Our method generalizes to different collision geometries as shown in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' We show rich cloth interaction with a rigid block, pushing the cloth through the center of a torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' These colliding geometries are completely novel with respect to those at training time, yet, the method produces correct results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' We show that our method generalizes to different collision geometries during inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' From left to right, we show the original low resolution input, a mesh obtained using Loop subdivision [1987], our method, and a high resolution simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Note how our method adds detail compared to a subdivision which simply smooths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Our method maintains the overall shape of the low resolution but adds detail, preserving artistic intent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' The high resolution simulation is very distinct from the low resolution making it difficult to obtain desired results when iterating on lower resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' 0 10 20 30 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content='2 iteration potential adam 10−2 adam 10−3 adam 10−4 gd 10−1 gd 100 gd 101 PhysGraph Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' We demonstrate that our method converges faster and to a lower po- tential than several gradient-descent optimizers, with varying base learning rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' For each baseline optimizer, we show increased learning rates until they divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content='5 Self Collisions Interactions of fabric with itself are ubiquitous for cloth simulations as garments are often layered and display complex interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Therefore, it is essential to model them appropriately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' We demon- strate the importance of including self collisions in Figures 1, 3 and 10, where we show that our model is capable of including these forces, providing clean, intersection free meshes as shown on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content='6 Convergence Analysis We highlight the effectiveness of PhysGraph by comparing conver- gence with respect to two baseline optimizers: Adam and gradient decent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' We showcase various learning rates in Figure 8 for which they still converge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Note how our learned integrator is the most effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content='7 Comparisons To Related Work We compare PhysGraph (ours) to two recent ML-approaches for gar- ment draping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Both SSCH [Santesteban et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' 2021] and SNUG [San- testeban et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' 2022a] are skinning-based approaches, trained with ground-truth simulations and self-supervision, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' We use kb = 1 kb = 10 kb = 100 kb = 100 ks = 1e4 ks = 1e4 ks = 1e3 ks = 1e4PhysGraph: Physics-Based Integration Using Graph Neural Networks 7 their publicly released t-shirt models for comparison, and emphasize that both models were trained for this specific garment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' In compar- ison, our method has not seen this garment during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Our method takes the skinned mesh as input, and predicts a refined draping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Figure 9 shows a qualitative comparison and Table 1 pro- vides quantitative comparison of the system potentials for each method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' While both SNUG and SSCH are limited to a fixed topology and resolution since both models are specialized for this specific garment, we are still able to generate detail at several resolutions without having trained on this topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Furthermore, our method provides qualitatively and quantitatively better results with a more general method which has not been optimized for this particular setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Table 2 provides a functional comparison which additionally includes ULNeF [Santesteban et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' 2022b], Hood [Grigorev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' 2022] and MeshGraphNet (MGN) [Pfaff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' SNUG SSCH PhysGraph Low Resolution PhysGraph High Resolution Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Comparison to skinning-based ML-draping methods of three differ- ent frames of the CMU-07-02-poses sequence from Mahmood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' [2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' From left to right: SNUG [Santesteban et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' 2022a];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' SSCH [Santesteban et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' 2021];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' PhysGraph low resolution prediction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' PhysGraph high resolution refinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Note how SSCH and SNUG show similar wrinkling regardless of the pose, whereas PhysGraph is capable of adding fine detail in a more physically plausible way at several resolutions where detail varies with pose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Also note that SNUG results in intersections with the body as seen in the bottom row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' 6 DISCUSSION, LIMITATIONS, AND FUTURE WORK We propose a novel method for the integration of physics-based forces using a graph neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Our method demonstrates excellent generalization capabilities and we show a successful appli- cation to the detail enhancement of coarse cloth geometry for both tight and loose clothing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Our method is capable of modeling gar- ment interactions with itself and the body and other collision objects and we are the first to support collisions with multiple garments simultaneously using their geometry directly without needing an SDF or other representation which introduce several limitations for Total Potential [erg] ↓ Body Collision(%) ↓ Skinning 243.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content='2094 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content='5088 SSCH 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content='2101 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content='8257 SNUG 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content='3766 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content='7818 Ours 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content='4635 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content='4813 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Quantitative comparison with state-of-the-art methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' ↓ means a lower value is better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' We report the potential of the cloth in [erg] units, using a mass-spring model under gravity, using 𝑘𝑠 = 1𝑒4 erg/cm2, 𝑘𝑏 = 10 erg, and 𝜌 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content='0187 gr/cm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' To obtain the rest edge lengths, we use SNUG’s rest mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' The collisions with the body are reported as the percentage of cloth vertices admitting negative values when used as a query to the body SDF, which is defined with respect to the body with a collision margin of 2 mm, similar to SNUG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' All the compared potentials are calculated over the same mesh topology taken from SNUG’s shirt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Topology Invariant Pose Invariant Force Agnostic Body- Garment Collisions Garment self Collisions Multi Garment Collisions Unsupervised SSCH ✗ ✓ ✗ ✓ ✗ ✗ ✗ SNUG ✗ ✓ ✗ ✗ ✗ ✗ ✓ ULNeF ✓ ✗ ✗ ✗ ✓ ✓ ✗ MGN ✓ ✓ ✗ ✓ ✓ ✗ ✗ Hood ✓ ✓ ✗ ✓ ✓ ✗ ✓ PhysGraph ✓ ✓ ✓ ✓ ✓ ✓ ✓ Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Summary of Related Work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Our work achieves all desirable features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' modeling layered clothing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Our implementation is not optimized to enhance cloth geometry at real-time rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' However, currently the bulk of the computation is within the force module, which is implemented in Python.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' We believe that given the engineering re- sources, the method has the potential to run at interactive rates due to the parallel nature of the force module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Additionally, there are active research efforts on improving message-passing architectures efficiency [Rahman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' 2022] which have already demonstrated the ability to accelerate the computation by two or- ders of magnitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Like most other methods, our current model is unable to resolve pre-existing self-intersections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' However, future work could include untangling forces [Baraff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' 2003] as part of the force module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' In Figure 11, we show how our method performs on a simu- lated t-shirt sequence, where our generated detail is already mostly temporally coherent with the exception of areas with clustered col- lisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' This is encouraging given that the method operates per frame independently 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Meng Zhang, Duygu Ceylan, and Niloy J Mitra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Motion guided deep dynamic 3d garments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' ACM Transactions on Graphics (TOG) 41, 6 (2022), 1–12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Meng Zhang, Tuanfeng Wang, Duygu Ceylan, and Niloy J Mitra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' 2021a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Deep detail enhancement for any garment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' In Computer Graphics Forum, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Wiley Online Library, 399–411.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Meng Zhang, Tuanfeng Y Wang, Duygu Ceylan, and Niloy J Mitra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' 2021b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Dynamic neural garments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' ACM Transactions on Graphics (TOG) 40, 6 (2021), 1–15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' PhysGraph: Physics-Based Integration Using Graph Neural Networks 9 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' We show several more examples of the cloth enhancement process using PhysGraph to demonstrate that our method scales to complicated multi-layer outfits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' We demonstrate a tucked in shirt with belt and an outfit consisting of a hoodie, shirt and pants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Note how our model is capable of resolving collisions with small geometric features such as the belt loops and pockets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' 10 Halimi, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' et al Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' We show several consecutive frames of an enhanced cloth sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Despite operating per frame, our method shows mostly temporally consistent results with the exception of collision heavy regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' Please refer to the supplemental video for the full result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} +page_content=' 33333333 9 8 16 Coarse Enhanced Coarse Enhanced ndno Input Input ndno' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFKT4oBgHgl3EQfiS5U/content/2301.11841v1.pdf'} diff --git 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0000000000000000000000000000000000000000..e469f6a49ab991e638d8e7aa2d86f78043412a65 --- /dev/null +++ b/V9E0T4oBgHgl3EQfVgDO/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:b4a82ccd4940e8755ce350b004f662fd91e51849072d7b3976e0c55bcf72073f +size 1900589 diff --git a/WNE3T4oBgHgl3EQf0wtX/content/tmp_files/2301.04740v1.pdf.txt b/WNE3T4oBgHgl3EQf0wtX/content/tmp_files/2301.04740v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..627cda4365d9ae9dcccb81665f4de91ed5c0300c --- /dev/null +++ b/WNE3T4oBgHgl3EQf0wtX/content/tmp_files/2301.04740v1.pdf.txt @@ -0,0 +1,3617 @@ +arXiv:2301.04740v1 [cs.LG] 11 Jan 2023 +The Berkelmans-Pries Feature Importance +Method: A Generic Measure of +Informativeness of Features +Joris Pries1,*, Guus Berkelmans1, Sandjai Bhulai2, and Rob van +der Mei1 +1Centrum Wiskunde & Informatica, Department of Stochastics, +Science Park 123, Amsterdam 1098 XG, Netherlands +2Vrije Universiteit, Department of Mathematics, De Boelelaan +1111, Amsterdam 1081 HV, Netherlands +*Corresponding author: Joris Pries, joris.pries@cwi.nl +January 13, 2023 +Abstract +Over the past few years, the use of machine learning models has +emerged as a generic and powerful means for prediction purposes. At +the same time, there is a growing demand for interpretability of predic- +tion models. To determine which features of a dataset are important to +predict a target variable Y , a Feature Importance (FI) method can be +used. By quantifying how important each feature is for predicting Y , +irrelevant features can be identified and removed, which could increase +the speed and accuracy of a model, and moreover, important features +can be discovered, which could lead to valuable insights. +A major +problem with evaluating FI methods, is that the ground truth FI is +often unknown. As a consequence, existing FI methods do not give the +exact correct FI values. This is one of the many reasons why it can be +hard to properly interpret the results of an FI method. Motivated by +this, we introduce a new global approach named the Berkelmans-Pries +FI method, which is based on a combination of Shapley values and the +Berkelmans-Pries dependency function. We prove that our method has +many useful properties, and accurately predicts the correct FI values +for several cases where the ground truth FI can be derived in an exact +1 + +manner. We experimentally show for a large collection of FI methods +(468) that existing methods do not have the same useful properties. +This shows that the Berkelmans-Pries FI method is a highly valuable +tool for analyzing datasets with complex interdependencies. +1 +Introduction +How important are you? This is a question that researchers (especially data +scientists) have wondered for many years. Researchers need to understand +how important a random variable (RV) X is for determining Y . +Which +features are important for predicting the weather? Can indicators be found +as symptoms for a specific disease? Can redundant variables be discarded +to increase performance? These kinds of questions are relevant in almost +any research area. +Especially nowadays, as the rise of machine learning +models generates the need to demystify prediction models. Altmann et al. [3] +state that “In life sciences, interpretability of machine learning models is as +important as their prediction accuracy.” Although this might not hold for all +research areas, interpretability is very useful. Knowing how predictions are +made and why, is crucial for adapting these methods in everyday life. +Determining Feature Importance (FI) is the art of discovering the impor- +tance of each feature Xi when predicting Y . The following two cases are +particularly useful. (I) Finding variables that are not important: redundant +variables can be discovered using FI methods. Irrelevant features could de- +grade the performance of a prediction model due to high dimensionality and +irrelevant information [26]. Eliminating redundant features could therefore +increase both the speed and the accuracy of a prediction model. (II) Find- +ing variables that are important: important features could reveal underlying +structures that give valuable insights. Observing that variable X is impor- +tant for predicting Y could steer research efforts into the right direction. +Although it is critical to keep in mind that high FI does not mean causation. +However, FI values do, for example, “enable an anaesthesiologist to better +formulate a diagnosis by knowing which attributes of the patient and pro- +cedure contributed to the current risk predicted” [36]. In this way, an FI +method can have really meaningful impact. +Over the years, many FI methods have been suggested, which results in +a wide range of FI values for the same dataset. +For example, stochastic +methods do not even repeatedly predict the same FI values. This makes +interpretation difficult. Examine e.g., a result of Fryer et al. [17], where one +measure assigns an FI of 3.19 to a variable, whereas another method gives the +2 + +same variable an FI value of 0.265. This raises a lot of questions: ‘Which FI +method is correct?’, ’Is this variable deemed important?’, and more generally +‘What information does this give us?’. To assess the performance of an FI +method, the ground truth should be known, which is often not the case [1, +21, 56, 61]. Therefore, when FI methods were developed, the focus has not +yet lied on predicting the exact correct FI values. Additionally, many FI +methods do not have desirable properties. For example, two features that +contain the same amount of information should get the same FI. We later +show that this is often not the case. +To improve interpretability, we introduce a new FI method called Berkelmans- +Pries FI method, which is based on Shapley values [49] and the Berkelmans- +Pries dependency function [5]. Multiple existing methods already use Shap- +ley values, which has been shown to give many nice properties. However, +by additionally using the Berkelmans-Pries dependency function, even more +useful properties are obtained. Notably, we prove that this approach accu- +rately predicts the FI in some cases where the ground truth FI can be derived +in an exact manner. By combining Shapley values and the Berkelmans-Pries +dependency function a powerful FI method is created. This research is an im- +portant step forward for the field of FI, because of the following reasons: +• We introduce a new FI method; +• We prove multiple useful properties of this method; +• We provide some cases where the ground truth FI can be derived in an +exact manner; +• We prove for these cases that our FI method accurately predicts the +correct FI; +• We obtain the largest collection of existing FI methods; +• We test if these methods adhere to the same properties, which shows +that no method comes close to fulfilling all the useful properties; +• We provide Python code to determine the FI values [44]. +2 +Berkelmans-Pries FI +Kruskal [27] stated that “There are infinitely many possible measures of asso- +ciation, and it sometimes seems that almost as many have been proposed at +one time or another.” Although this quote was about dependency functions, +it could just as well have been about FI methods. Over the years, many FI +3 + +methods have been suggested, but it remains unclear which method should +be used when and why [21]. In this section, we propose yet another new +FI method named the Berkelmans-Pries FI method (BP-FI). Although it is +certainly subjective what it is that someone wants from an FI method, we +show in Section 3 that BP-FI has many useful and intuitive properties. The +BP-FI method is based on two key elements: (1) Shapley values and (2) the +Berkelmans-Pries dependency function. We will discuss these components +first to clarify how the BP-FI method works. +2.1 +Shapley value approach +The Shapley value is a unique game-theoretical way to assign value to each +player participating in a multiplayer game based on four axioms [49]. This +concept is widely used in FI methods, as it can be naturally adapted to +determine how important (value) each feature (player) is for predicting a +target variable (game). Let Nvars be the number of features, then the Shapley +value of feature i is defined by +φi(v) = +� +S⊆{1,...,Nvars}\{i} +|S|! · (Nvars − |S| − 1)! +Nvars! +· (v(S ∪ {i}) − v(S)) , +(1) +where v(S) can be interpreted as the ‘worth’ of the coalition S [49]. The +principle behind this formulation can also be explained in words: For every +possible sequence of features up to feature i, the added value of feature i is +the difference between the worth before it was included (i.e., v(S)) and after +(i.e., v(S ∪ {i})). Averaging these added values over all possible sequences of +features gives the final Shapley value for feature i. +SHAP +There are multiple existing FI methods that use Shapley values +[14, 17, 35], which immediately ensures some useful properties. The most +famous of these methods is SHAP [35]. This method is widely used for local +explanations (see Section 4.1). To measure the local FI for a specific sample x +and a prediction model f, the conditional expectation is used as characteristic +function (i.e., v in Equation (1)). +Let x = (x1, x2, . . . , xNvars), where xi +is the feature value of feature i, then SHAP FI values can be determined +using: +vx(S) := Ez [f(z)|zi = xi for all i ∈ S, where z = (z1, . . . , zNvars)] . +(2) +Observe that the characteristic function vx is defined locally for each x. To get +global FI values, an average can be taken over all local FI values. Our novel +4 + +FI method uses a different characteristic function, namely the Berkelmans- +Pries dependency function. This leads to many additional useful properties. +Furthermore, the focus of this research is not on local explanations, but global +FI values. +2.2 +Berkelmans-Pries dependency function +A new dependency measure, called the Berkelmans-Pries (BP) dependency +function, was introduced in [5], which is used in the formulation of the BP- +FI method. It is shown that the BP dependency function satisfies a list of +desirable properties, whereas existing dependency measures did not. It has +a measure-theoretical formulation, but this reduces to a simpler and more +intuitive version when all variables are discrete [5]. We want to highlight this +formulation to give some intuition behind the BP dependency function. It +is given by +Dep (Y |X) := + + + + + +UD(X,Y ) +UD(Y,Y ) +if Y is not a.s. constant, +undefined +if Y is a.s. constant, +(3) +where (in the discrete case) it holds that +UD (X, Y ) := +� +x +pX(x) · +� +y +��pY |X=x(y) − pY (y) +�� . +(4) +The BP dependency measure can be interpreted in the following manner. +The numerator is the expected absolute difference between the distribution +of Y and the distribution of Y given X. If Y is highly dependent on X, the +distribution changes as knowing X gives information about Y , whereas if Y +is independent of X, there is no difference between these two distributions. +The denominator is the maximal possible change in distribution of Y for any +variable, which is used to standardize the dependency function. Note that +the BP dependency function is asymmetric: Dep (Y |X) is the dependency +of Y on X, not vice versa. Due to the many desirable properties, the BP +dependency function is used for the BP-FI. +2.3 +Berkelmans-Pries FI method +One crucial component of translating the game-theoretical approach of Shap- +ley values to the domain of FI is choosing the function v in Equation (1). +5 + +This function assigns for each set of features S a value v(S) that character- +izes the ‘worth’ of the set S. How this function is defined, has a critical +impact on the resulting FI. We choose to define the ‘worth’ of a set S to be +the BP dependency of Y on the set S, which is denoted by Dep (Y |S) [5]. +Here, Dep (Y |S) = Dep (Y |ZS(D)) where D denotes the entire dataset with +all features and ZS(D) is the reduction of the dataset to include only the +subset of features S. Let Ωfeat be the set of all feature variables. Now, for +every S ⊆ Ωfeat, we define: +v(S) := Dep (Y |S) . +(5) +In other words, the value of set S is exactly how dependent the target variable +Y is on the features in S. The difference v(S ∪ {i}) − v(S) in Equation (1) +can now be viewed as the increase in dependency of Y on the set of features, +when feature i is also known. The resulting Shapley values using the BP +dependency function as characteristic function are defined to be the BP-FI +outcome. For each feature i, we get: +FI(i) := +� +S⊆Ωfeat\{i} +|S|! · (Nvars − |S| − 1)! +Nvars! +· (v(S ∪ {i}) − v(S)) += +� +S⊆Ωfeat\{i} +|S|! · (Nvars − |S| − 1)! +Nvars! +· (Dep (Y |S ∪ {i}) − Dep (Y |S)) . +(6) +Abbreviated notation improves readability of upcoming derivations, which is +why we define +w(S, Nvars) := |S|! · (Nvars − |S| − 1)! +Nvars! +, +(N1) +D(X, Y, S) := Dep (Y |S ∪ {X}) − Dep (Y |S) . +(N2) +Note that when Y is almost surely constant (i.e., P(Y = y) = 1), Dep (Y |S) +is undefined for any feature set S (see Equation (3)). We argue that it is +natural to assume that FI(i) is also undefined, as every feature attributes +everything and nothing at the same time. In the remainder of this paper, we +assume that Y is not a.s. constant. +6 + +3 +Properties of BP-FI +Recall that it is hard to evaluate FI methods, as the ground truth FI is +often unknown [1, 21, 56, 61]. With this in mind, we want to show that the +BP-FI method has many desirable properties. We also give some synthetic +cases where the BP-FI method gives a natural expected outcome. The BP- +FI method is stooled on Shapley values, which are a unique solution based +on four axioms [60]. These axioms already give many characteristics that +are preferable for an FI method. Additionally, using the BP dependency +function ensures that it has extra desirable properties. In this section, we +prove properties of the BP-FI method and discuss why these are relevant and +useful. +Property 1 (Efficiency). The sum of all FI scores is equal to the total +dependency of Y on all features: +� +i∈Ωfeat +FI(i) = Dep (Y |Ωfeat) . +Proof. Shapley values are efficient, meaning that all the value is distributed +among the players. Thus, +� +i∈Ωfeat +FI(i) = v(Ωfeat) = Dep (Y |Ωfeat) . +Relevance. With our approach, we try to answer the question ‘How much did +each feature contribute to the total dependency?’. The total ‘payoff’ is in our +case the total dependency. It is therefore natural to divide the entire payoff +(but not more than that) amongst all features. +Corollary 1.1. If adding a RV X to the dataset does not give any additional +information (i.e., Dep (Y |Ωfeat ∪ X) = Dep (Y |Ωfeat)), then the sum of all FI +remains the same. +Proof. This directly follows from Property 1. +Relevance. If the collective knowledge remains the same, the same amount +of credit is available to be divided amongst the features. Only when new +information is added, an increase in combined credit is warranted. A direct +result of this corollary is that adding a clone (i.e., Xclone := X) of a variable +X to the dataset will never increase the total sum of FI. +7 + +Property 2 (Symmetry). If for every S ⊆ Ωfeat \ {i, j} it holds that +Dep (Y |S ∪ {i}) = Dep (Y |S ∪ {j}), then FI(i) = FI(j). +Proof. Shapley values are symmetric, meaning that if v(S ∪ {i}) = v(S ∪ {j}) +for every S ⊆ Ωfeat \ {i, j}, it follows that FI(i) = FI(j). Thus, it automati- +cally follows that BP-FI is also symmetric. +Relevance. If two variables are interchangeable, meaning that they always +contribute equally to the dependency, it is only sensible that they obtain the +same FI. This is a desirable property for an FI method, as two features that +contribute equally should obtain the same FI. +Property 3 (Range). For any RV X, it holds that FI(X) ∈ [0, 1]. +Proof. The BP dependency function is non-increasing under functions of X +[5], which means that for any measurable function f it holds that +Dep (Y |f(X)) ≤ Dep (Y |X) . +Take f := ZS, which is the function that reduces D to the subset of features in +S. Using the non-increasing property of BP dependency function, it follows +that: +Dep (Y |S) = Dep (Y |ZS(D)) = Dep +� +Y |ZS(ZS∪{i}(D)) +� +≤ Dep +� +Y |ZS∪{i}(D) +� += Dep (Y |S ∪ {i}) . +(7) +Examining Equation (6), we observe that every FI value must be greater or +equal to zero, as Dep (Y |S ∪ {i}) − Dep (Y |S) ≥ 0. +One of the properties of the BP dependency function is that for any X, Y it +holds that Dep (Y |X) ∈ [0, 1] [5]. Using Property 1, the sum of all FI values +must therefore be in [0, 1], as � +i∈Ωfeat FI(i) = Dep (Y |Ωfeat) ∈ [0, 1]. This +gives an upper bound for the FI values, which is why we can now conclude +that FI(X) ∈ [0, 1] for any RV X. +Relevance. It is essential for interpretability that an FI method is bounded +by known bounds. For example, an FI score of 4.2 cannot be interpreted +properly, when the upper or lower bound is unknown. +Property 4 (Bounds). Every FI(X) with X ∈ Ωfeat is bounded by +Dep (Y |X) +Nvars +≤ FI(X) ≤ Dep (Y |Ωfeat) . +8 + +Proof. The upper bound follows from Properties 1 and 3, as +Dep (Y |Ωfeat) = +� +i∈Ωfeat +FI(i) ≥ FI(X), +where the last inequality follows since FI(i) ∈ [0, 1] for all i ∈ Ωfeat. +The lower bound can be established using the inequality from Equation (7) +within Equation (6). This gives (using Notation (N1)) +FI(X) = +� +S⊆Ωfeat\{X} +w(S, Nvars) · +� +Dep (Y |S ∪ {X}) − Dep (Y |S) +� +≥ w(0, Nvars) · (Dep (Y |∅ ∪ {X}) − Dep (Y |∅)) += 0! · (Nvars − 0 − 1)! +Nvars! +· Dep (Y |X) += Dep (Y |X) +Nvars +. +Relevance. These bounds are useful for upcoming proofs. +Property 5 (Zero FI). For any RV X, it holds that +FI(X) = 0 ⇔ Dep (Y |S ∪ {X}) = Dep (Y |S) for all S ∈ Ωfeat \ {X}. +Proof. ⇐: When Dep (Y |S ∪ {X}) = Dep (Y |S) for all S ∈ Ωfeat \ {X}, it +immediately follows from Equation (6) (with Notation (N1)) that +FI(X) = +� +S⊆Ωfeat\{X} +w(S, Nvars) · +� +Dep (Y |S ∪ {X}) − Dep (Y |S) +� += +� +S⊆Ωfeat\{X} +|S|! · (Nvars − |S| − 1)! +Nvars! +· 0 += 0. +⇒: +Assume that FI(X) = 0. +It follows from the proof of Property 3 +that Dep (Y |S ∪ {X}) − Dep (Y |S) ≥ 0 for every S ⊆ Ωfeat \ {X}. +If +9 + +Dep (Y |S∗ ∪ {X}) − Dep (Y |S∗) > 0 for some given S∗ ∈ Ωfeat \ {X}, it +follows from Equation (6) (with Notation (N1)) that +FI(X) = +� +S⊆Ωfeat\{X} +w(S, Nvars) · +� +Dep (Y |S ∪ {X}) − Dep (Y |S) +� +≥ w(S∗, Nvars) · (Dep (Y |S∗ ∪ {X}) − Dep (Y |S∗)) += |S∗|! · (Nvars − |S∗| − 1)! +Nvars! +· (Dep (Y |S∗ ∪ {X}) − Dep (Y |S∗)) +> 0. +This gives a contradiction with the assumption that FI(X) = 0, thus it is +not possible that such an S∗ exists. This means that Dep (Y |S ∪ {X}) = +Dep (Y |S) for all S ∈ Ωfeat \ {X}. +Relevance. When a feature never contributes any information, it is only fair +that it does not receive any FI. The feature can be removed from the dataset, +as it has no effect on the target variable. On the other hand, when a feature +has an FI of zero, it would be unfair to this feature if it does in fact contribute +information somewhere. It should then be rewarded some FI, albeit small it +should be larger than zero. +Null-independence +The property that a feature receives zero FI, when +Dep (Y |S ∪ {X}) = Dep (Y |S) for all S ∈ Ωfeat \ {X}, is the same notion as +a null player in game theory. Berkelmans et al. [5] show that Dep (Y |X) = 0, +when Y is independent of X. To be a null player requires a stricter definition +of independence, which we call null-independence. Y is null-independent on +X if Dep (Y |S ∪ {X}) = Dep (Y |S) for all S ∈ Ωfeat \ {X}. In other words, +X is null-independent if and only if FI(X) = 0. +Corollary 5.1. Independent feature ̸⇒ null-independent feature. +Proof. Take e.g., the dataset consisting of two binary features X1, X2 ∼ +U({0, 1}) and a target variable Y = X1 · (1 − X2) + X2 · (1 − X1) which is +the XOR of X1 and X2. Individually, the variables do not give any infor- +mation about Y , whereas collectively they fully determine Y . In the proof +of Property 15, we show that this leads to FI(X1) = FI(X2) = +1 +2, whilst +Dep (Y |X1) = Dep (Y |X2) = 0. Thus, X1 and X2 are independent, but not +null-independent. +10 + +Corollary 5.2. Independent feature ⇐ null-independent feature. +Proof. When X is null-independent, it holds that FI(X) = 0. Using Prop- +erty 4, we obtain +0 = FI(X) ≥ Dep (Y |X) +Nvars +⇔ Dep (Y |X) = 0. +Thus, when X is null-independent, it is also independent. +Corollary 5.3. Almost surely constant variables get zero FI. +Proof. If X is almost surely constant (i.e., P(X = x) = 1), it immediately +follows that Dep (Y |S ∪ {X}) = Dep (Y |S) for any S ⊆ Ωfeat \ {X}, as the +distribution of Y is not affected by X. +Property 6 (FI equal to one). When FI(X) = 1, it holds that Dep (Y |X) = +1 and all other features are null-independent. +Proof. As the BP dependency function is bounded by [0, 1] [5], it follows +from Property 1 that � +i∈Ωfeat FI(i) ≤ 1. Noting that each FI must be in +[0, 1] due to Property 3, we find that +FI(X) = 1 ⇒ FI(X′) = 0 for all X′ ∈ Ωfeat \ {X}. +Thus all other features are null-independent. Next, we show that Dep (Y |X) = +1 must also hold, when FI(X) = 1. Assume that Dep (Y |X) < 1. Using +11 + +Equation (6) (with Notations (N1) and (N2)) we find that +1 = FI(X) = +� +S⊆Ωfeat\{X} +w(S, Nvars) · D(X, Y, S) += +� +S⊆Ωfeat\{X}:|S|>0 +(w(S, Nvars) · D(X, Y, S)) + w(∅, Nvars) · D(X, Y, ∅) +≤ +� +S⊆Ωfeat\{X}:|S|>0 +(w(S, Nvars) · (1 − 0)) + w(∅, Nvars) · (Dep (Y |X) − 0) +< +� +S⊆Ωfeat\{X} +w(S, Nvars) += +Nvars−1 +� +k=0 +�Nvars − 1 +k +� +· k! · (Nvars − k − 1)! +Nvars! += +Nvars−1 +� +k=0 +(Nvars − 1)! +k! · (Nvars − 1 − k)! · k! · (Nvars − k − 1)! +Nvars! += +Nvars−1 +� +k=0 +1 +Nvars += 1. +Note that the inequality step follows from the range of the BP dependency +function (i.e., [0, 1]). The largest possible addition is when Dep (Y |S ∪ {X})− +Dep (Y |S) = 1 − 0 = 1. This result gives a contradiction, as 1 < 1 cannot +be true, which means that Dep (Y |X) = 1. +Relevance. When a variable gets an FI of one, the rest of the variables should +be zero. Additionally, it should mean that this variable contains the necessary +information to fully determine Y , which is why Dep (Y |X) = 1 should hold. +Property 7. Dep (Y |X) = 1 ̸⇒ FI(X) = 1. +Proof. As counterexample, examine the case where there are multiple vari- +ables that fully determine Y . Properties 1 and 3 must still hold. Thus, if FI +12 + +is one for every variable that fully determines Y , we get +� +i∈Ωfeat +FI(i) ≥ 1 + 1 ̸= 1 = Dep (Y |Ωfeat) , +which is a contradiction. +Relevance. This property is important for interpretation of the FI score. +When FI(X) ̸= 1, it cannot be automatically concluded that Y is not fully +determined by X. +If Y is fully determined by X, we call X fully informative, as it gives all +information that is necessary to determine Y . +Property 8 (Max FI when fully informative). If X is fully informative, it +holds that FI(i) ≤ FI(X) for any i ∈ Ωfeat. +Proof. Assume that there exists a feature i such that FI(i) > FI(X), when +Y is fully determined by X. To attain a higher FI, somewhere in the sum +of Equation (6), a higher gain must be made by i compared to X. Observe +that for any S ⊆ Ωfeat \ {i, X} it holds that +Dep (Y |S ∪ {i}) − Dep (Y |S) ≤ 1 − Dep (Y |S) += Dep (Y |S ∪ {X}) − Dep (Y |S) . +For any S ⊆ Ωfeat \ {i} with X ∈ S, it holds that +Dep (Y |S ∪ {i}) − Dep (Y |S) = Dep (Y |S ∪ {i}) − 1 += 0. +The last step follows from Equation (7), as the dependency function is in- +creasing, thus Dep (Y |S ∪ {i}) = 1. In other words, no possible gain can be +achieved with respect to X in the Shapley values. Therefore, it cannot hold +that FI(i) > FI(X). +Relevance. Whenever a variable fully determines Y , it should attain the high- +est FI. What would an FI higher than such a score mean? It gives more +information than the maximal information? When this property would not +hold, it would result in a confusing and difficult interpretation process. +13 + +Property 9 (Limiting the outcome space). For any measurable function f +and RV X, replacing X with f(X) never increases the assigned FI to this +variable. +Proof. The BP dependency function is non-increasing under functions of X +[5]. This means that for any measurable function g, it holds that +Dep (Y |g(X)) ≤ Dep (Y |X) . +Choose g to be the function that maps the union of any feature set S and +the original RV X to the union of S and the replacement f(X). In other +words g(S ∪ {X}) = S ∪ {f(X)} for any feature set S. It then follows that: +Dep (Y |S ∪ {f(X)}) = Dep (Y |g(S ∪ {X})) ≤ Dep (Y |S ∪ {X}) , +and +Dep (Y |S ∪ {f(X)}) − Dep (Y |S) ≤ Dep (Y |S ∪ {X}) − Dep (Y |S) +for any S ⊆ Ωfeat \ {X}. Thus, using Equation (6), we can conclude that +replacing X with f(X) never increases the assigned FI. +Relevance. This is an important observation for preprocessing. Whenever a +variable is binned, it would receive less (or equal) FI when less bins are used. +It could also potentially provide a useful upper bound, when the FI is already +known before replacing X with f(X). +Corollary 9.1. For any measurable function f and RV X, when X = f(X′) +for another RV X′, replacing feature X by feature X′ will never decrease the +assigned FI. +Proof. When X = f(X′) holds, it follows again (similar to Property 9) that +Dep (Y |S ∪ {X}) = Dep (Y |S ∪ {f(X′)}) ≤ Dep (Y |S ∪ {X′}) +for any S ⊆ Ωfeat\{X}. Therefore, using Equation (6), observe that replacing +X with X′ never decreases the assigned FI. +Shapley values have additional properties when the characteristic function v +is subadditive and/or superadditive [49]. We show that our function, defined +by Equation (5), is neither. +14 + +Property 10 (Neither subadditive nor superadditive). Our characteristic +function v(S) = Dep (Y |S) is neither subadditive nor superadditive. +Proof. Consider the following two counterexamples. +Counterexample subadditive: A function f is subadditive if for any S, T ∈ Ωfeat +it holds that +f(S ∪ T) ≤ f(S) + f(T). +Examine the dataset consisting of two binary features X1, X2 ∼ U({0, 1}) +and a target variable Y = X1 · (1 − X2) + X2 · (1 − X1) which is the XOR of +X1 and X2. Both X1 and X2 do not individually give any new information +about the distribution of Y , thus v(X1) = v(X2) = 0 (see properties of the +BP dependency function [5]). However, collectively they fully determine Y +and thus v(X1∪X2) = 1. We can therefore conclude that v is not subadditive, +as +v(X1 ∪ X2) = 1 ̸≤ 0 + 0 = v(X1) + v(X2). +Counterexample superadditive: A function f is superadditive if for any S, T ∈ +Ωfeat it holds that +f(S ∪ T) ≥ f(S) + f(T). +Consider the dataset consisting of two binary features X ∼ U({0, 1}) and a +clone Xclone := X, where the target variable Y is defined as Y := X. Note +that both X and Xclone fully determine Y , thus v(X) = v(Xclone) = 1 (see +properties of the BP dependency function [5]). Combining X and Xclone also +fully determines Y , which leads to: +v(X ∪ Xclone) = 1 ̸≥ 1 + 1 = v(X) + v(Xclone). +Thus, v is also not superadditive. +Relevance. If the characteristic function v is subadditive, it would hold that +FI(X) ≤ v(X) for any X ∈ Ωfeat. When v is superadditive, it follows that +FI(X) ≥ v(X) for any X ∈ Ωfeat. +This is sometimes also referred to as +individual rationality, which means that no player receives less, than what +he could get on his own. This makes sense in a game-theoretic scenario with +human players that can decide to not play when one could gain more by not +15 + +cooperating. In our case, features do not have a free will, which makes this +property not necessary. The above proof shows that v is in our case neither +subadditive nor superadditive, which is why we cannot use their corresponding +bounds. +Property 11 (Adding features can increase FI). When an extra feature is +added to the dataset, the FI of X can increase. +Proof. Consider the previously mentioned XOR dataset, where X1, X2 ∼ +U({0, 1}) and Y = X1 · (1 − X2) + X2 · (1 − X1). If at first, X2 was not in +the dataset, the FI of X1 would be zero, as Dep (Y |X1) = 0. However, if X2 +is added to the dataset, the FI of X1 increases to 1 +2 (see Property 15). The +FI of a feature can thus increase if another feature is added. +Property 12 (Adding features can decrease FI). When an extra feature is +added to the dataset, the FI of X can decrease. +Proof. Consider the dataset given by X ∼ U({0, 1}) and Y := X. It im- +mediately follows that FI(X) = 1. +However, when a clone is introduced +(Xclone := X), it holds that FI(X) = FI(Xclone), because of Property 8. Ad- +ditionally, it follows from Property 1 that FI(X) + FI(Xclone) = 1. Thus, +FI(X) = 1 +2, and the FI of a variable can therefore be decreased if another +variable is added. +Relevance. It is important to observe that the FI of a variable is dependent +on the other features (Properties 11 and 12). Adding or removing features +could change the FI, which one needs to be aware of. +Property 13 (Cloning does not increase FI). For any RV X ∈ Ωfeat, adding +an identical variable Xclone := X (cloning) to the dataset, does not increase +the FI of X. +Proof. Let FIwith clone(X) denote the FI of X after the clone Xclone is added. +16 + +Using Equation (6) (with Notations (N1) and (N2)), we find +FIwith clone(X) = +� +S⊆Ωfeat∪{Xclone}\{X} +w(S, Nvars + 1) · D(X, Y, S) +(a) += +� +S⊆Ωfeat∪{Xclone}\{X}:Xclone∈S +w(S, Nvars + 1) · D(X, Y, S) ++ +� +S⊆Ωfeat∪{Xclone}\{X}:Xclone̸∈S +w(S, Nvars + 1) · D(X, Y, S) +(b)= +� +S⊆Ωfeat∪{Xclone}\{X}:Xclone∈S +w(S, Nvars + 1) · 0 ++ +� +S⊆Ωfeat∪{Xclone}\{X}:Xclone̸∈S +w(S, Nvars + 1) · D(X, Y, S) += +� +S⊆Ωfeat\{X} +w(S, Nvars + 1) · D(X, Y, S). +Equality (a) follows by splitting the sum over all subsets of Ωfeat ∪ {Xclone} \ +{X} whether Xclone is part of the subset or not. Adding X to a subset that +already contains the clone Xclone does not change the BP dependency func- +tion, which is why Equality (b) follows. The takeaway from this derivation +is that the sum over all subsets S ⊆ Ωfeat ∪ {Xclone} \ {X} reduces to the +sum over S ⊆ Ωfeat \ {X}. +Comparing the new FIwith clone(X) with the original FI(X) gives +FI(X) − FIwith clone(X) = +� +S⊆Ωfeat\{X} +w(S, Nvars) · D(X, Y, S) +− +� +S⊆Ωfeat\{X} +w(S, Nvars + 1) · D(X, Y, S). +Using Notation (N1), we find that +w(S, Nvars + 1) +w(S, Nvars) += +|S|!·(Nvars+1−|S|−1)! +(Nvars+1)! +|S|!·(Nvars−|S|−1)! +Nvars! += Nvars − |S| +Nvars + 1 < 1, +17 + +thus FI(X) − FIwith clone(X) ≥ 0 with equality if and only if FI(X) = 0. +Therefore, we can conclude that cloning a variable cannot increase the FI of +X and will decrease the FI when X is null-independent. +Relevance. We consider this a natural property of a good FI method, as no +logical reason can be found why adding the exact same information would +lead to an increase in FI for the original variable. The information a variable +contains only becomes less valuable, as it becomes common knowledge. +Property 14 (Order does not change FI). The order of the features does not +affect the individually assigned FI. Consider the datasets [X1, X2, . . . , XNvars] +and [Z1, Z2, . . . , ZNvars], where Zπ(i) = Xi for some permutation π. It holds +that FI(Xi) = FI(Zπ(i)) for any i ∈ {1, . . . , Nvars}. +Proof. Note that the order of features nowhere plays a roll in the definition +of BP-FI (Equation (6)). The BP dependency function is also independent +of the given order, which is why this property trivially holds. +Relevance. This is a very natural property of a good FI. Consider what would +happen if the FI is dependent on the order in the dataset. Should all possible +orders be evaluated and averaged to receive a final FI? We cannot find any +arguments why someone should want FI to be dependent on the order of +features. +Datasets +Next, we consider a few datasets, where we derive the theoretical outcome +for the BP-FI. These datasets are also used in Section 4.3 to test FI methods. +It is very hard to evaluate FI methods, as the ground truth is often unknown. +However, we believe that the FI outcomes on these datasets are all natural +and defendable. However, it remains subjective what one considers to be the +‘correct’ FI values. +Property 15 (XOR dataset). Consider the following dataset consisting of +two binary features X1, X2 ∼ U({0, 1}) and a target variable Y = X1 · (1 − +X2) + X2 · (1 − X1) which is the XOR of X1 and X2. It holds that +FI(X1) = FI(X2) = 1 +2. +Proof. Observe that Dep (Y |X1) = Dep (Y |X2) = 0 and Dep (Y |X1 ∪ X2) = +18 + +1. With Equation (6), it follows that +FI(X1) = +� +S⊆{1,2}\X1 +|S|! · (1 − |S|)! +2! +· (Dep (Y |S ∪ X1) − Dep (Y |S)) += |{∅}|! · (1 − |{∅}|)! +2! +· (Dep (Y |{∅} ∪ X1) − Dep (Y |{∅})) ++ |{X2}|! · (1 − |{X2}|)! +2! +· (Dep (Y |X1 ∪ X2) − Dep (Y |X2)) += 1 +2 · (Dep (Y |X1) − 0) + 1 +2 · (Dep (Y |X1 ∪ X2) − Dep (Y |X2)) += 1 +2 · 0 + 1 +2 · (1 − 0) += 1 +2. +Using Property 1, it follows that FI(X2) = 1 − FI(X1) = 1 +2. +Relevance. This XOR formula is discussed and used to test FI methods in +[17]. However, they only test for equality (FI(X1) = FI(X2)), not the specific +value. Due to symmetry, we would also argue that both X1 and X2 should +get the same FI, as they fulfill the same role. Together, they fully determine +Y , which is why the total FI should be one (see Property 6). Dividing this +equally amongst the two variables, gives a logical desirable FI outcome of 1 +2 +for each variable. +Property 16 (Probability dataset). Consider the following dataset consist- +ing of Y = ⌊XS/2⌋ and Xi = Zi + (S − 1) with Zi ∼ U ({0, 2}) for i = 1, 2 +and P(S = 1) = p, P(S = 2) = 1 − p. It holds that +FI(X1) = p and FI(X2) = 1 − p. +19 + +Proof. Observe that by Equation (4) +UD (X1, Y ) = +� +x1∈{0,1,2,3} +pX1(x1) · +� +y∈{0,1} +��pY |X1=x1(y) − pY (y) +�� += +� +x1∈{0,2} +pX1(x1) · +� +y∈{0,1} +����pY |X1=x1(y) − 1 +2 +���� ++ +� +x1∈{1,3} +pX1(x1) · +� +y∈{0,1} +����pY |X1=x1(y) − 1 +2 +���� += +� +x1∈{0,2} +p +2 · +�����1 − 1 +2 +���� + +����0 − 1 +2 +���� +� ++ +� +x1∈{1,3} +1 − p +2 +· +� +y∈{0,1} +|pY (y) − pY (y)| += p. +Similarly, it follows that UD (X2, Y ) = 1 − p. +UD (Y, Y ) = +� +y′∈{0,1} +pY (y′) · +� +y∈{0,1} +��pY |Y =y′(y) − pY (y) +�� += +� +y′∈{0,1} +1 +2 · +�����1 − 1 +2 +���� + +����0 − 1 +2 +���� +� += 1. +From Equation (3), it follows that Dep (Y |X1) = p and Dep (Y |X2) = 1 − +p. +Additionally, note that knowing X1 and X2 fully determines Y , thus +20 + +Dep (Y |X1 ∪ X2) = 1. With Equation (6), we now find +FI(X1) = +� +S⊆{X1,X2}\X1 +|S|! · (1 − |S|)! +2! +· (Dep (Y |S ∪ X1) − Dep (Y |S)) += |{∅}|! · (1 − |{∅}|)! +2! +· (Dep (Y |{∅} ∪ X1) − Dep (Y |{∅})) ++ |{X2}|! · (1 − |{X2}|)! +2! +· (Dep (Y |X1 ∪ X2) − Dep (Y |X2)) += 1 +2 · (Dep (Y |X1) − 0) + 1 +2 · (Dep (Y |X1 ∪ X2) − Dep (Y |X2)) += 1 +2 · (p − 0) + 1 +2 · (1 − (1 − p)) += p +2 + p +2 = p. +Using Property 1, it follows that FI(X2) = 1 − FI(X1) = 1 − p. +Relevance. At first glance, it is not immediately clear why these FI values +are natural, which is why we discuss this dataset in more detail. S can be +considered a selection parameter that determines if X1 or X2 is used for Y +with probability p and 1 − p, respectively. Xi is constructed in such a way +that it is uniformly drawn from {0, 2} or {1, 3} depending on S. However, +as Y = ⌊XS/2⌋, it holds that XS = 0 and XS = 1 give the same outcome +for Y . The same holds for XS = 2 and XS = 3. Therefore, note that the +distribution of Y is independent of the selection parameter S. Knowing X1 +gives the following information. First, S can be derived from the value of +X1. When X1 ∈ {0, 2} it must hold that S = 1, and if X1 ∈ {1, 3} it follows +that S = 2. Second, when S = 1 it means that Y is fully determined by X1. +If S = 2, knowing that X1 = 1 or X1 = 3 does not provide any additional +information about Y . With probability p knowing X1 will fully determine +Y , whereas with probability 1 − p, it will provide no information about the +distribution of Y . The outcome FI(X1) = p, is therefore very natural. The +same argumentation applies for X2, which leads to FI(X2) = 1 − p. +21 + +4 +Comparing with existing methods +In the previous section, we showed that BP-FI has many desirable proper- +ties. Next, we evaluate for a large collection of FI methods if the properties +hold for several synthetic datasets. Note that these datasets can only be +used as counterexample, not as proof of a property. First, we discuss the +in Section 4.1 the FI methods that are investigated. Second, we give the +datasets (Section 4.2) and explain how they are used to test the properties +(Section 4.3). The results are discussed in Section 4.4. +4.1 +Alternative FI methods +A wide range of FI methods have been suggested for all kinds of situations. It +is therefore first necessary to discuss the major categorical differences between +them. +Global vs. +local +An important distinction to make for FI methods is +whether they are constructed for local or global explanations. +Global FI +methods give an importance score for each feature over the entire dataset, +whereas local FI methods explain which variables were important for a single +example [18]. The global and local scores do not have to coincide: “features +that are globally important may not be important in the local context, and +vice versa” [46]. This research is focussed on global FI methods, but some- +times a local FI approach can be averaged out to obtain a global FI. For +example, in [34] a local FI method is introduced called Tree SHAP. It is also +used globally, by averaging the absolute values of the local FI. +Model-specific vs. +model-agnostic +A distinction within FI methods +can be made between model-specific and -agnostic methods. Model-specific +methods aim to find the FI using a prediction model such as a neural network +or random forest, whereas model-agnostic methods do not use a prediction +model. The BP-FI is model-agnostic, which therefore gives insights into the +dataset. Whenever a model-specific method is used, the focus lies more on +gaining information about the prediction model, not the dataset. In our tests, +we use both model-specific and -agnostic methods. +Classification vs. regression +Depending on the exact dataset, the target +variable is either categorical or numerical, which is precisely the difference +between classification and regression. Not all existing FI methods can handle +both cases. In this research, we generate synthetic classification datasets, so +22 + +we only examine FI methods that are intended for these cases. An additional +problem with regression datasets, is that continuous variables need to be +converted to discrete bins. This conversion could drastically change the FI +scores, which makes it harder to draw fair conclusions. +Collection +We have gathered the largest known collection of FI methods +from various sources [2, 4, 6, 8, 11–13, 17, 18, 20, 22, 28, 35, 38, 40, 42, 43, +45, 47, 48, 57, 58] or implemented them ourselves. This has been done with +the following policy: Whenever code of a classification FI method was avail- +able in R or Python or the implementation was relatively straightforward, it +was added to the collection. This resulted in 196 base methods and 468 to- +tal methods, as some base methods can be combined with multiple machine +learning approaches or selection objectives, see Table 1. However, beware +that most methods also contain additional parameters, which are not inves- +tigated in this research. The default values for these parameters are always +used. +4.2 +Synthetic datasets +Next, we briefly discuss the datasets that are used to test the properties de- +scribed in Section 3 for alternative FI methods. In Appendix A, we introduce +each dataset and explain how they are generated. To draw fair conclusions, +the datasets are not drawn randomly, but fixed. To give an example of how +we do generate a dataset, we examine Dataset 1 Binary system (see Ap- +pendix A), where the target variable Y is defined as Y := �3 +i=1 2i−1 ·Xi with +Xi ∼ U ({0, 1}) for all i ∈ {1, 2, 3}. To get interpretable results, we draw each +combination of X and Y values the same number of times. An example can +be seen in Table 2. For most datasets, we draw 1,000 samples in total. How- +ever Datasets 6 and 7 consist of 2,000 samples to ensure null-independence. +The datasets have been selected to be computationally inexpensive and to +test many properties (see Section 4.3) with a limited number of datasets. An +overview of the generated datasets can be found in Table 3 including the cor- +responding outcome of BP-FI. Appendix A provides more technical details +about the features and target variables. +4.3 +Property evaluation +In Section 4.1, we gathered a collection of existing FI methods. +In this +section, we evaluate if these FI methods have the same desirable and proven +properties of the BP-FI method (see Section 3). Due to the sheer number of +FI methods (468), it is unfeasible to prove each property for every method. +23 + +Table 1: +All evaluated FI methods: +List of all FI methods that +are evaluated in the experiments. +The colored methods work in com- +bination with multiple options: +Logistic RegressionI, II, III, RidgeI, II, Linear +RegressionI, II, LassoI, II, SGD ClassifierI, III, MLP ClassifierI, II, K Neighbors ClassifierI, II, +Gradient Boosting ClassifierI, II, IV, AdaBoost ClassifierI, II, Gaussian NBI, II, Bernoulli +NBI, II, Linear Discriminant AnalysisI, II, Decision Tree ClassifierI, II, IV, V, Random +Forest ClassifierI, II, IV, V, SVCI, CatBoost ClassifierI, II, LGBM ClassifierI, II, IV, XGB +ClassifierI, II, IV, VII, XGBRF ClassifierI, II, IV, VII, ExtraTree ClassifierIV, V, ExtraTrees +ClassifierIV, V, plsdaVI, splsdaVI, giniVIII, entropyVIII, NN1IX, NN2IX. This leads to a +total of 468 FI methods from various sources [2, 4, 6, 8, 11–13, 17, 18, 20, 22, +28, 35, 38, 40, 42, 43, 45, 47, 48, 57, 58] or self-implemented. +Feature Importance methods +1. AdaBoost Classifier +2. Random Forest ClassifierVIII +3. Extra Trees ClassifierVIII +4. Gradient Boosting Classifier +5. SVR absolute weights +6. EL absolute weights +7. Permutation Importance ClassifierI +8. PCA sum +9. PCA weighted +10. chi2 +11. f classif +12. mutual info classif +13. KL divergence +14. R Mutual Information +15. Fisher Score +16. FeatureVec +17. R Varimp Classifier +18. R PIMP Classifier +19. Treeinterpreter ClassifierV +20. DIFFI +21. Tree ClassifierIV +22. Linear ClassifierIII +23. Permutation ClassifierI +24. Partition ClassifierI +25. Sampling ClassifierI +26. Kernel ClassifierI +27. Exact ClassifierI +28. RFI ClassifierI +29. CFI ClassifierI +30. Sum ClassifierVI +31. Weighted X ClassifierVI +32. Weighted Y ClassifierVI +33. f oneway +34. alexandergovern +35. pearsonr +36. spearmanr +37. pointbiserialr +38. kendalltau +39. weightedtau +40. somersd +41. linregress +42. siegelslopes +43. theilslopes +44. multiscale graphcorr +45. booster weightVII +46. booster gainVII +47. booster coverVII +48. snn +49. knn +50. bayesglm +51. lssvmRadial +52. rocc +53. ownn +54. ORFpls +55. rFerns +56. treebag +57. RRF +58. svmRadial +59. ctree2 +60. evtree +61. pda +62. rpart +63. cforest +64. svmLinear +65. xyf +66. C5.0Tree +67. avNNet +68. kknn +69. svmRadialCost +70. gaussprRadial +71. FH.GBML +72. svmLinear2 +73. bstSm +74. LogitBoost +75. wsrf +76. plr +77. xgbLinear +78. rf +79. null +80. protoclass +81. monmlp +82. Rborist +83. mlpWeightDecay +84. svmRadialWeights +85. mlpML +86. ctree +87. loclda +88. sdwd +89. mlpWeightDecayML +90. svmRadialSigma +91. bstTree +92. dnn +93. ordinalRF +94. pda2 +95. BstLm +96. RRFglobal +97. mlp +98. rpart1SE +99. pcaNNet +100. ORFsvm +101. parRF +102. rpart2 +103. gaussprPoly +104. C5.0Rules +105. rda +106. rbfDDA +107. multinom +108. gaussprLinear +109. svmPoly +110. knn +111. treebag +112. RRF +113. ctree2 +114. evtree +115. pda +116. rpart +117. cforest +118. xyf +119. C5.0Tree +120. kknn +121. gaussprRadial +122. LogitBoost +123. wsrf +124. xgbLinear +125. rf +126. null +127. monmlp +128. Rborist +129. mlpWeightDecay +130. mlpML +131. ctree +132. mlpWeightDecayML +133. dnn +134. pda2 +135. RRFglobal +136. mlp +137. rpart1SE +138. parRF +139. rpart2 +140. gaussprPoly +141. C5.0Rules +142. rbfDDA +143. multinom +144. gaussprLinear +145. binaryConsistency +146. chiSquared +147. cramer +148. gainRatio +149. giniIndex +150. IEConsistency +151. IEPConsistency +152. mutualInformation +153. roughsetConsistency +154. ReliefFeatureSetMeasure +155. symmetricalUncertain +156. IteratedEstimatorII +157. PermutationEstimatorII +158. KernelEstimatorII +159. SignEstimatorII +160. ShapleyI +161. BanzhafI +162. RF +163. GarsonIX +164. VIANNIX +165. LOFOIX +166. Relief +167. ReliefF +168. RReliefF +169. fit criterion measure +170. f ratio measure +171. gini index +172. su measure +173. spearman corr +174. pearson corr +175. fechner corr +176. kendall corr +177. chi2 measure +178. anova +179. laplacian score +180. information gain +181. modified t score +182. MIM +183. MRMR +184. JMI +185. Add: CIFE +186. CMIM +187. ICAP +188. DCSF +189. CFR +190. MRI +191. IWFS +192. NDFS +193. RFS +194. SPEC +195. MCFS +196. UDFS +197. R2 +198. DC +199. BCDC +200. AIDC +201. HSIC +202. BP-FI +Legend +1-12 +sklearn +[42] +13-20 +Additional methods +[2, 8, 11, 18, 22, 38, 45, 48] +21-27 +shap explainer +[35] +28-29 +Relative feature importance +[6] +30-32 +R vip +[20] +33-44 +scipy stats +[58] +45-47 +booster classifier +[12] +48-109 +R caret classifier +[28] +110-144 +R firm classifier +[20] +145-155 +R FSinR Classifier +[4] +156-159 +Sage Classifier +[13] +160-161 +QII Averaged Classifier +[57] +162-165 +Rebelosa Classifier +[47] +166-168 +Relief Classifier +[40] +169-196 +ITMO +[43] +197-201 +Sunnies +[17] +202 +BP-FI +- +24 + +Table 2: Fixed draw: Example of how the datasets are drawn. Instead of +drawing each possible outcome uniformly at random, we draw each combina- +tion an equal fixed number of times. +Outcome +# Drawn +X1 +X2 +X3 +Y +Fixed +Uniform +0 +0 +0 +0 +125 +133 +0 +0 +1 +4 +125 +129 +0 +1 +0 +2 +125 +121 +0 +1 +1 +6 +125 +109 +1 +0 +0 +1 +125 +136 +1 +0 +1 +5 +125 +124 +1 +1 +0 +3 +125 +115 +1 +1 +1 +7 +125 +133 +Instead, we devise tests to find counterexamples of these properties using +generated datasets (see Section 4.2). Due to the number of tests (18), we +only discuss the parts that are not straightforward, as most test directly +measure the corresponding property. An overview of each test can be found +in Appendix B. A summary of the tests can be found in Table 4, where it is +outlined for each test which property is tested on which datasets. +Computational errors +To allow for computational errors, we tolerate a +margin of ǫ = 0.01 in each test. If, e.g., an FI value should be zero, a score of +0.01 or −0.01 is still considered a pass, whereas an FI value of 0.05 is counted +as a fail. Usually, this works in the favor of the FI method. However, in Test 9 +we evaluate if the FI method assigns zero FI to variables that are not null- +independent. In this case, we consider |FI(X)| ≤ ǫ to be zero, as the datasets +are constructed in such a way that variables are either null-independent or +far from being null-independent. +Running time +We limit the running time to one hour per dataset on an +i7-12700K processor, whilst four algorithms are running simultaneously. The +datasets consist of a small number of features with a very limited outcome +space and the number of samples is either 1,000 or 2,000, which is why one +hour is a reasonable amount of time. +NaN or infinite values +In some cases, an FI method assigns NaN or ±∞ +to a feature. How we handle these values depends on the test. +E.g., we +consider NaN to fall outside the range [0, 1] (Tests 4 and 55), but when we +evaluate if the sum of FI values remains stable (Test 2) or if two symmetric +25 + +Table 3: Overview of datasets: An overview of the generated datasets and the corresponding BP-FI outcome. +The details of these datasets can be found in Appendix A. They are used to evaluate if existing FI methods adhere +to the same properties as BP-FI (see Section 4.3). +Dataset +Variables +BP-FI outcome +Binary system +1. +- base +(X1, X2, X3) +(0.333, 0.333, 0.333) +2. +- clone +(Xclone +1 +, X1, X2, X3) +(0.202, 0.202, 0.298, 0.298) +3. +- clone + 1x fully info. +(Xclone +1 +, X1, X2, X3, Xfull +4 ) +(0.148, 0.148, 0.183, 0.183, 0.338) +4. +- clone + 2x fully info. +(Xclone +1 +, X1, X2, X3, Xfull +4 , Xfull +5 ) +(0.117, 0.117, 0.136, 0.136, 0.248, 0.248) +5. +- clone + 2x fully info. (different order) +(X3, Xfull +4 , Xfull +5 , Xclone +1 +, X1, X2) +(0.136, 0.248, 0.248, 0.117, 0.117, 0.136) +Null-independent system +6. +- base +(Xnull-indep. +1 +, Xnull-indep. +2 +, Xnull-indep. +3 +) +(0.000, 0.000, 0.000) +7. +- constant variable +(Xnull-indep. +1 +, Xnull-indep. +2 +, Xnull-indep. +3 +, Xconst, null-indep. +4 +) +(0.000, 0.000, 0.000, 0.000) +Increasing bins +8. +- base +(Xbins=10 +1 +, Xbins=50 +2 +, Xbins=1,000, full +3 +) +(0.297, 0.342, 0.361) +9. +- more variables +(Xbins=10 +1 +, Xbins=20 +2 +, Xbins=50 +3 +, Xbins=100 +4 +, Xbins=1,000, full +5 +) +(0.179, 0.193, 0.204, 0.208, 0.216) +10. +- clone (different order) +(Xbins=1,000, full +3 +, Xbins=50 +2 +, Xbins=10 +1 +, Xclone, full +3 +) +(0.262, 0.253, 0.223, 0.262) +Dependent system +11. +- 1x fully info. +(Xfull +1 , Xnull-indep. +2 +, Xnull-indep. +3 +) +(1.000, 0.000, 0.000) +12. +- 2x fully info. +(Xfull +1 , Xfull +2 , Xnull-indep. +3 +) +(0.500, 0.500, 0.000) +13. +- 3x fully info. +(Xfull +1 , Xfull +2 , Xfull +3 ) +(0.333, 0.333, 0.333) +XOR dataset +14. +- base +(X1, X2) +(0.500, 0.500) +15. +- single variable +(Xnull-indep. +1 +) +(0.000) +16. +- clone +(Xclone +1 +, X1, X2) +(0.167, 0.167, 0.667) +17. +- null-independent +(X1, X2, Xnull-indep. +3 +) +(0.500, 0.500, 0.000) +Probability dataset +18-28. +- for p ∈ {0, 0.1, . . . , 1} +(X1, X2) +(p, 1 − p) +26 + +Table 4: Overview of experiments: To evaluate if existing FI methods have the same properties as the BP-FI, +we use the tests from Appendix B on the datasets from Appendix A. ✓means that the test is performed on this +dataset. ↕(i) denotes that this dataset is used as baseline or in conjunction with dataset i. The details of the tests +and datasets can be found in the appendix. +Test +Evaluates: +Dataset (Appendix A) +(Appendix B) +Property/Corollary +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +13 +14 +15 +16 +17 +18 +19 +20 +21 +22 +23 +24 +25 +26 +27 +28 +1 +1 +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +2 +1.1 +↕(2-5) +✓ +✓ +✓ +✓ +↕(7) +✓ +↕(9-10) +✓ +✓ +↕(12-13) +✓ +✓ +↕(16-17) +✓ +✓ +3 +2 +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +4 +3 +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +5 +3 +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +6 +4 +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +7 +4 +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +8 +5 +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +9 +5 +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +10 +6 +✓ +11 +8 +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +12 +9 +✓ +✓ +✓ +13 +11 +↕(2) +✓↕(3) +✓↕(4) +✓ +↕(7) +✓ +↕(9-10) +✓ +✓ +✓↕(16-17) +↕(14) +✓ +✓ +14 +12 +↕(2) +✓↕(3) +✓↕(4) +✓ +↕(7) +✓ +↕(9-10) +✓ +✓ +✓↕(16-17) +↕(14) +✓ +✓ +15 +13 +↕(2) +✓ +↕(10) +✓ +↕(16) +✓ +16 +14 +↕(5) +✓ +↕(28) +↕(27) +↕(26) +↕(25) +↕(24) +✓ +✓ +✓ +✓ +✓ +17 +15 +✓ +✓ +18 +16 +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +27 + +features receive the same FI (Test 3), we consider twice NaN or twice ±∞ +to be the same. +Property 9 (Limiting the outcome space) +Property 9 states that ap- +plying any measurable function f to a RV X cannot increase the FI. In other +words, FI(X) ≥ FI(f(X)) holds. This property is tested using Datasets 8 +to 10 (see Table 4). These datasets contain variables that are the outcome +of binning the target variable using different number of bins. This is how +Property 9 is tested, as it should hold that FI(Xi) ≥ FI(Xj), whenever Xi +has more bins than Xj. +Properties 11 and 12 (Adding features can increase/decrease FI) +In all other tests, the goal is to find a counterexample of the property. How- +ever, Tests 13 and 14 are designed to evaluate if a feature gets an increased/de- +creased FI when a feature is added. This increase/decrease should be more +than ǫ. The datasets are chosen in such a way that both an increase and de- +crease could occur (according to the BP-FI). Only for these tests, we consider +the test failed if no counterexample (increase/decrease) is found. +4.4 +Evaluation results +An overview of the general results can be seen in Table 5, where the number +of methods that pass and fail is given per test. Next, we highlight additional +insights into the results of the experiments. +Best performing methods +The top 20 FI methods that pass the most +tests are given in Table 6. Out of 18 tests, the BP-FI passes all tests, which is +as expected as we have proven in Section 3 that the BP-FI actually has these +properties. Classifiers from R FSinR Classifier and ITMO fill 11 of the top +20 spots. Out of 11 R FSinR Classifier methods, six are in the top 20, which is +quite remarkable. However, observe that the gap between the BP-FI method +and the second best method is 18−11 = 7 passed tests. Additionally, 424 out +of 468 methods fail more than half of the tests. Figure 1 shows how frequently +each number of passed tests occurs. A detailed overview of where each top +20 method fails, can be seen in Table 5. Note again that in Tests 13 and 14 +it is considered a fail if adding features never increase or decrease the FI, +respectively. It could be that these methods are in fact capable of increasing +or decreasing, but for some reason do not with our datasets. +Strikingly, +most of these methods perform bad on the datasets with a desirable outcome +28 + +Table 5: Overview of the results: Each FI method is evaluated using +the tests outlined in Appendix B, which evaluates if the method adheres to +the same properties as the BP-FI (see Section 3). This table summarizes out +of 468 FI methods how many pass or fail the test. A distinction is made for +the top 20 passing methods. Failing the test means that a counterexample +is found. Note that passing the test does not ‘prove’ that the FI method +actually has the property. No result indicates that the test could not be +executed, because the running time of the FI method was too long or an +error occurred. +Test +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +13 +14 +15 +16 +17 +18 +Overall +# Passed +1 +92 +45 +438 +200 +97 +132 +283 +97 +31 +141 +241 +243 +314 +365 +172 +13 +5 +# Failed +466 +369 +421 +29 +267 +370 +335 +184 +370 +413 +326 +98 +216 +145 +58 +288 +421 +459 +# No result +1 +7 +2 +1 +1 +1 +1 +1 +1 +24 +1 +129 +9 +9 +45 +8 +34 +4 +Top 20 +# Passed +1 +10 +15 +20 +19 +7 +18 +18 +2 +13 +17 +20 +4 +6 +20 +17 +2 +4 +# Failed +19 +10 +5 +0 +1 +13 +2 +2 +18 +7 +3 +0 +16 +14 +0 +3 +17 +16 +# No result +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +1 +0 +(Tests 17 and 18). Adding a variable without additional information (Test 2), +also often leads to a change in total FI. +Test 1 +In this test, it is evaluated if the sum of FI values is the same as +the sum for BP-FI. At first, this seems a rather strict requirement. However, +it holds for all datasets that were used that Dep (Y |Ωfeat) is either zero or +one. Thus, we essentially evaluate if the sum of FI is equal to one, when +all variables collectively fully determine Y and zero if all variables are null- +independent. The tests show that no FI method is able to pass this test, +except for the BP-FI. To highlight some of the methods that came close: +162. Rebelosa Classifier RF, 2. Random Forest Classifier entropy, 2. Ran- +dom Forest Classifier gini only fail for the datasets where the sum should be +zero (because of null-independence) and 1. AdaBoost Classifier only does +not pass on three of the four datasets based on the XOR function (see Ap- +pendix A), where the sum should be one, but was zero instead. FI method 51. +lssvmRadial came closest with two fails. For the null-independent datasets +(Datasets 6 and 7), it gives each feature an FI of 0.5, making the sum larger +than zero. +Test 2 +In Figure 2, a breakdown is given of where the sum of the FI values +is unstable. +The most errors are made with the Binary system datasets, +29 + +Table 6: +Top 20: Out of 468 FI methods, these 20 methods pass the +18 tests given in Appendix B the most often. These tests are designed to +examine if an FI method adheres to the same properties as the BP-FI , given +in Section 3. Passed means that the datasets from Appendix A do not give a +counterexample. Certainly, this does not mean that the FI method is proven +to actually have this property. +Failed means that a counterexample was +found. No result indicates that the test could not be executed, because the +running time of the FI method was too long or an error occurred. +Combined result: +Method +# Passed +# Failed +# No result +202. +BP-FI +18 +0 +0 +147. +cramer +11 +7 +0 +148. +gainRatio +11 +7 +0 +153. +roughsetConsistency +11 +7 +0 +155. +symmetricalUncertain +11 +7 +0 +172. +su measure +11 +7 +0 +88. +sdwd +10 +7 +1 +3. +Extra Trees Classifier +10 +8 +0 +116. +rpart +10 +8 +0 +126. +null +10 +8 +0 +145. +binaryConsistency +10 +8 +0 +152. +mutualInformation +10 +8 +0 +161. +Banzhaf Ridge +10 +8 +0 +197. +R2 +10 +8 +0 +162. +RF +10 +8 +0 +166. +Relief +10 +8 +0 +173. +spearman corr +10 +8 +0 +188. +DCSF +10 +8 +0 +189. +CFR +10 +8 +0 +191. +IWFS +10 +8 +0 +0 +20 +40 +60 +80 +100 +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 11 12 13 14 15 16 17 18 +# Passed tests +Frequency +BP-FI +1 +0 +0 +0 +0 +0 +0 +5 +13 +25 +74 +78 +78 +101 +57 +24 +7 +4 +1 +Figure 1: Frequency of total passed test: Histogram of the number of +passed tests (out of 18) for the 468 FI methods. +30 + +0 +50 +100 +150 +200 +250 +300 +1↕2 +1↕3 +1↕4 +1↕5 +6↕7 +8↕9 +8↕10 11↕12 11↕13 14↕16 14↕17 +Compared datasets +# Unstable sum FI +188 +302 +311 +299 +163 +190 +95 +203 +194 +124 +117 +Figure 2: Unstable sum FI: Whenever a variable is added that does not +give any additional information, the sum of all FI should remain stable. For +each comparison, we determine how often this is not the case out of 468 FI +methods. +when a fully informative feature is added. In total, 92 methods passed the +test, whereas 369 failed. From these 369 methods, 279 fail with at least one +increase of the sum, whereas 232 methods fail with at least one decrease. +An alarming number of FI methods thus assign significantly more or less FI +when a variable is added that does not contain any additional information. +More or less credit is given out, whilst the collective knowledge is stable and +does not warrant an increase or decrease in credit. Additionally, when the +initial and final sum both contain a NaN value, it is considered as a pass. +Three out of 92 would have not passed without this rule. If only the initial +or the final sum contained NaN, it is considered a fail, because the sum is +not the same. Only five methods fail solely by this rule: 15. Fisher Score, +11. f classif, 178. anova, 179. laplacian score and 192. NDFS. +Test 11 +Figure 3 shows how often each variable is within an ǫ-bound of the +largest FI in the dataset. Fully informative variables should attain the largest +FI, according to Property 8. In total, we observe that the fully informative +variables are often the largest FI with respect to the other variables. However, +there still remain many cases where they are not. 326 FI methods fail this test, +thus definitively not having Property 8. This makes interpretation difficult, +when a variable can get more FI than a variable which fully determines the +target variable. What does it mean, when a variable is more important than +a variable that gives perfect information? +31 + +Variables within dataset (i) +# Variable in arg max +0 +50 +100 +150 +200 +250 +300 +350 +400 +450 +500 +Theoretical maximum: 468 (# FI methods) +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +13 +14 +15 +16 +17 +18 +19 +20 +21 +22 +23 +24 +25 +26 +27 +28 +Figure 3: Argmax FI: For each variable in every dataset, we determine +how often it receives the largest FI (within an ǫ-bound for ǫ = 0.01) with +respect to the other variables in the dataset. +Fully informative variables +should attain the largest FI (see Property 8). All fully informative variables +are shaded in the figure. +Test 10, 17, 18 +These tests all evaluate if the FI method assigns a specific +value to a feature. From Table 5, we observe that not many methods are able +to pass these tests. This is not surprising, as they have not been thoroughly +tested yet to give a specific value. This is one of the important contributions +of this research, which is why we want to elaborate on the attempts that have +been made in previous research. A lot of synthetic datasets for FI have been +proposed [1–3, 6, 7, 9, 15–17, 19, 21, 23–25, 30, 32–34, 37, 39, 41, 50–52, 55, +56, 59, 61, 62], but no specific desirable FI values were given. Most commonly, +synthetic datasets are generated to evaluate the ability of an FI method to +find noisy features [3, 7, 19, 21, 23, 24, 30, 50, 52, 55, 59, 61]. The common +general concept of such a dataset is that the target variable is independent +of certain variables. The FI values are commonly evaluated by comparing +the FI values of independent variables with dependent variables with the +goal to establish if the FI method is able to find independent variables. If +the FI method actually predicts the exact desirable FI is not considered. +Next, we highlight the papers where some comment about the desired FI +is made. Lundberg et al. [34] give two similar datasets, where one variable +increases in importance. They evaluate multiple FI methods to see if the +same behavior is reflected in the outcome of these methods. This shows that +some commonly used methods could assign lower importance to a variable, +when it should actually be increasing. Giles et al. [19] also design multiple +32 + +artificial datasets to represent different scenarios, where comments are made +about which variables should obtain more FI. Sundararajan et al. [55] remark +that if every feature value is unique, that all variables get equal attributions +for an FI method (CES) even if the function is not symmetric in the variables. +If a tiny amount of noise is added to each feature, all features would get +identical attributions. +However, no assessment is done on the validity of +this outcome. Owen et al. [41] give the following example. Let f(x1, x2) = +106x1+x2 with x1 = 106x2, where they argue that, despite the larger variance +of x1, both variables are equally important, as the function can be written +as a function of x1 alone, but also only as a function of x2. Although we +have previously seen that ‘written as a function of’ is not a good criterion +(due to dependencies), we agree with the authors that the FI should be equal. +Another example is given by Owen et al. [41], where P(x1 = 0, x2 = 0, y = +y0) = p0, P(x1 = 1, x2 = 0, y = y1) = p1, and P(x1 = 0, x2 = 1, y = y2) = p2 +are the possible outcomes. If p0 = 0, it is stated in [41] that the Shapley +relative importance of x1 is 1 +2, which is “what it must be because there is +then a bijection between x1 and x2”. This is an interesting observation, as +most papers do not comment about the validity of an outcome. Additionally, +when y1 = y2 (and y0 ̸= y1), Owen et al. [41] argue that the most important +variable, is the one with the largest variance. Fryer et al. [17] also create a +binary XOR dataset (see Dataset 14). They evaluate seven FI methods for +this specific dataset. The role of X1 and X2 is symmetric, thus the assigned FI +should also be identical. It is shown that six out of seven methods do indeed +give a symmetrical result. However, the exact FI value varies greatly. SHAP +gives FI of 3.19, whereas Shapley DC assigns 0.265 as FI. Only symmetry is +checked, not the accuracy of the FI method. In conclusion, existing research +was not focussed on predicting the exact accurate FI values. It is therefore +not surprising that FI methods fail these accuracy tests so often. Table 7 +outlines in more detail how often the variables are assigned an FI value +outside an ǫ-bound (with ǫ = 0.01) of the desired outcome. With Dataset 11, +the FI methods mostly struggle with assigning 1 to the fully informative +variable. In total, 413 methods failed Test 10. For Datasets 14 and 17, the +two XOR variables fail about as often. Comparing these two datasets, it +is interesting to note that the XOR variables fail more often, when a null- +independent variable is added. In total, 421 methods failed Test 17. Test 18 +is hard, as the FI method should assign the correct values for all probability +datasets (see Appendix A). Only five methods are able to pass this test: 152. +mutualInformation, 153. roughsetConsistency, 162. RF, 175. fechner corr, +and 202. BP-FI. These five methods also pass Test 10. However, besides +BP-FI, there is only one method that also satisfies Test 17, which is 162. RF. +The other three methods all assign only zeros for Datasets 14 and 17, not +33 + +0 +300 +350 +400 +450 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +Probability dataset (p) +Frequency failed +399 +438 +441 +445 +442 +427 +451 +453 +450 +450 +405 +Figure 4: Breakdown Test 18 per dataset: In Test 18 an FI method +needs to assign the correct FI values for every probability dataset (see Ap- +pendix A). In this figure, we breakdown per dataset how often an FI method +fails. +identifying the value that the XOR variables hold, when their information +is combined. In Figure 4, a breakdown is given for each probability dataset +how often FI methods fail. An unexpected result, is that the dataset with +probability p < 1 +2 and the dataset with probability 1 − p do not fail as often. +Consistently, p < 1 +2 fails less often than its counterpart 1 − p, although the +datasets are the same up to a reordering of the features and the samples. +This effect can also be seen in Table 7. +No result +Focussing on the no result row of Table 5, there is one base +method named 158. KernelEstimator in combination with Lasso that in all +cases did not work or exceeded running time. The large number of no results +in Test 12 stem mostly from slow running times on the three datasets that +are used in the test. At least 63 methods were too slow for each dataset, +which automatically means that the test cannot be executed. +5 +Discussion and future research +Whilst it is recommended to use our new FI method, it is important to +understand the limitations and potential pitfalls. +Below we elaborate on +both the shortcomings of the approach proposed, and the related challenges +for further research. We start by discussing by some matters that one needs +34 + +Table 7: Specific outcomes: Tests 10, 17 and 18 all evaluate if an FI +method gives a specific outcome for certain dataset. In this table, it is out- +lined how often each variable of these datasets is assigned a value outside an +ǫ-bound (with ǫ = 0.01) of the desired outcome. +# Non desirable outcome +not NaN +NaN +Dataset +Desirable +outcome +X1 +X2 +X3 +X1 +X2 +X3 +11 +(1, 0, 0) +360 +89 +88 +4 +4 +4 +14 +( 1 +2, 1 +2) +353 +351 +- +5 +5 +- +17 +( 1 +2, 1 +2, 0) +369 +364 +90 +5 +5 +5 +18 +(0, 1) +82 +352 +- +4 +4 +- +19 +( 1 +10, 9 +10) +412 +434 +- +3 +3 +- +20 +( 2 +10, 8 +10) +434 +438 +- +3 +3 +- +21 +( 3 +10, 7 +10) +435 +441 +- +3 +3 +- +22 +( 4 +10, 6 +10) +439 +436 +- +3 +3 +- +23 +( 5 +10, 5 +10) +423 +422 +- +3 +3 +- +24 +( 6 +10, 4 +10) +448 +447 +- +3 +3 +- +25 +( 7 +10, 3 +10) +449 +446 +- +3 +3 +- +26 +( 8 +10, 2 +10) +446 +444 +- +3 +3 +- +27 +( 9 +10, 1 +10) +444 +435 +- +3 +3 +- +28 +(1, 0) +352 +86 +- +5 +5 +- +35 + +to be aware of when applying the BP-FI (Section 5.1). Next, we discuss +some choices that were made for the experiments in Section 5.2. Finally, we +elaborate on other possible research avenues in Section 5.3. +5.1 +Creating awareness +Binning +Berkelmans et al. [5] explained that the way in which continuous +data is discretized can have a considerable effect on the BP dependency func- +tion, which is why all datasets that were used in our research are discrete. If +a feature has too many unique values (due to poor binning), it will receive a +higher FI from BP-FI, as more information can be stored in the unique values +(see Property 9). On the other hand, when too few bins are chosen, an impor- +tant feature can receive low FI, as the information is lost due to the binning. +Future research should investigate and test which binning algorithms give +the closest results to the underlying FI. +Too few samples +Consider the following dataset: Xi, Y ∼ U ({0, 1, . . ., 9}) +i.i.d. for i ∈ {1, . . . , 5}. Note that all features are null-independent, as Y +is just uniformly drawn without considering the features in any way. +If +nsamples = ∞, the desired outcome would therefore be (0, 0, 0, 0, 0). How- +ever, when not enough samples are given in the dataset, the features will +get nonzero FI. Considering that the total number of different feature values +is 105, combining all features does actually give information about Y , when +nsamples ≪ 105. For any possible combination of features, it is unlikely that it +occurs more than once in the dataset. Therefore, knowing all feature values +would (almost surely) determine the value of Y . Property 1 gives that the +sum of all FI should therefore be one. All feature variables are also symmet- +ric (Property 2), which is why the desired outcome is ( 1 +5, 1 +5, 1 +5, 1 +5, 1 +5) instead. +This example shows that one should be aware of the influence of the number +of samples on the resulting FI. Variables that do not influence Y can still +contain information, when not enough samples are provided. In this way, +insufficient samples could lead to wrong conclusions, if one is not wary of +this phenomenon. +Counterintuitive dependency case +The Berkelmans-Pries dependency +of Y on X measures how much probability mass of Y is shifted by know- +ing X. However, two similar shifts in probability mass could lead to dif- +ferent predictive power. To explain this, we examine the following dataset. +36 + +X1, X2 ∼ U ({0, 1}) with +P(Y = y|X1 = x1, X2 = x2) = + + + + + + + + + + + + + + + + + + + + + +1/4 +if (x2, y) = (0, 0), +3/4 +if (x2, y) = (0, 1), +5/8 +if (x1, x2, y) = (0, 1, 0), +3/8 +if (x1, x2, y) = (0, 1, 1), +7/8 +if (x1, x2, y) = (1, 1, 0), +1/8 +if (x1, x2, y) = (1, 1, 1). +Knowing the value of X2 shifts the distribution of Y . Before, Y was split +50/50, but when the value of X2 is known, the labels are either split 25/75 +or 75/25, depending on the value of X2. Knowing X1 gives even more in- +formation, as e.g., knowing X1 = X2 = 1 makes it more likely that Y = 0. +However, the shift in distribution of Y is the same for knowing only X2 and +X1 combined with X2, which results in Dep (Y |X2) = Dep (Y |X1 ∪ X2). This +is a counterintuitive result. Globally, knowing X2 or X1 ∪ X2 gives the same +shift in distribution, but locally we can predict Y much better if we know +X1 as well. We are unsure how this effects the BP-FI. In this case, it follows +that FI(X1 ∪ X2) > FI(X2), which is desirable. It is not unthinkable that +a solution can be found to modify the dependency function in order to get +a more intuitive result for such a case. Think e.g., of a different distance +metric, that incorporates the local accuracy given the feature values or a +conditional variant, which not only tests for independence, but also for con- +ditional independence. These are all critical research paths that should be +investigated. +Using FI for feature selection +Feature selection (FS) is “the problem +of choosing a small subset of features that ideally is necessary and sufficient +to describe the target concept” [26]. +Basically, the objective is to find a +subset of all features that gives the best performance for a given model, as +larger feature sets could decrease the accuracy of a model [29]. Many FI +methods actually stem from a FS procedure. However, it is important to +stress that high FI means that it should automatically be selected as feature. +Shared knowledge with other features could render the feature less useful than +expected. The other way around, low FI features should not automatically +be discarded. In combination with other features, it could still give some +additional insights that other features are not able to provide. Calculation +of BP-FI values could also provide insight into which group of K features Y +is most dependent on. To derive the result of BP-FI, all dependencies of Y +37 + +on a subset S ⊆ Ωfeat are determined. If only K variables are selected, it is +natural to choose +S∗ +K ∈ +arg max +S⊆Ωfeat:|S|=K +{Dep (Y |S)}. +These values are stored as an intermediate step in BP-FI, thus S∗ +K can be +derived quickly thereafter. +Larger outcome space leads to higher FI +We have proven that a larger +outcome space can never lead to a decrease in FI for BP-FI. This means, that +features with more possible outcomes are more likely to attain a higher FI, +depending on the distribution. There is a difference between a feature that +has many possible outcomes that are almost never attained, and a feature +where many possible outcomes are regularly observed. We do not find this +property undesirable, as some articles suggest [53, 61], as we would argue +that a feature can contain more information by storing the information in +additional outcomes, which would lead to an non-decreasing FI. +5.2 +Experimental design choices +Regression +To avoid binning issues, we only considered classification mod- +els and datasets. There are many more regression FI methods, that should +be considered in a similar fashion. However, to draw clear and accurate con- +clusions, it is first necessary to understand how binning affects the results. +Sometimes counterintuitive results can occur due to binning, that are not +necessarily wrong. In such a case, it is crucial that the FI method is not +depreciated. +Runtime +In the experiments, it could happen that an FI method had no +result, due to an excessive runtime or incompatible FI scores. The maximum +runtime for each algorithm was set to one hour per dataset on an i7-12700K +processor with 4 algorithms running simultaneously. The maximum runtime +was necessary due to the sheer number of FI methods and datasets. Run- +ning four algorithms in parallel could unfairly penalize the runtime, as the +processor is sometimes limited by other algorithms. In some occurrences, +other parallel processes were already finished, which could potentially lower +the runtime of an algorithm. There is a potential risk here, that accurate +(but slow) FI methods are not showing up in the results. However, our syn- +thetic datasets are relatively small with respect to the number of samples and +the number of features, and we argue that one hour should be reasonable. +38 + +Depending on the use case, sometimes a long time can be used to deter- +mine an FI value, whereas in other cases it could be essential to determine +it rather quickly. Especially for larger datasets, it could even be unfeasible +to run some FI methods. BP-FI uses Shapley values, which are exponen- +tially harder to compute when the number of features grow. Approximation +algorithms should be developed to faster estimate the true BP-FI outcome. +Quick approximations could be useful if the runtime is much faster and the +approximation is decent enough. Already, multiple papers have suggested +approaches to approximate Shapley values faster [1, 10, 24, 31, 54]. These +approaches save time, but at what cost? A study could be done to find the +best FI method given a dataset and an allowed running time. +Stochasticity methods +One factor we did not incorporate, is the stochas- +ticity of some FI methods. Some methods do not predict the same FI values, +when it is repeatedly used. As example, 79. rf predicted for Dataset 3 (12.1, +11.7, 17.9, 15.2, 37.7) rounded to the first decimal. Running the method +again gives a different result: (11.4, 12.0, 17.4, 15.6, 37.1), as this method +uses a stochastic random forest. In principle, it is undesirable that an FI +method is stochastic, as we believe that there should be a unique assignment +of FI given a dataset. Due to the number of FI methods and datasets, we did +not repeat and averaged each FI method. This would however give a better +view on the performance of stochastic FI methods. +Parameter tuning +All FI methods were used with default parameter val- +ues. Different parameter values could lead to more or less failed tests. How- +ever, the ideal parameter setting is not known beforehand, making it nec- +essary to search a wide range of parameters. +This was not the focus of +our research, but future research could try to understand and learn which +parameter values should be chosen for a given dataset. +Ranking FI methods +In Table 6, the 20 FI methods that passed the most +tests were highlighted. However, it is important to stress that not every test +is equally difficult. Depending on the user, some properties could be more or +less relevant. It is e.g., much harder to accurately predict the specific values +for 11 datasets (Test 18), than to always predict non-negatively (Test 4). +Every test is weighed equally, but this does not necessarily represent the +difficulty of passing each test accurately. However, we note that 175. fechner +corr is the only FI method that passed Test 18, that ended up outside the +top 20. We stress that we focussed on finding out if FI methods adhere to +the properties, not necessarily finding the best and most fair ranking. +39 + +5.3 +Additional matters +Global vs. local +BP-FI is designed to determine the FI globally. How- +ever, another important research area focusses on local explanations. These +explanations should provide information about why a specific sample has a +certain target value instead of a different value. They provide the necessary +interpretability that is increasingly demanded for practical applications. This +could give insights for questions like: ‘If my income would be higher, could I +get a bigger loan?’, ‘Does race play a role in this prediction?’, and ‘For this +automated machine learning decision, what were the critical factors?’. Many +local FI methods have been proposed, and some even use Shapley values. A +structured review should be made about all proposed local methods, simi- +lar to our approach for global FI methods to find which local FI methods +actually produce accurate explanations. +BP-FI can be modified to provide local explanations. For example, we can +make the characteristic function localized in the following way. Let YS,z be +Y restricted to the event that Xi = zi for i /∈ S, let us similarly define XS,z. +Then, we can define a localized characteristic function by: +vz(S) := Dep (YS,z|XS,z) . +(8) +When dealing with continuous data, assuming equality could be too strict. +In this case, a precision vector parameter ǫ can be used, where we define +YS,z,ǫ to be Y restricted to the event that |Xi − zi| ≤ ǫi for i /∈ S, and in the +same way we define XS,z,ǫ. We then get the following localized characteristic +function: +vz,ǫ(S) := Dep (YS,z,ǫ|XS,z,ǫ) . +Additionally, there are at least two possible ways how BP-FI can be adapted +to be used for local explanations if some distance function d(i, j) and param- +eter δ are available to determine if sample j is close enough to i to be consid- +ered ‘local’. We can (I) discard all samples where d(i, j) > δ and/or (II) gen- +erate samples, such that d(i, j) ≤ δ for all generated samples. Then, we can +use BP-FI on the remaining samples and/or the generated samples, which +would give local FI. Note that there should still be enough samples, as we +have previously discussed that too few samples could lead to different FI +outcomes. However, there are many more ways how BP-FI can be modified +to be used for local explanations. +Model-specific FI +BP-FI is in principle model-agnostic, as the FI is deter- +mined of the dataset, not the FI for a prediction model. However, BP-FI can +40 + +still provide insights for any specific model. By replacing the target variable +with the predicted outcomes of the model, we can apply BP-FI to this new +dataset, which gives insight into which features are useful in the prediction +model. Additionally, one can compare these FI results with the original FI +(before replacing the target variable with the predicted outcomes) to see in +what way the model changed the FI. +Additional properties +In this research, we have proven properties of BP- +FI. However, an in-depth study could lead to finding more useful properties. +This holds both for BP-FI as well as the dependency function it is based +on. Applying isomorphisms e.g., does not change the dependency function. +Therefore, the BP-FI is also stable under isomorphisms. Understanding what +properties BP-FI has is a double-edged sword. +Finding useful properties +shows the power of BP-FI and finding undesirable behavior could lead to a +future improvement. +Additional datasets +Ground truths are often unknown for FI. In this re- +search, we have given two kinds of datasets where the desirable outcomes are +natural. It would however, be useful to create a larger collection of datasets +both for global and local FI with an exact ground truth. We recognize that +this could be a tall order, but we believe that it is essential to further improve +FI methods. +Human labeling +In some articles [35, 46], humans are used to evaluate +explanations. An intriguing question to investigate is if humans are good at +predicting FI. The BP-FI can be used as baseline to validate the values that +are given by the participants. Are humans able to identify the correct order of +FI? Even more difficult, can they predict close to the actual FI values? +6 +Summary +We started by introducing a novel FI method named Berkelmans-Pries FI +(BP-FI), which combines Shapley values and the Berkelmans-Pries depen- +dency function [5]. In Section 3, we proved many useful properties of BP-FI. +We discussed which FI methods already exist and introduced datasets to +evaluate if these methods adhere to the same properties. In Section 4.3, we +explain how the properties are tested. The results show that BP-FI is able +to pass many more tests than any other FI method from a large collection of +FI methods (468), which is a significant step forwards. Most methods have +not previously been tested to give exact results due to missing ground truths. +41 + +In this research, we provide several specific datasets, where the desired FI +can be derived. From the tests, it follows that previous methods are not +able to accurately predict the desired FI values. In Section 5, we extensively +discussed the shortcomings of this paper, and the challenges for further re- +search. There are many challenging research opportunities that should be +explored to further improve interpretability and explainability of datasets +and machine learning models. +A +Datasets +In this appendix, we discuss how the datasets are generated that are used +in the experiments. We use fixed draw instead of uniformly random to draw +each dataset exactly according to its distribution. This is done to remove +stochasticity from the dataset in order to get precise and interpretable re- +sults. An example of the difference between fixed draw and uniformly ran- +dom can be seen in Table 2. The datasets consist of 1,000 samples, except for +Datasets 6 and 7 which contains 2,000 samples to ensure null-independence. +The datasets are designed to be computationally inexpensive, whilst still be- +ing able to test many properties (see Section 4.3). Below, we outline the +formulas that are used to generate the datasets and give the corresponding +FI values of our novel method BP-FI. +Dataset 1: Binary system +Feature variable(s): Xi ∼ U ({0, 1}) i.i.d. for i ∈ {1, 2, 3} +Target variable: Y := �3 +i=1 2i−1 · Xi. +Order: (X1, X2, X3). +BP-FI: (0.333, 0.333, 0.333). +Dataset 2: Binary system with clone +Feature variable(s): Xi ∼ U ({0, 1}) i.i.d. for i ∈ {1, 2, 3} and Xclone +1 +:= +X1. +Target variable: Y := �3 +i=1 2i−1 · Xi. +Order: (Xclone +1 +, X1, X2, X3). +BP-FI: (0.202, 0.202, 0.298, 0.298). +Dataset 3: Binary system with clone and one fully informative +variable +Feature variable(s): Xi ∼ U ({0, 1}) i.i.d. for i ∈ {1, 2, 3} and Xclone +1 +:= +X1 and Xfull +4 +:= Y 2. +Target variable: Y := �3 +i=1 2i−1 · Xi. +Order: (Xclone +1 +, X1, X2, X3, Xfull +4 ). +BP-FI: (0.148, 0.148, 0.183, 0.183, 0.338). +42 + +Dataset 4: Binary system with clone and two fully informative +variables +Feature variable(s): Xi ∼ U ({0, 1}) i.i.d. for i ∈ {1, 2, 3} and Xclone +1 +:= +X1 and Xfull +4 +:= Y 2, Xfull +5 +:= Y 3. +Target variable: Y := �3 +i=1 2i−1 · Xi. +Order: (Xclone +1 +, X1, X2, X3, Xfull +4 , Xfull +5 ). +BP-FI: (0.117, 0.117, 0.136, 0.136, 0.248, 0.248). +Dataset 5: Binary system with clone and two fully informative +variables different order +Feature variable(s): Xi ∼ U ({0, 1}) i.i.d. for i ∈ {1, 2, 3} and Xclone +1 +:= +X1 and Xfull +4 +:= Y 2, Xfull +5 +:= Y 3. +Target variable: Y := �3 +i=1 2i−1 · Xi. +Order: (X3, Xfull +4 , Xfull +5 , Xclone +1 +, X1, X2). +BP-FI: (0.136, 0.248, 0.248, 0.117, 0.117, 0.136). +Dataset 6: Null-independent system +Feature variable(s): Xnull-indep. +i +∼ U ({0, 1}) i.i.d. for i ∈ {1, 2, 3}. +Target variable: Y ∼ U ({0, 1}). +Order: (Xnull-indep. +1 +, Xnull-indep. +2 +, Xnull-indep. +3 +). +BP-FI: (0.000, 0.000, 0.000). +Dataset 7: Null-independent system with constant variable +Feature variable(s): Xnull-indep. +i +∼ U ({0, 1}) i.i.d. for i ∈ {1, 2, 3} and +Xconst, null-indep. +4 +:= 1. +Target variable: Y ∼ U ({0, 1}). +Order: (Xnull-indep. +1 +, Xnull-indep. +2 +, Xnull-indep. +3 +, Xconst, null-indep. +4 +). +BP-FI: (0.000, 0.000, 0.000, 0.000). +43 + +Dataset 8: Uniform system increasing bins +Feature variable(s): Let Li := {0, 1/(i−1), . . . , 1} be an equally spaced +set. Define: +Xbins=10 +1 +:= arg max +x1∈L10 +{Y ≥ x1}, +Xbins=50 +2 +:= arg max +x2∈L50 +{Y ≥ x2}, +Xbins=1,000, full +3 +:= arg max +x3∈L1,000 +{Y ≥ x3}. +Target variable: Y ∼ U (L1,000). +Order: (Xbins=10 +1 +, Xbins=50 +2 +, Xbins=1,000, full +3 +). +BP-FI: (0.297, 0.342, 0.361). +Dataset 9: Uniform system increasing bins more variables +Feature variable(s): Let Li := {0, 1/(i−1), . . . , 1} be an equally spaced +set. Define: +Xbins=10 +1 +:= arg max +x1∈L10 +{Y ≥ x1}, +Xbins=20 +2 +:= arg max +x2∈L20 +{Y ≥ x2}, +Xbins=50 +3 +:= arg max +x3∈L50 +{Y ≥ x3}, +Xbins=100 +4 +:= arg max +x4∈L100 +{Y ≥ x4}, +Xbins=1,000, full +5 +:= arg max +x5∈L1,000 +{Y ≥ x5}. +Target variable: Y ∼ U (L1,000). +Order: (Xbins=10 +1 +, Xbins=20 +2 +, Xbins=50 +3 +, Xbins=100 +4 +, Xbins=1,000, full +5 +). +BP-FI: (0.179, 0.193, 0.204, 0.208, 0.216). +44 + +Dataset 10: Uniform system increasing bins with clone differ- +ent order +Feature variable(s): Let Li := {0, 1/(i−1), . . . , 1} be an equally spaced +set. Define: +Xbins=10 +1 +:= arg max +x1∈L10 +{Y ≥ x1}, +Xbins=50 +2 +:= arg max +x2∈L50 +{Y ≥ x2}, +Xbins=1,000, full +3 +:= arg max +x3∈L1,000 +{Y ≥ x3}, +Xclone, full +3 +:= Xbins=1,000, full +3 +. +Target variable: Y ∼ U (L1,000). +Order: (Xbins=1,000, full +3 +, Xbins=50 +2 +, Xbins=10 +1 +, Xclone, full +3 +). +BP-FI: (0.262, 0.253, 0.223, 0.262). +Dataset 11: Dependent system: 1x fully informative variable +Feature variable(s): Xfull +1 , Xnull-indep. +2 +, Xnull-indep. +3 +∼ U ({1, 2}). +Target variable: Y := Xfull +1 . +Order: (Xfull +1 , Xnull-indep. +2 +, Xnull-indep. +3 +). +BP-FI: (1.000, 0.000, 0.000). +Dataset 12: Dependent system: 2x fully informative variable +Feature variable(s): Xfull +1 , Xnull-indep. +3 +∼ U ({1, 2}) and Xfull +2 +:= Y 2. +Target variable: Y := Xfull +1 . +Order: (Xfull +1 , Xfull +2 , Xnull-indep. +3 +). +BP-FI: (0.500, 0.500, 0.000). +Dataset 13: Dependent system: 3x fully informative variable +Feature variable(s): Xfull +1 +∼ U ({1, 2}) and Xfull +2 +:= Y 2, Xfull +3 +:= Y 3. +Target variable: Y := Xfull +1 . +Order: (Xfull +1 , Xfull +2 , Xfull +3 ). +BP-FI: (0.333, 0.333, 0.333). +Dataset 14: XOR dataset +Feature variable(s): X1, X2 ∼ U ({1, 2}). +Target variable: Y := X1 · (1 − X2) + X2 · (1 − X1). +Order: (X1, X2). +BP-FI: (0.500, 0.500). +45 + +Dataset 15: XOR dataset one variable +Feature variable(s): Xnull-indep. +1 +∼ U ({1, 2}). +Target variable: Y := Xnull-indep. +1 +· (1 − X2) + X2 · (1 − Xnull-indep. +1 +) with +X2 ∼ U ({1, 2}). +Order: (Xnull-indep. +1 +). +BP-FI: (0.000). +Dataset 16: XOR dataset with clone +Feature variable(s): X1, X2 ∼ U ({1, 2}) and Xclone +1 +:= X1. +Target variable: Y := X1 · (1 − X2) + X2 · (1 − X1). +Order: (Xclone +1 +, X1, X2). +BP-FI: (0.167, 0.167, 0.667). +Dataset 17: XOR dataset with null independent +Feature variable(s): X1, X2 ∼ U ({1, 2}) and Xnull-indep. +3 +∼ U ({0, 3}). +Target variable: Y := X1 · (1 − X2) + X2 · (1 − X1). +Order: (X1, X2, Xnull-indep. +3 +). +BP-FI: (0.500, 0.500, 0.000). +Dataset 18-28: Probability datasets +Feature variable(s): Xi = Zi + S with Zi ∼ U ({0, 2}) i.i.d. for i = 1, 2 +and P(S = 1) = p, P(S = 2) = 1 − p. +Target variable: Y = ⌊XS/2⌋. +Order: (X1, X2). +BP-FI: (p, 1 − p). +B +Tests +This appendix gives an overview of the tests that are used for each FI method +to evaluate if they adhere to the properties given in Section 3. Most tests are +straightforward, but additional explanations are given in Section 4.3. +Test 1: Efficiency sum BP-FI +Evaluates: Property 1. +Explanation: We evaluate if the sum of all FI is equal to the sum of +the Berkelmans-Pries dependency function of Y on all features. When +an FI value of NaN or infinite is assigned, the sum is automatically not +equal to the sum for BP-FI. +46 + +Test 2: Efficiency stable +Evaluates: Corollary 1.1. +Explanation: Whenever a variable is added to a dataset, we examine +if the sum of all FI changes. If a variable does not give any additional +information compared to the other variables, the sum of all FI should +stay the same. +Test 3: Symmetry +Evaluates: Property 2. +Explanation: In some datasets, there are symmetrical variables (see +Property 2). We determine for all symmetrical variables if they receive +identical FI. +Test 4: Range (lower) +Evaluates: Property 3. +Explanation: We examine for all FI outcomes if they are greater or +equal to zero. +Test 5: Range (upper) +Evaluates: Property 3. +Explanation: We examine for all FI outcomes if they are smaller or +equal to one. +Test 6: Bounds BP-FI (lower) +Evaluates: Property 4. +Explanation: We evaluate if the bounds given in Property 4 also hold +for other FI methods. +Every FI(X) with X ∈ Ωfeat can be lower +bounded for BP-FI by Dep(Y |X) +Nvars +≤ FI(X). +Test 7: Bounds BP-FI (upper) +Evaluates: Property 4. +Explanation: We evaluate if the bounds given in Property 4 also hold +for other FI methods. +Every FI(X) with X ∈ Ωfeat can be upper +bounded for BP-FI by X ≤ Dep (Y |Ωfeat) . +Test 8: Null-independent implies zero FI +Evaluates: Property 5. +Explanation: In some datasets, there are null-independent variables. +In these cases, we investigate if they also receive zero FI. +Test 9: Zero FI implies null-independent +Evaluates: Property 5. +Explanation: When a variable gets zero FI, it should hold that such a +feature is null-independent. +47 + +Test 10: One fully informative, two null-independent +Evaluates: Property 6. +Explanation: +feature importance: +appendix: +datasets) consists +of a fully dependent target variable Y +:= +Xfull +1 +and two null- +independent variables Xnull-indep. +2 +, Xnull-indep. +3 +. We test if FI(Xfull +1 ) = 1 +and FI(Xnull-indep. +2 +) = FI(Xnull-indep. +3 +) = 0. +Test 11: Fully informative variable in argmax FI +Evaluates: Property 8. +Explanation: Whenever a fully informative feature exists in a dataset, +there should not be a feature that attains a higher FI. +Test 12: Limiting the outcome space +Evaluates: Property 9. +Explanation: +To evaluate if applying a measurable function f to a RV +X could increase the FI, we examine the datasets where the same RV +is binned using different bins. The binning can be viewed as applying +a function f. Whenever less bins are used, the FI should not increase. +Test 13: Adding features can increase FI +Evaluates: Property 11. +Explanation: Whenever a feature is added to a dataset, we examine +if this ever increases the FI of an original variable. If the FI never +increases, we consider this a fail. +Test 14: Adding features can decrease FI +Evaluates: Property 12. +Explanation: Whenever a feature is added to a dataset, we examine +if this ever decreases the FI of an original variable. If the FI never +decreases, we consider this a fail. +Test 15: Cloning does not increase FI +Evaluates: Property 13. +Explanation: We evaluate if adding a clone to a dataset increase the +FI of the original variable. +Test 16: Order does not change FI +Evaluates: Property 14. +Explanation: We check if the order of the variables changes the assigned +FI. +48 + +Test 17: Outcome XOR +Evaluates: Property 15. +Explanation: This test evaluates the specific outcome of two datasets. +For Dataset 14 the desired outcome is (1/2, 1/2) and (1/2, 1/2, 0) for +Dataset 17. 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Berlin, Heidelberg: Springer Berlin Heidelberg, +2009, pages 694–709. isbn: 978-3-642-04174-7. +55 + diff --git a/WNE3T4oBgHgl3EQf0wtX/content/tmp_files/load_file.txt b/WNE3T4oBgHgl3EQf0wtX/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f4a7205b85bfbd28552de224e1cc8dbbd7d9ba74 --- /dev/null +++ b/WNE3T4oBgHgl3EQf0wtX/content/tmp_files/load_file.txt @@ -0,0 +1,2838 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf,len=2837 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='04740v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='LG] 11 Jan 2023 The Berkelmans-Pries Feature Importance Method: A Generic Measure of Informativeness of Features Joris Pries1,*, Guus Berkelmans1, Sandjai Bhulai2, and Rob van der Mei1 1Centrum Wiskunde & Informatica, Department of Stochastics, Science Park 123, Amsterdam 1098 XG, Netherlands 2Vrije Universiteit, Department of Mathematics, De Boelelaan 1111, Amsterdam 1081 HV, Netherlands Corresponding author: Joris Pries, joris.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='pries@cwi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='nl January 13, 2023 Abstract Over the past few years, the use of machine learning models has emerged as a generic and powerful means for prediction purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' At the same time, there is a growing demand for interpretability of predic- tion models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' To determine which features of a dataset are important to predict a target variable Y , a Feature Importance (FI) method can be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' By quantifying how important each feature is for predicting Y , irrelevant features can be identified and removed, which could increase the speed and accuracy of a model, and moreover, important features can be discovered, which could lead to valuable insights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' A major problem with evaluating FI methods, is that the ground truth FI is often unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' As a consequence, existing FI methods do not give the exact correct FI values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' This is one of the many reasons why it can be hard to properly interpret the results of an FI method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Motivated by this, we introduce a new global approach named the Berkelmans-Pries FI method, which is based on a combination of Shapley values and the Berkelmans-Pries dependency function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' We prove that our method has many useful properties, and accurately predicts the correct FI values for several cases where the ground truth FI can be derived in an exact 1 manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' We experimentally show for a large collection of FI methods (468) that existing methods do not have the same useful properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' This shows that the Berkelmans-Pries FI method is a highly valuable tool for analyzing datasets with complex interdependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' 1 Introduction How important are you?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' This is a question that researchers (especially data scientists) have wondered for many years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Researchers need to understand how important a random variable (RV) X is for determining Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Which features are important for predicting the weather?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Can indicators be found as symptoms for a specific disease?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Can redundant variables be discarded to increase performance?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' These kinds of questions are relevant in almost any research area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Especially nowadays, as the rise of machine learning models generates the need to demystify prediction models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Altmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' [3] state that “In life sciences, interpretability of machine learning models is as important as their prediction accuracy.” Although this might not hold for all research areas, interpretability is very useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Knowing how predictions are made and why, is crucial for adapting these methods in everyday life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Determining Feature Importance (FI) is the art of discovering the impor- tance of each feature Xi when predicting Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' The following two cases are particularly useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' (I) Finding variables that are not important: redundant variables can be discovered using FI methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Irrelevant features could de- grade the performance of a prediction model due to high dimensionality and irrelevant information [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Eliminating redundant features could therefore increase both the speed and the accuracy of a prediction model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' (II) Find- ing variables that are important: important features could reveal underlying structures that give valuable insights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Observing that variable X is impor- tant for predicting Y could steer research efforts into the right direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Although it is critical to keep in mind that high FI does not mean causation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' However, FI values do, for example, “enable an anaesthesiologist to better formulate a diagnosis by knowing which attributes of the patient and pro- cedure contributed to the current risk predicted” [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' In this way, an FI method can have really meaningful impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Over the years, many FI methods have been suggested, which results in a wide range of FI values for the same dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' For example, stochastic methods do not even repeatedly predict the same FI values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' This makes interpretation difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Examine e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=', a result of Fryer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' [17], where one measure assigns an FI of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='19 to a variable, whereas another method gives the 2 same variable an FI value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='265.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' This raises a lot of questions: ‘Which FI method is correct?’' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=', ’Is this variable deemed important?’' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=', and more generally ‘What information does this give us?’' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' To assess the performance of an FI method, the ground truth should be known, which is often not the case [1, 21, 56, 61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Therefore, when FI methods were developed, the focus has not yet lied on predicting the exact correct FI values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Additionally, many FI methods do not have desirable properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' For example, two features that contain the same amount of information should get the same FI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' We later show that this is often not the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' To improve interpretability, we introduce a new FI method called Berkelmans- Pries FI method, which is based on Shapley values [49] and the Berkelmans- Pries dependency function [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Multiple existing methods already use Shap- ley values, which has been shown to give many nice properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' However, by additionally using the Berkelmans-Pries dependency function, even more useful properties are obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Notably, we prove that this approach accu- rately predicts the FI in some cases where the ground truth FI can be derived in an exact manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' By combining Shapley values and the Berkelmans-Pries dependency function a powerful FI method is created.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' This research is an im- portant step forward for the field of FI, because of the following reasons: We introduce a new FI method;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' We prove multiple useful properties of this method;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' We provide some cases where the ground truth FI can be derived in an exact manner;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' We prove for these cases that our FI method accurately predicts the correct FI;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' We obtain the largest collection of existing FI methods;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' We test if these methods adhere to the same properties, which shows that no method comes close to fulfilling all the useful properties;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' We provide Python code to determine the FI values [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' 2 Berkelmans-Pries FI Kruskal [27] stated that “There are infinitely many possible measures of asso- ciation, and it sometimes seems that almost as many have been proposed at one time or another.” Although this quote was about dependency functions, it could just as well have been about FI methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Over the years, many FI 3 methods have been suggested, but it remains unclear which method should be used when and why [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' In this section, we propose yet another new FI method named the Berkelmans-Pries FI method (BP-FI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Although it is certainly subjective what it is that someone wants from an FI method, we show in Section 3 that BP-FI has many useful and intuitive properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' The BP-FI method is based on two key elements: (1) Shapley values and (2) the Berkelmans-Pries dependency function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' We will discuss these components first to clarify how the BP-FI method works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='1 Shapley value approach The Shapley value is a unique game-theoretical way to assign value to each player participating in a multiplayer game based on four axioms [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' This concept is widely used in FI methods, as it can be naturally adapted to determine how important (value) each feature (player) is for predicting a target variable (game).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Let Nvars be the number of features, then the Shapley value of feature i is defined by φi(v) = � S⊆{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=',Nvars}\\{i} |S|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' · (Nvars − |S| − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Nvars!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' (v(S ∪ {i}) − v(S)) , (1) where v(S) can be interpreted as the ‘worth’ of the coalition S [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' The principle behind this formulation can also be explained in words: For every possible sequence of features up to feature i, the added value of feature i is the difference between the worth before it was included (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=', v(S)) and after (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=', v(S ∪ {i})).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Averaging these added values over all possible sequences of features gives the final Shapley value for feature i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' SHAP There are multiple existing FI methods that use Shapley values [14, 17, 35], which immediately ensures some useful properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' The most famous of these methods is SHAP [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' This method is widely used for local explanations (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' To measure the local FI for a specific sample x and a prediction model f, the conditional expectation is used as characteristic function (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=', v in Equation (1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Let x = (x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' , xNvars), where xi is the feature value of feature i, then SHAP FI values can be determined using: vx(S) := Ez [f(z)|zi = xi for all i ∈ S, where z = (z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' , zNvars)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' (2) Observe that the characteristic function vx is defined locally for each x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' To get global FI values, an average can be taken over all local FI values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Our novel 4 FI method uses a different characteristic function, namely the Berkelmans- Pries dependency function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' This leads to many additional useful properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Furthermore, the focus of this research is not on local explanations, but global FI values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='2 Berkelmans-Pries dependency function A new dependency measure, called the Berkelmans-Pries (BP) dependency function, was introduced in [5], which is used in the formulation of the BP- FI method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' It is shown that the BP dependency function satisfies a list of desirable properties, whereas existing dependency measures did not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' It has a measure-theoretical formulation, but this reduces to a simpler and more intuitive version when all variables are discrete [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' We want to highlight this formulation to give some intuition behind the BP dependency function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' It is given by Dep (Y |X) := \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 UD(X,Y ) UD(Y,Y ) if Y is not a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' constant, undefined if Y is a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' constant, (3) where (in the discrete case) it holds that UD (X, Y ) := � x pX(x) · � y ��pY |X=x(y) − pY (y) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' (4) The BP dependency measure can be interpreted in the following manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' The numerator is the expected absolute difference between the distribution of Y and the distribution of Y given X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' If Y is highly dependent on X, the distribution changes as knowing X gives information about Y , whereas if Y is independent of X, there is no difference between these two distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' The denominator is the maximal possible change in distribution of Y for any variable, which is used to standardize the dependency function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Note that the BP dependency function is asymmetric: Dep (Y |X) is the dependency of Y on X, not vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Due to the many desirable properties, the BP dependency function is used for the BP-FI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='3 Berkelmans-Pries FI method One crucial component of translating the game-theoretical approach of Shap- ley values to the domain of FI is choosing the function v in Equation (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' 5 This function assigns for each set of features S a value v(S) that character- izes the ‘worth’ of the set S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' How this function is defined, has a critical impact on the resulting FI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' We choose to define the ‘worth’ of a set S to be the BP dependency of Y on the set S, which is denoted by Dep (Y |S) [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Here, Dep (Y |S) = Dep (Y |ZS(D)) where D denotes the entire dataset with all features and ZS(D) is the reduction of the dataset to include only the subset of features S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Let Ωfeat be the set of all feature variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Now, for every S ⊆ Ωfeat, we define: v(S) := Dep (Y |S) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' (5) In other words, the value of set S is exactly how dependent the target variable Y is on the features in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' The difference v(S ∪ {i}) − v(S) in Equation (1) can now be viewed as the increase in dependency of Y on the set of features, when feature i is also known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' The resulting Shapley values using the BP dependency function as characteristic function are defined to be the BP-FI outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' For each feature i, we get: FI(i) := � S⊆Ωfeat\\{i} |S|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' · (Nvars − |S| − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Nvars!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' (v(S ∪ {i}) − v(S)) = � S⊆Ωfeat\\{i} |S|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' · (Nvars − |S| − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Nvars!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' (Dep (Y |S ∪ {i}) − Dep (Y |S)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' (6) Abbreviated notation improves readability of upcoming derivations, which is why we define w(S, Nvars) := |S|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' · (Nvars − |S| − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Nvars!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' , (N1) D(X, Y, S) := Dep (Y |S ∪ {X}) − Dep (Y |S) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' (N2) Note that when Y is almost surely constant (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=', P(Y = y) = 1), Dep (Y |S) is undefined for any feature set S (see Equation (3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' We argue that it is natural to assume that FI(i) is also undefined, as every feature attributes everything and nothing at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' In the remainder of this paper, we assume that Y is not a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' 6 3 Properties of BP-FI Recall that it is hard to evaluate FI methods, as the ground truth FI is often unknown [1, 21, 56, 61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' With this in mind, we want to show that the BP-FI method has many desirable properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' We also give some synthetic cases where the BP-FI method gives a natural expected outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' The BP- FI method is stooled on Shapley values, which are a unique solution based on four axioms [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' These axioms already give many characteristics that are preferable for an FI method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Additionally, using the BP dependency function ensures that it has extra desirable properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' In this section, we prove properties of the BP-FI method and discuss why these are relevant and useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Property 1 (Efficiency).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' The sum of all FI scores is equal to the total dependency of Y on all features: � i∈Ωfeat FI(i) = Dep (Y |Ωfeat) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Shapley values are efficient, meaning that all the value is distributed among the players.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Thus, � i∈Ωfeat FI(i) = v(Ωfeat) = Dep (Y |Ωfeat) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Relevance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' With our approach, we try to answer the question ‘How much did each feature contribute to the total dependency?’' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' The total ‘payoff’ is in our case the total dependency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' It is therefore natural to divide the entire payoff (but not more than that) amongst all features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' If adding a RV X to the dataset does not give any additional information (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=', Dep (Y |Ωfeat ∪ X) = Dep (Y |Ωfeat)), then the sum of all FI remains the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' This directly follows from Property 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Relevance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' If the collective knowledge remains the same, the same amount of credit is available to be divided amongst the features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Only when new information is added, an increase in combined credit is warranted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' A direct result of this corollary is that adding a clone (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=', Xclone := X) of a variable X to the dataset will never increase the total sum of FI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' 7 Property 2 (Symmetry).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' If for every S ⊆ Ωfeat \\ {i, j} it holds that Dep (Y |S ∪ {i}) = Dep (Y |S ∪ {j}), then FI(i) = FI(j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Shapley values are symmetric, meaning that if v(S ∪ {i}) = v(S ∪ {j}) for every S ⊆ Ωfeat \\ {i, j}, it follows that FI(i) = FI(j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Thus, it automati- cally follows that BP-FI is also symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Relevance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' If two variables are interchangeable, meaning that they always contribute equally to the dependency, it is only sensible that they obtain the same FI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' This is a desirable property for an FI method, as two features that contribute equally should obtain the same FI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Property 3 (Range).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' For any RV X, it holds that FI(X) ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' The BP dependency function is non-increasing under functions of X [5], which means that for any measurable function f it holds that Dep (Y |f(X)) ≤ Dep (Y |X) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Take f := ZS, which is the function that reduces D to the subset of features in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Using the non-increasing property of BP dependency function, it follows that: Dep (Y |S) = Dep (Y |ZS(D)) = Dep � Y |ZS(ZS∪{i}(D)) � ≤ Dep � Y |ZS∪{i}(D) � = Dep (Y |S ∪ {i}) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' (7) Examining Equation (6), we observe that every FI value must be greater or equal to zero, as Dep (Y |S ∪ {i}) − Dep (Y |S) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' One of the properties of the BP dependency function is that for any X, Y it holds that Dep (Y |X) ∈ [0, 1] [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Using Property 1, the sum of all FI values must therefore be in [0, 1], as � i∈Ωfeat FI(i) = Dep (Y |Ωfeat) ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' This gives an upper bound for the FI values, which is why we can now conclude that FI(X) ∈ [0, 1] for any RV X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Relevance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' It is essential for interpretability that an FI method is bounded by known bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' For example, an FI score of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='2 cannot be interpreted properly, when the upper or lower bound is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Property 4 (Bounds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Every FI(X) with X ∈ Ωfeat is bounded by Dep (Y |X) Nvars ≤ FI(X) ≤ Dep (Y |Ωfeat) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' 8 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' The upper bound follows from Properties 1 and 3, as Dep (Y |Ωfeat) = � i∈Ωfeat FI(i) ≥ FI(X), where the last inequality follows since FI(i) ∈ [0, 1] for all i ∈ Ωfeat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' The lower bound can be established using the inequality from Equation (7) within Equation (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' This gives (using Notation (N1)) FI(X) = � S⊆Ωfeat\\{X} w(S, Nvars) · � Dep (Y |S ∪ {X}) − Dep (Y |S) � ≥ w(0, Nvars) · (Dep (Y |∅ ∪ {X}) − Dep (Y |∅)) = 0!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' · (Nvars − 0 − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Nvars!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Dep (Y |X) = Dep (Y |X) Nvars .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Relevance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' These bounds are useful for upcoming proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Property 5 (Zero FI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' For any RV X, it holds that FI(X) = 0 ⇔ Dep (Y |S ∪ {X}) = Dep (Y |S) for all S ∈ Ωfeat \\ {X}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' ⇐: When Dep (Y |S ∪ {X}) = Dep (Y |S) for all S ∈ Ωfeat \\ {X}, it immediately follows from Equation (6) (with Notation (N1)) that FI(X) = � S⊆Ωfeat\\{X} w(S, Nvars) · � Dep (Y |S ∪ {X}) − Dep (Y |S) � = � S⊆Ωfeat\\{X} |S|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' · (Nvars − |S| − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Nvars!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' ⇒: Assume that FI(X) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' It follows from the proof of Property 3 that Dep (Y |S ∪ {X}) − Dep (Y |S) ≥ 0 for every S ⊆ Ωfeat \\ {X}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' If 9 Dep (Y |S∗ ∪ {X}) − Dep (Y |S∗) > 0 for some given S∗ ∈ Ωfeat \\ {X}, it follows from Equation (6) (with Notation (N1)) that FI(X) = � S⊆Ωfeat\\{X} w(S, Nvars) · � Dep (Y |S ∪ {X}) − Dep (Y |S) � ≥ w(S∗, Nvars) · (Dep (Y |S∗ ∪ {X}) − Dep (Y |S∗)) = |S∗|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' · (Nvars − |S∗| − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Nvars!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' (Dep (Y |S∗ ∪ {X}) − Dep (Y |S∗)) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' This gives a contradiction with the assumption that FI(X) = 0, thus it is not possible that such an S∗ exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' This means that Dep (Y |S ∪ {X}) = Dep (Y |S) for all S ∈ Ωfeat \\ {X}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Relevance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' When a feature never contributes any information, it is only fair that it does not receive any FI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' The feature can be removed from the dataset, as it has no effect on the target variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' On the other hand, when a feature has an FI of zero, it would be unfair to this feature if it does in fact contribute information somewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' It should then be rewarded some FI, albeit small it should be larger than zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Null-independence The property that a feature receives zero FI, when Dep (Y |S ∪ {X}) = Dep (Y |S) for all S ∈ Ωfeat \\ {X}, is the same notion as a null player in game theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Berkelmans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' [5] show that Dep (Y |X) = 0, when Y is independent of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' To be a null player requires a stricter definition of independence, which we call null-independence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Y is null-independent on X if Dep (Y |S ∪ {X}) = Dep (Y |S) for all S ∈ Ωfeat \\ {X}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' In other words, X is null-independent if and only if FI(X) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Independent feature ̸⇒ null-independent feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Take e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=', the dataset consisting of two binary features X1, X2 ∼ U({0, 1}) and a target variable Y = X1 · (1 − X2) + X2 · (1 − X1) which is the XOR of X1 and X2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Individually, the variables do not give any infor- mation about Y , whereas collectively they fully determine Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' In the proof of Property 15, we show that this leads to FI(X1) = FI(X2) = 1 2, whilst Dep (Y |X1) = Dep (Y |X2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Thus, X1 and X2 are independent, but not null-independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' 10 Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Independent feature ⇐ null-independent feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' When X is null-independent, it holds that FI(X) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Using Prop- erty 4, we obtain 0 = FI(X) ≥ Dep (Y |X) Nvars ⇔ Dep (Y |X) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Thus, when X is null-independent, it is also independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Almost surely constant variables get zero FI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' If X is almost surely constant (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=', P(X = x) = 1), it immediately follows that Dep (Y |S ∪ {X}) = Dep (Y |S) for any S ⊆ Ωfeat \\ {X}, as the distribution of Y is not affected by X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Property 6 (FI equal to one).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' When FI(X) = 1, it holds that Dep (Y |X) = 1 and all other features are null-independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' As the BP dependency function is bounded by [0, 1] [5], it follows from Property 1 that � i∈Ωfeat FI(i) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Noting that each FI must be in [0, 1] due to Property 3, we find that FI(X) = 1 ⇒ FI(X′) = 0 for all X′ ∈ Ωfeat \\ {X}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Thus all other features are null-independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Next, we show that Dep (Y |X) = 1 must also hold, when FI(X) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Assume that Dep (Y |X) < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Using 11 Equation (6) (with Notations (N1) and (N2)) we find that 1 = FI(X) = � S⊆Ωfeat\\{X} w(S, Nvars) · D(X, Y, S) = � S⊆Ωfeat\\{X}:|S|>0 (w(S, Nvars) · D(X, Y, S)) + w(∅, Nvars) · D(X, Y, ∅) ≤ � S⊆Ωfeat\\{X}:|S|>0 (w(S, Nvars) · (1 − 0)) + w(∅, Nvars) · (Dep (Y |X) − 0) < � S⊆Ωfeat\\{X} w(S, Nvars) = Nvars−1 � k=0 �Nvars − 1 k � k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' · (Nvars − k − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Nvars!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' = Nvars−1 � k=0 (Nvars − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' · (Nvars − 1 − k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' · k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' · (Nvars − k − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Nvars!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' = Nvars−1 � k=0 1 Nvars = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Note that the inequality step follows from the range of the BP dependency function (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=', [0, 1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' The largest possible addition is when Dep (Y |S ∪ {X})− Dep (Y |S) = 1 − 0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' This result gives a contradiction, as 1 < 1 cannot be true, which means that Dep (Y |X) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Relevance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' When a variable gets an FI of one, the rest of the variables should be zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Additionally, it should mean that this variable contains the necessary information to fully determine Y , which is why Dep (Y |X) = 1 should hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Property 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Dep (Y |X) = 1 ̸⇒ FI(X) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' As counterexample, examine the case where there are multiple vari- ables that fully determine Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Properties 1 and 3 must still hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Thus, if FI 12 is one for every variable that fully determines Y , we get � i∈Ωfeat FI(i) ≥ 1 + 1 ̸= 1 = Dep (Y |Ωfeat) , which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Relevance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' This property is important for interpretation of the FI score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' When FI(X) ̸= 1, it cannot be automatically concluded that Y is not fully determined by X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' If Y is fully determined by X, we call X fully informative, as it gives all information that is necessary to determine Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Property 8 (Max FI when fully informative).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' If X is fully informative, it holds that FI(i) ≤ FI(X) for any i ∈ Ωfeat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Assume that there exists a feature i such that FI(i) > FI(X), when Y is fully determined by X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' To attain a higher FI, somewhere in the sum of Equation (6), a higher gain must be made by i compared to X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Observe that for any S ⊆ Ωfeat \\ {i, X} it holds that Dep (Y |S ∪ {i}) − Dep (Y |S) ≤ 1 − Dep (Y |S) = Dep (Y |S ∪ {X}) − Dep (Y |S) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' For any S ⊆ Ωfeat \\ {i} with X ∈ S, it holds that Dep (Y |S ∪ {i}) − Dep (Y |S) = Dep (Y |S ∪ {i}) − 1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' The last step follows from Equation (7), as the dependency function is in- creasing, thus Dep (Y |S ∪ {i}) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' In other words, no possible gain can be achieved with respect to X in the Shapley values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Therefore, it cannot hold that FI(i) > FI(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Relevance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Whenever a variable fully determines Y , it should attain the high- est FI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' What would an FI higher than such a score mean?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' It gives more information than the maximal information?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' When this property would not hold, it would result in a confusing and difficult interpretation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' 13 Property 9 (Limiting the outcome space).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' For any measurable function f and RV X, replacing X with f(X) never increases the assigned FI to this variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' The BP dependency function is non-increasing under functions of X [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' This means that for any measurable function g, it holds that Dep (Y |g(X)) ≤ Dep (Y |X) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Choose g to be the function that maps the union of any feature set S and the original RV X to the union of S and the replacement f(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' In other words g(S ∪ {X}) = S ∪ {f(X)} for any feature set S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' It then follows that: Dep (Y |S ∪ {f(X)}) = Dep (Y |g(S ∪ {X})) ≤ Dep (Y |S ∪ {X}) , and Dep (Y |S ∪ {f(X)}) − Dep (Y |S) ≤ Dep (Y |S ∪ {X}) − Dep (Y |S) for any S ⊆ Ωfeat \\ {X}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Thus, using Equation (6), we can conclude that replacing X with f(X) never increases the assigned FI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Relevance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' This is an important observation for preprocessing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Whenever a variable is binned, it would receive less (or equal) FI when less bins are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' It could also potentially provide a useful upper bound, when the FI is already known before replacing X with f(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Corollary 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' For any measurable function f and RV X, when X = f(X′) for another RV X′, replacing feature X by feature X′ will never decrease the assigned FI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' When X = f(X′) holds, it follows again (similar to Property 9) that Dep (Y |S ∪ {X}) = Dep (Y |S ∪ {f(X′)}) ≤ Dep (Y |S ∪ {X′}) for any S ⊆ Ωfeat\\{X}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Therefore, using Equation (6), observe that replacing X with X′ never decreases the assigned FI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Shapley values have additional properties when the characteristic function v is subadditive and/or superadditive [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' We show that our function, defined by Equation (5), is neither.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' 14 Property 10 (Neither subadditive nor superadditive).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Our characteristic function v(S) = Dep (Y |S) is neither subadditive nor superadditive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Consider the following two counterexamples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Counterexample subadditive: A function f is subadditive if for any S, T ∈ Ωfeat it holds that f(S ∪ T) ≤ f(S) + f(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Examine the dataset consisting of two binary features X1, X2 ∼ U({0, 1}) and a target variable Y = X1 · (1 − X2) + X2 · (1 − X1) which is the XOR of X1 and X2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Both X1 and X2 do not individually give any new information about the distribution of Y , thus v(X1) = v(X2) = 0 (see properties of the BP dependency function [5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' However, collectively they fully determine Y and thus v(X1∪X2) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' We can therefore conclude that v is not subadditive, as v(X1 ∪ X2) = 1 ̸≤ 0 + 0 = v(X1) + v(X2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Counterexample superadditive: A function f is superadditive if for any S, T ∈ Ωfeat it holds that f(S ∪ T) ≥ f(S) + f(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Consider the dataset consisting of two binary features X ∼ U({0, 1}) and a clone Xclone := X, where the target variable Y is defined as Y := X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Note that both X and Xclone fully determine Y , thus v(X) = v(Xclone) = 1 (see properties of the BP dependency function [5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Combining X and Xclone also fully determines Y , which leads to: v(X ∪ Xclone) = 1 ̸≥ 1 + 1 = v(X) + v(Xclone).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Thus, v is also not superadditive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Relevance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' If the characteristic function v is subadditive, it would hold that FI(X) ≤ v(X) for any X ∈ Ωfeat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' When v is superadditive, it follows that FI(X) ≥ v(X) for any X ∈ Ωfeat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' This is sometimes also referred to as individual rationality, which means that no player receives less, than what he could get on his own.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' This makes sense in a game-theoretic scenario with human players that can decide to not play when one could gain more by not 15 cooperating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' In our case, features do not have a free will, which makes this property not necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' The above proof shows that v is in our case neither subadditive nor superadditive, which is why we cannot use their corresponding bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Property 11 (Adding features can increase FI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' When an extra feature is added to the dataset, the FI of X can increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Consider the previously mentioned XOR dataset, where X1, X2 ∼ U({0, 1}) and Y = X1 · (1 − X2) + X2 · (1 − X1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' If at first, X2 was not in the dataset, the FI of X1 would be zero, as Dep (Y |X1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' However, if X2 is added to the dataset, the FI of X1 increases to 1 2 (see Property 15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' The FI of a feature can thus increase if another feature is added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Property 12 (Adding features can decrease FI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' When an extra feature is added to the dataset, the FI of X can decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Consider the dataset given by X ∼ U({0, 1}) and Y := X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' It im- mediately follows that FI(X) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' However, when a clone is introduced (Xclone := X), it holds that FI(X) = FI(Xclone), because of Property 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Ad- ditionally, it follows from Property 1 that FI(X) + FI(Xclone) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Thus, FI(X) = 1 2, and the FI of a variable can therefore be decreased if another variable is added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Relevance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' It is important to observe that the FI of a variable is dependent on the other features (Properties 11 and 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Adding or removing features could change the FI, which one needs to be aware of.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Property 13 (Cloning does not increase FI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' For any RV X ∈ Ωfeat, adding an identical variable Xclone := X (cloning) to the dataset, does not increase the FI of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Let FIwith clone(X) denote the FI of X after the clone Xclone is added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' 16 Using Equation (6) (with Notations (N1) and (N2)), we find FIwith clone(X) = � S⊆Ωfeat∪{Xclone}\\{X} w(S, Nvars + 1) · D(X, Y, S) (a) = � S⊆Ωfeat∪{Xclone}\\{X}:Xclone∈S w(S, Nvars + 1) · D(X, Y, S) + � S⊆Ωfeat∪{Xclone}\\{X}:Xclone̸∈S w(S, Nvars + 1) · D(X, Y, S) (b)= � S⊆Ωfeat∪{Xclone}\\{X}:Xclone∈S w(S, Nvars + 1) · 0 + � S⊆Ωfeat∪{Xclone}\\{X}:Xclone̸∈S w(S, Nvars + 1) · D(X, Y, S) = � S⊆Ωfeat\\{X} w(S, Nvars + 1) · D(X, Y, S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Equality (a) follows by splitting the sum over all subsets of Ωfeat ∪ {Xclone} \\ {X} whether Xclone is part of the subset or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Adding X to a subset that already contains the clone Xclone does not change the BP dependency func- tion, which is why Equality (b) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' The takeaway from this derivation is that the sum over all subsets S ⊆ Ωfeat ∪ {Xclone} \\ {X} reduces to the sum over S ⊆ Ωfeat \\ {X}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Comparing the new FIwith clone(X) with the original FI(X) gives FI(X) − FIwith clone(X) = � S⊆Ωfeat\\{X} w(S, Nvars) · D(X, Y, S) − � S⊆Ωfeat\\{X} w(S, Nvars + 1) · D(X, Y, S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Using Notation (N1), we find that w(S, Nvars + 1) w(S, Nvars) = |S|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='·(Nvars+1−|S|−1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' (Nvars+1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' |S|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='·(Nvars−|S|−1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Nvars!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' = Nvars − |S| Nvars + 1 < 1, 17 thus FI(X) − FIwith clone(X) ≥ 0 with equality if and only if FI(X) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Therefore, we can conclude that cloning a variable cannot increase the FI of X and will decrease the FI when X is null-independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Relevance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' We consider this a natural property of a good FI method, as no logical reason can be found why adding the exact same information would lead to an increase in FI for the original variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' The information a variable contains only becomes less valuable, as it becomes common knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Property 14 (Order does not change FI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' The order of the features does not affect the individually assigned FI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Consider the datasets [X1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' , XNvars] and [Z1, Z2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' , ZNvars], where Zπ(i) = Xi for some permutation π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' It holds that FI(Xi) = FI(Zπ(i)) for any i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' , Nvars}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Note that the order of features nowhere plays a roll in the definition of BP-FI (Equation (6)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' The BP dependency function is also independent of the given order, which is why this property trivially holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Relevance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' This is a very natural property of a good FI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Consider what would happen if the FI is dependent on the order in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Should all possible orders be evaluated and averaged to receive a final FI?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' We cannot find any arguments why someone should want FI to be dependent on the order of features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Datasets Next, we consider a few datasets, where we derive the theoretical outcome for the BP-FI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' These datasets are also used in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='3 to test FI methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' It is very hard to evaluate FI methods, as the ground truth is often unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' However, we believe that the FI outcomes on these datasets are all natural and defendable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' However, it remains subjective what one considers to be the ‘correct’ FI values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Property 15 (XOR dataset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Consider the following dataset consisting of two binary features X1, X2 ∼ U({0, 1}) and a target variable Y = X1 · (1 − X2) + X2 · (1 − X1) which is the XOR of X1 and X2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' It holds that FI(X1) = FI(X2) = 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Observe that Dep (Y |X1) = Dep (Y |X2) = 0 and Dep (Y |X1 ∪ X2) = 18 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' With Equation (6), it follows that FI(X1) = � S⊆{1,2}\\X1 |S|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' · (1 − |S|)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' 2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' (Dep (Y |S ∪ X1) − Dep (Y |S)) = |{∅}|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' · (1 − |{∅}|)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' 2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' (Dep (Y |{∅} ∪ X1) − Dep (Y |{∅})) + |{X2}|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' · (1 − |{X2}|)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' 2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' (Dep (Y |X1 ∪ X2) − Dep (Y |X2)) = 1 2 · (Dep (Y |X1) − 0) + 1 2 · (Dep (Y |X1 ∪ X2) − Dep (Y |X2)) = 1 2 · 0 + 1 2 · (1 − 0) = 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Using Property 1, it follows that FI(X2) = 1 − FI(X1) = 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Relevance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' This XOR formula is discussed and used to test FI methods in [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' However, they only test for equality (FI(X1) = FI(X2)), not the specific value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Due to symmetry, we would also argue that both X1 and X2 should get the same FI, as they fulfill the same role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Together, they fully determine Y , which is why the total FI should be one (see Property 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Dividing this equally amongst the two variables, gives a logical desirable FI outcome of 1 2 for each variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Property 16 (Probability dataset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Consider the following dataset consist- ing of Y = ⌊XS/2⌋ and Xi = Zi + (S − 1) with Zi ∼ U ({0, 2}) for i = 1, 2 and P(S = 1) = p, P(S = 2) = 1 − p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' It holds that FI(X1) = p and FI(X2) = 1 − p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' 19 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Observe that by Equation (4) UD (X1, Y ) = � x1∈{0,1,2,3} pX1(x1) · � y∈{0,1} ��pY |X1=x1(y) − pY (y) �� = � x1∈{0,2} pX1(x1) · � y∈{0,1} ����pY |X1=x1(y) − 1 2 ���� + � x1∈{1,3} pX1(x1) · � y∈{0,1} ����pY |X1=x1(y) − 1 2 ���� = � x1∈{0,2} p 2 · �����1 − 1 2 ���� + ����0 − 1 2 ���� � + � x1∈{1,3} 1 − p 2 � y∈{0,1} |pY (y) − pY (y)| = p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Similarly, it follows that UD (X2, Y ) = 1 − p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' UD (Y, Y ) = � y′∈{0,1} pY (y′) · � y∈{0,1} ��pY |Y =y′(y) − pY (y) �� = � y′∈{0,1} 1 2 · �����1 − 1 2 ���� + ����0 − 1 2 ���� � = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' From Equation (3), it follows that Dep (Y |X1) = p and Dep (Y |X2) = 1 − p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Additionally, note that knowing X1 and X2 fully determines Y , thus 20 Dep (Y |X1 ∪ X2) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' With Equation (6), we now find FI(X1) = � S⊆{X1,X2}\\X1 |S|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' · (1 − |S|)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' 2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' (Dep (Y |S ∪ X1) − Dep (Y |S)) = |{∅}|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' · (1 − |{∅}|)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' 2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' (Dep (Y |{∅} ∪ X1) − Dep (Y |{∅})) + |{X2}|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' · (1 − |{X2}|)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' 2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' (Dep (Y |X1 ∪ X2) − Dep (Y |X2)) = 1 2 · (Dep (Y |X1) − 0) + 1 2 · (Dep (Y |X1 ∪ X2) − Dep (Y |X2)) = 1 2 · (p − 0) + 1 2 · (1 − (1 − p)) = p 2 + p 2 = p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Using Property 1, it follows that FI(X2) = 1 − FI(X1) = 1 − p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Relevance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' At first glance, it is not immediately clear why these FI values are natural, which is why we discuss this dataset in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' S can be considered a selection parameter that determines if X1 or X2 is used for Y with probability p and 1 − p, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Xi is constructed in such a way that it is uniformly drawn from {0, 2} or {1, 3} depending on S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' However, as Y = ⌊XS/2⌋, it holds that XS = 0 and XS = 1 give the same outcome for Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' The same holds for XS = 2 and XS = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Therefore, note that the distribution of Y is independent of the selection parameter S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Knowing X1 gives the following information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' First, S can be derived from the value of X1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' When X1 ∈ {0, 2} it must hold that S = 1, and if X1 ∈ {1, 3} it follows that S = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Second, when S = 1 it means that Y is fully determined by X1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' If S = 2, knowing that X1 = 1 or X1 = 3 does not provide any additional information about Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' With probability p knowing X1 will fully determine Y , whereas with probability 1 − p, it will provide no information about the distribution of Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' The outcome FI(X1) = p, is therefore very natural.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' The same argumentation applies for X2, which leads to FI(X2) = 1 − p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' 21 4 Comparing with existing methods In the previous section, we showed that BP-FI has many desirable proper- ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Next, we evaluate for a large collection of FI methods if the properties hold for several synthetic datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Note that these datasets can only be used as counterexample, not as proof of a property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' First, we discuss the in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='1 the FI methods that are investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Second, we give the datasets (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='2) and explain how they are used to test the properties (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' The results are discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='1 Alternative FI methods A wide range of FI methods have been suggested for all kinds of situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' It is therefore first necessary to discuss the major categorical differences between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Global vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' local An important distinction to make for FI methods is whether they are constructed for local or global explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Global FI methods give an importance score for each feature over the entire dataset, whereas local FI methods explain which variables were important for a single example [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' The global and local scores do not have to coincide: “features that are globally important may not be important in the local context, and vice versa” [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' This research is focussed on global FI methods, but some- times a local FI approach can be averaged out to obtain a global FI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' For example, in [34] a local FI method is introduced called Tree SHAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' It is also used globally, by averaging the absolute values of the local FI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Model-specific vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' model-agnostic A distinction within FI methods can be made between model-specific and -agnostic methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Model-specific methods aim to find the FI using a prediction model such as a neural network or random forest, whereas model-agnostic methods do not use a prediction model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' The BP-FI is model-agnostic, which therefore gives insights into the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Whenever a model-specific method is used, the focus lies more on gaining information about the prediction model, not the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' In our tests, we use both model-specific and -agnostic methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Classification vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' regression Depending on the exact dataset, the target variable is either categorical or numerical, which is precisely the difference between classification and regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Not all existing FI methods can handle both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' In this research, we generate synthetic classification datasets, so 22 we only examine FI methods that are intended for these cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' An additional problem with regression datasets, is that continuous variables need to be converted to discrete bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' This conversion could drastically change the FI scores, which makes it harder to draw fair conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Collection We have gathered the largest known collection of FI methods from various sources [2, 4, 6, 8, 11–13, 17, 18, 20, 22, 28, 35, 38, 40, 42, 43, 45, 47, 48, 57, 58] or implemented them ourselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' This has been done with the following policy: Whenever code of a classification FI method was avail- able in R or Python or the implementation was relatively straightforward, it was added to the collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' This resulted in 196 base methods and 468 to- tal methods, as some base methods can be combined with multiple machine learning approaches or selection objectives, see Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' However, beware that most methods also contain additional parameters, which are not inves- tigated in this research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' The default values for these parameters are always used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='2 Synthetic datasets Next, we briefly discuss the datasets that are used to test the properties de- scribed in Section 3 for alternative FI methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' In Appendix A, we introduce each dataset and explain how they are generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' To draw fair conclusions, the datasets are not drawn randomly, but fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' To give an example of how we do generate a dataset, we examine Dataset 1 Binary system (see Ap- pendix A), where the target variable Y is defined as Y := �3 i=1 2i−1 ·Xi with Xi ∼ U ({0, 1}) for all i ∈ {1, 2, 3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' To get interpretable results, we draw each combination of X and Y values the same number of times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' An example can be seen in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' For most datasets, we draw 1,000 samples in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' How- ever Datasets 6 and 7 consist of 2,000 samples to ensure null-independence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' The datasets have been selected to be computationally inexpensive and to test many properties (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='3) with a limited number of datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' An overview of the generated datasets can be found in Table 3 including the cor- responding outcome of BP-FI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Appendix A provides more technical details about the features and target variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='3 Property evaluation In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='1, we gathered a collection of existing FI methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' In this section, we evaluate if these FI methods have the same desirable and proven properties of the BP-FI method (see Section 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Due to the sheer number of FI methods (468), it is unfeasible to prove each property for every method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' 23 Table 1: All evaluated FI methods: List of all FI methods that are evaluated in the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' The colored methods work in com- bination with multiple options: Logistic RegressionI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' II,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' III,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' RidgeI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' II,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Linear RegressionI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' II,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' LassoI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' II,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' SGD ClassifierI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' III,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' MLP ClassifierI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' II,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' K Neighbors ClassifierI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' II,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Gradient Boosting ClassifierI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' II,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' IV,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' AdaBoost ClassifierI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' II,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Gaussian NBI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' II,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Bernoulli NBI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' II,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Linear Discriminant AnalysisI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' II,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Decision Tree ClassifierI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' II,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' IV,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Random Forest ClassifierI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' II,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' IV,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' SVCI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' CatBoost ClassifierI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' II,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' LGBM ClassifierI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' II,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' IV,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' XGB ClassifierI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' II,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' IV,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' VII,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' XGBRF ClassifierI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' II,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' IV,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' VII,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' ExtraTree ClassifierIV,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' ExtraTrees ClassifierIV,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' plsdaVI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' splsdaVI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' giniVIII,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' entropyVIII,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' NN1IX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' NN2IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' This leads to a total of 468 FI methods from various sources [2, 4, 6, 8, 11–13, 17, 18, 20, 22, 28, 35, 38, 40, 42, 43, 45, 47, 48, 57, 58] or self-implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Feature Importance methods 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' AdaBoost Classifier 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Random Forest ClassifierVIII 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Extra Trees ClassifierVIII 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Gradient Boosting Classifier 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' SVR absolute weights 6.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' modified t score 182.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' MIM 183.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' MRMR 184.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' JMI 185.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Add: CIFE 186.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' CMIM 187.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' ICAP 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SPEC 195.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' MCFS 196.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' UDFS 197.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' R2 198.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' DC 199.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' BCDC 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' AIDC 201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' 45,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' 48] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='21-27 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='shap explainer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='[35] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='28-29 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='Relative feature importance ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='[6] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='30-32 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='R vip ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='[20] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='33-44 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='scipy stats ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='[58] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='45-47 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='booster classifier ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='[12] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='48-109 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='R caret classifier ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='[28] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='110-144 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='R firm classifier ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='[20] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='145-155 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='R FSinR Classifier ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='[4] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='156-159 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='Sage Classifier ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='[13] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='160-161 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='QII Averaged Classifier ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='[57] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='162-165 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='Rebelosa Classifier ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='[47] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='166-168 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='Relief Classifier ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='[40] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='169-196 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='ITMO ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='[43] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='197-201 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='Sunnies ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='[17] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='202 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='BP-FI 24 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='Table 2: Fixed draw: Example of how the datasets are drawn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Instead of drawing each possible outcome uniformly at random, we draw each combina- tion an equal fixed number of times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Outcome # Drawn X1 X2 X3 Y Fixed Uniform 0 0 0 0 125 133 0 0 1 4 125 129 0 1 0 2 125 121 0 1 1 6 125 109 1 0 0 1 125 136 1 0 1 5 125 124 1 1 0 3 125 115 1 1 1 7 125 133 Instead, we devise tests to find counterexamples of these properties using generated datasets (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Due to the number of tests (18), we only discuss the parts that are not straightforward, as most test directly measure the corresponding property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' An overview of each test can be found in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' A summary of the tests can be found in Table 4, where it is outlined for each test which property is tested on which datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Computational errors To allow for computational errors, we tolerate a margin of ǫ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='01 in each test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' If, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=', an FI value should be zero, a score of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='01 or −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='01 is still considered a pass, whereas an FI value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='05 is counted as a fail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Usually, this works in the favor of the FI method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' However, in Test 9 we evaluate if the FI method assigns zero FI to variables that are not null- independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' In this case, we consider |FI(X)| ≤ ǫ to be zero, as the datasets are constructed in such a way that variables are either null-independent or far from being null-independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Running time We limit the running time to one hour per dataset on an i7-12700K processor, whilst four algorithms are running simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' The datasets consist of a small number of features with a very limited outcome space and the number of samples is either 1,000 or 2,000, which is why one hour is a reasonable amount of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' NaN or infinite values In some cases, an FI method assigns NaN or ±∞ to a feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' How we handle these values depends on the test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=', we consider NaN to fall outside the range [0, 1] (Tests 4 and 55), but when we evaluate if the sum of FI values remains stable (Test 2) or if two symmetric 25 Table 3: Overview of datasets: An overview of the generated datasets and the corresponding BP-FI outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' The details of these datasets can be found in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' They are used to evaluate if existing FI methods adhere to the same properties as BP-FI (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Dataset Variables BP-FI outcome Binary system 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' base (X1, X2, X3) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='333, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='333, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='333) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' clone (Xclone 1 , X1, X2, X3) (0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' (different order) (X3, Xfull 4 , Xfull 5 , Xclone 1 , X1, X2) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='136, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='248, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='248, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='117, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='117, 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='000) Probability dataset 18-28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' for p ∈ {0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' , 1} (X1, X2) (p, 1 − p) 26 Table 4: Overview of experiments: To evaluate if existing FI methods have the same properties as the BP-FI, we use the tests from Appendix B on the datasets from Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' ✓means that the test is performed on this dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' ↕(i) denotes that this dataset is used as baseline or in conjunction with dataset i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' The details of the tests and datasets can be found in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Test Evaluates: Dataset (Appendix A) (Appendix B) Property/Corollary 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 1 1 ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='27 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='features receive the same FI (Test 3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' we consider twice NaN or twice ±∞ to be the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Property 9 (Limiting the outcome space) Property 9 states that ap- plying any measurable function f to a RV X cannot increase the FI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' In other words, FI(X) ≥ FI(f(X)) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' This property is tested using Datasets 8 to 10 (see Table 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' These datasets contain variables that are the outcome of binning the target variable using different number of bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' This is how Property 9 is tested, as it should hold that FI(Xi) ≥ FI(Xj), whenever Xi has more bins than Xj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Properties 11 and 12 (Adding features can increase/decrease FI) In all other tests, the goal is to find a counterexample of the property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' How- ever, Tests 13 and 14 are designed to evaluate if a feature gets an increased/de- creased FI when a feature is added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' This increase/decrease should be more than ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' The datasets are chosen in such a way that both an increase and de- crease could occur (according to the BP-FI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Only for these tests, we consider the test failed if no counterexample (increase/decrease) is found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='4 Evaluation results An overview of the general results can be seen in Table 5, where the number of methods that pass and fail is given per test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Next, we highlight additional insights into the results of the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Best performing methods The top 20 FI methods that pass the most tests are given in Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Out of 18 tests, the BP-FI passes all tests, which is as expected as we have proven in Section 3 that the BP-FI actually has these properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Classifiers from R FSinR Classifier and ITMO fill 11 of the top 20 spots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Out of 11 R FSinR Classifier methods, six are in the top 20, which is quite remarkable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' However, observe that the gap between the BP-FI method and the second best method is 18−11 = 7 passed tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Additionally, 424 out of 468 methods fail more than half of the tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Figure 1 shows how frequently each number of passed tests occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' A detailed overview of where each top 20 method fails, can be seen in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Note again that in Tests 13 and 14 it is considered a fail if adding features never increase or decrease the FI, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' It could be that these methods are in fact capable of increasing or decreasing, but for some reason do not with our datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Strikingly, most of these methods perform bad on the datasets with a desirable outcome 28 Table 5: Overview of the results: Each FI method is evaluated using the tests outlined in Appendix B, which evaluates if the method adheres to the same properties as the BP-FI (see Section 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' This table summarizes out of 468 FI methods how many pass or fail the test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' A distinction is made for the top 20 passing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Failing the test means that a counterexample is found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Note that passing the test does not ‘prove’ that the FI method actually has the property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' No result indicates that the test could not be executed, because the running time of the FI method was too long or an error occurred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='Test ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='Overall ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='# Passed ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='92 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='45 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='438 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='97 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='132 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='283 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='97 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='31 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='141 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='241 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='243 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='314 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='365 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='172 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='# Failed ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='466 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='369 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='421 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='29 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='267 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='370 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='335 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='184 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='370 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='413 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='326 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='98 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='216 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='145 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='58 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='288 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='421 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='459 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='# No result ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='24 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='129 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='45 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='34 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='Top 20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='# Passed ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='19 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='# Failed ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='19 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='# No result ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='(Tests 17 and 18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Adding a variable without additional information (Test 2), also often leads to a change in total FI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Test 1 In this test, it is evaluated if the sum of FI values is the same as the sum for BP-FI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' At first, this seems a rather strict requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' However, it holds for all datasets that were used that Dep (Y |Ωfeat) is either zero or one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Thus, we essentially evaluate if the sum of FI is equal to one, when all variables collectively fully determine Y and zero if all variables are null- independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' The tests show that no FI method is able to pass this test, except for the BP-FI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' To highlight some of the methods that came close: 162.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Rebelosa Classifier RF, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Random Forest Classifier entropy, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Ran- dom Forest Classifier gini only fail for the datasets where the sum should be zero (because of null-independence) and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' AdaBoost Classifier only does not pass on three of the four datasets based on the XOR function (see Ap- pendix A), where the sum should be one, but was zero instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' FI method 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' lssvmRadial came closest with two fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' For the null-independent datasets (Datasets 6 and 7), it gives each feature an FI of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='5, making the sum larger than zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Test 2 In Figure 2, a breakdown is given of where the sum of the FI values is unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' The most errors are made with the Binary system datasets, 29 Table 6: Top 20: Out of 468 FI methods, these 20 methods pass the 18 tests given in Appendix B the most often.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' These tests are designed to examine if an FI method adheres to the same properties as the BP-FI , given in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Passed means that the datasets from Appendix A do not give a counterexample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Certainly, this does not mean that the FI method is proven to actually have this property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Failed means that a counterexample was found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' No result indicates that the test could not be executed, because the running time of the FI method was too long or an error occurred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Combined result: Method # Passed # Failed # No result 202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' BP-FI 18 0 0 147.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' cramer 11 7 0 148.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' gainRatio 11 7 0 153.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' roughsetConsistency 11 7 0 155.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' symmetricalUncertain 11 7 0 172.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' su measure 11 7 0 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' sdwd 10 7 1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Extra Trees Classifier 10 8 0 116.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' rpart 10 8 0 126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' null 10 8 0 145.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' binaryConsistency 10 8 0 152.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' mutualInformation 10 8 0 161.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Banzhaf Ridge 10 8 0 197.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' R2 10 8 0 162.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' RF 10 8 0 166.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Relief 10 8 0 173.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' spearman corr 10 8 0 188.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' DCSF 10 8 0 189.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' CFR 10 8 0 191.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' IWFS 10 8 0 0 20 40 60 80 100 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 # Passed tests Frequency BP-FI 1 0 0 0 0 0 0 5 13 25 74 78 78 101 57 24 7 4 1 Figure 1: Frequency of total passed test: Histogram of the number of passed tests (out of 18) for the 468 FI methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' 30 0 50 100 150 200 250 300 1↕2 1↕3 1↕4 1↕5 6↕7 8↕9 8↕10 11↕12 11↕13 14↕16 14↕17 Compared datasets # Unstable sum FI 188 302 311 299 163 190 95 203 194 124 117 Figure 2: Unstable sum FI: Whenever a variable is added that does not give any additional information, the sum of all FI should remain stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' For each comparison, we determine how often this is not the case out of 468 FI methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' when a fully informative feature is added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' In total, 92 methods passed the test, whereas 369 failed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' From these 369 methods, 279 fail with at least one increase of the sum, whereas 232 methods fail with at least one decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' An alarming number of FI methods thus assign significantly more or less FI when a variable is added that does not contain any additional information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' More or less credit is given out, whilst the collective knowledge is stable and does not warrant an increase or decrease in credit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Additionally, when the initial and final sum both contain a NaN value, it is considered as a pass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Three out of 92 would have not passed without this rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' If only the initial or the final sum contained NaN, it is considered a fail, because the sum is not the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Only five methods fail solely by this rule: 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Fisher Score, 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' f classif, 178.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' anova, 179.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' laplacian score and 192.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' NDFS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Test 11 Figure 3 shows how often each variable is within an ǫ-bound of the largest FI in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Fully informative variables should attain the largest FI, according to Property 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' In total, we observe that the fully informative variables are often the largest FI with respect to the other variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' However, there still remain many cases where they are not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' 326 FI methods fail this test, thus definitively not having Property 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' This makes interpretation difficult, when a variable can get more FI than a variable which fully determines the target variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' What does it mean, when a variable is more important than a variable that gives perfect information?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' 31 Variables within dataset (i) # Variable in arg max 0 50 100 150 200 250 300 350 400 450 500 Theoretical maximum: 468 (# FI methods) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 Figure 3: Argmax FI: For each variable in every dataset, we determine how often it receives the largest FI (within an ǫ-bound for ǫ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='01) with respect to the other variables in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Fully informative variables should attain the largest FI (see Property 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' All fully informative variables are shaded in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Test 10, 17, 18 These tests all evaluate if the FI method assigns a specific value to a feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' From Table 5, we observe that not many methods are able to pass these tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' This is not surprising, as they have not been thoroughly tested yet to give a specific value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' This is one of the important contributions of this research, which is why we want to elaborate on the attempts that have been made in previous research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' A lot of synthetic datasets for FI have been proposed [1–3, 6, 7, 9, 15–17, 19, 21, 23–25, 30, 32–34, 37, 39, 41, 50–52, 55, 56, 59, 61, 62], but no specific desirable FI values were given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Most commonly, synthetic datasets are generated to evaluate the ability of an FI method to find noisy features [3, 7, 19, 21, 23, 24, 30, 50, 52, 55, 59, 61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' The common general concept of such a dataset is that the target variable is independent of certain variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' The FI values are commonly evaluated by comparing the FI values of independent variables with dependent variables with the goal to establish if the FI method is able to find independent variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' If the FI method actually predicts the exact desirable FI is not considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Next, we highlight the papers where some comment about the desired FI is made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Lundberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' [34] give two similar datasets, where one variable increases in importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' They evaluate multiple FI methods to see if the same behavior is reflected in the outcome of these methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' This shows that some commonly used methods could assign lower importance to a variable, when it should actually be increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Giles et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' [19] also design multiple 32 artificial datasets to represent different scenarios, where comments are made about which variables should obtain more FI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Sundararajan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' [55] remark that if every feature value is unique, that all variables get equal attributions for an FI method (CES) even if the function is not symmetric in the variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' If a tiny amount of noise is added to each feature, all features would get identical attributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' However, no assessment is done on the validity of this outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Owen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' [41] give the following example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Let f(x1, x2) = 106x1+x2 with x1 = 106x2, where they argue that, despite the larger variance of x1, both variables are equally important, as the function can be written as a function of x1 alone, but also only as a function of x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Although we have previously seen that ‘written as a function of’ is not a good criterion (due to dependencies), we agree with the authors that the FI should be equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Another example is given by Owen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' [41], where P(x1 = 0, x2 = 0, y = y0) = p0, P(x1 = 1, x2 = 0, y = y1) = p1, and P(x1 = 0, x2 = 1, y = y2) = p2 are the possible outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' If p0 = 0, it is stated in [41] that the Shapley relative importance of x1 is 1 2, which is “what it must be because there is then a bijection between x1 and x2”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' This is an interesting observation, as most papers do not comment about the validity of an outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Additionally, when y1 = y2 (and y0 ̸= y1), Owen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' [41] argue that the most important variable, is the one with the largest variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Fryer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' [17] also create a binary XOR dataset (see Dataset 14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' They evaluate seven FI methods for this specific dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' The role of X1 and X2 is symmetric, thus the assigned FI should also be identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' It is shown that six out of seven methods do indeed give a symmetrical result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' However, the exact FI value varies greatly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' SHAP gives FI of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='19, whereas Shapley DC assigns 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='265 as FI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Only symmetry is checked, not the accuracy of the FI method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' In conclusion, existing research was not focussed on predicting the exact accurate FI values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' It is therefore not surprising that FI methods fail these accuracy tests so often.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Table 7 outlines in more detail how often the variables are assigned an FI value outside an ǫ-bound (with ǫ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='01) of the desired outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' With Dataset 11, the FI methods mostly struggle with assigning 1 to the fully informative variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' In total, 413 methods failed Test 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' For Datasets 14 and 17, the two XOR variables fail about as often.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Comparing these two datasets, it is interesting to note that the XOR variables fail more often, when a null- independent variable is added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' In total, 421 methods failed Test 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Test 18 is hard, as the FI method should assign the correct values for all probability datasets (see Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Only five methods are able to pass this test: 152.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' mutualInformation, 153.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' roughsetConsistency, 162.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' RF, 175.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' fechner corr, and 202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' BP-FI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' These five methods also pass Test 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' However, besides BP-FI, there is only one method that also satisfies Test 17, which is 162.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' RF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' The other three methods all assign only zeros for Datasets 14 and 17, not 33 0 300 350 400 450 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='9 1 Probability dataset (p) Frequency failed 399 438 441 445 442 427 451 453 450 450 405 Figure 4: Breakdown Test 18 per dataset: In Test 18 an FI method needs to assign the correct FI values for every probability dataset (see Ap- pendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' In this figure, we breakdown per dataset how often an FI method fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' identifying the value that the XOR variables hold, when their information is combined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' In Figure 4, a breakdown is given for each probability dataset how often FI methods fail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' An unexpected result, is that the dataset with probability p < 1 2 and the dataset with probability 1 − p do not fail as often.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Consistently, p < 1 2 fails less often than its counterpart 1 − p, although the datasets are the same up to a reordering of the features and the samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' This effect can also be seen in Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' No result Focussing on the no result row of Table 5, there is one base method named 158.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' KernelEstimator in combination with Lasso that in all cases did not work or exceeded running time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' The large number of no results in Test 12 stem mostly from slow running times on the three datasets that are used in the test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' At least 63 methods were too slow for each dataset, which automatically means that the test cannot be executed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' 5 Discussion and future research Whilst it is recommended to use our new FI method, it is important to understand the limitations and potential pitfalls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Below we elaborate on both the shortcomings of the approach proposed, and the related challenges for further research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' We start by discussing by some matters that one needs 34 Table 7: Specific outcomes: Tests 10, 17 and 18 all evaluate if an FI method gives a specific outcome for certain dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' In this table, it is out- lined how often each variable of these datasets is assigned a value outside an ǫ-bound (with ǫ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='01) of the desired outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' # Non desirable outcome not NaN NaN Dataset Desirable outcome X1 X2 X3 X1 X2 X3 11 (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' 0) 360 89 88 4 4 4 14 ( 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' 1 2) 353 351 5 5 17 ( 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' 0) 369 364 90 5 5 5 18 (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' 1) 82 352 4 4 19 ( 1 10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' 9 10) 412 434 3 3 20 ( 2 10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' 8 10) 434 438 3 3 21 ( 3 10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' 7 10) 435 441 3 3 22 ( 4 10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' 6 10) 439 436 3 3 23 ( 5 10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' 5 10) 423 422 3 3 24 ( 6 10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' 4 10) 448 447 3 3 25 ( 7 10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' 3 10) 449 446 3 3 26 ( 8 10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' 2 10) 446 444 3 3 27 ( 9 10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' 1 10) 444 435 3 3 28 (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' 0) 352 86 5 5 35 to be aware of when applying the BP-FI (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Next, we discuss some choices that were made for the experiments in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Finally, we elaborate on other possible research avenues in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='1 Creating awareness Binning Berkelmans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' [5] explained that the way in which continuous data is discretized can have a considerable effect on the BP dependency func- tion, which is why all datasets that were used in our research are discrete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' If a feature has too many unique values (due to poor binning), it will receive a higher FI from BP-FI, as more information can be stored in the unique values (see Property 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' On the other hand, when too few bins are chosen, an impor- tant feature can receive low FI, as the information is lost due to the binning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Future research should investigate and test which binning algorithms give the closest results to the underlying FI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Too few samples Consider the following dataset: Xi, Y ∼ U ({0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=', 9}) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' for i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' , 5}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Note that all features are null-independent, as Y is just uniformly drawn without considering the features in any way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' If nsamples = ∞, the desired outcome would therefore be (0, 0, 0, 0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' How- ever, when not enough samples are given in the dataset, the features will get nonzero FI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Considering that the total number of different feature values is 105, combining all features does actually give information about Y , when nsamples ≪ 105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' For any possible combination of features, it is unlikely that it occurs more than once in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Therefore, knowing all feature values would (almost surely) determine the value of Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Property 1 gives that the sum of all FI should therefore be one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' All feature variables are also symmet- ric (Property 2), which is why the desired outcome is ( 1 5, 1 5, 1 5, 1 5, 1 5) instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' This example shows that one should be aware of the influence of the number of samples on the resulting FI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Variables that do not influence Y can still contain information, when not enough samples are provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' In this way, insufficient samples could lead to wrong conclusions, if one is not wary of this phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Counterintuitive dependency case The Berkelmans-Pries dependency of Y on X measures how much probability mass of Y is shifted by know- ing X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' However, two similar shifts in probability mass could lead to dif- ferent predictive power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' To explain this, we examine the following dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' 36 X1, X2 ∼ U ({0, 1}) with P(Y = y|X1 = x1, X2 = x2) = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 1/4 if (x2, y) = (0, 0), 3/4 if (x2, y) = (0, 1), 5/8 if (x1, x2, y) = (0, 1, 0), 3/8 if (x1, x2, y) = (0, 1, 1), 7/8 if (x1, x2, y) = (1, 1, 0), 1/8 if (x1, x2, y) = (1, 1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Knowing the value of X2 shifts the distribution of Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Before, Y was split 50/50, but when the value of X2 is known, the labels are either split 25/75 or 75/25, depending on the value of X2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Knowing X1 gives even more in- formation, as e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=', knowing X1 = X2 = 1 makes it more likely that Y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' However, the shift in distribution of Y is the same for knowing only X2 and X1 combined with X2, which results in Dep (Y |X2) = Dep (Y |X1 ∪ X2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' This is a counterintuitive result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Globally, knowing X2 or X1 ∪ X2 gives the same shift in distribution, but locally we can predict Y much better if we know X1 as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' We are unsure how this effects the BP-FI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' In this case, it follows that FI(X1 ∪ X2) > FI(X2), which is desirable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' It is not unthinkable that a solution can be found to modify the dependency function in order to get a more intuitive result for such a case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Think e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=', of a different distance metric, that incorporates the local accuracy given the feature values or a conditional variant, which not only tests for independence, but also for con- ditional independence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' These are all critical research paths that should be investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Using FI for feature selection Feature selection (FS) is “the problem of choosing a small subset of features that ideally is necessary and sufficient to describe the target concept” [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Basically, the objective is to find a subset of all features that gives the best performance for a given model, as larger feature sets could decrease the accuracy of a model [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Many FI methods actually stem from a FS procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' However, it is important to stress that high FI means that it should automatically be selected as feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Shared knowledge with other features could render the feature less useful than expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' The other way around, low FI features should not automatically be discarded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' In combination with other features, it could still give some additional insights that other features are not able to provide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Calculation of BP-FI values could also provide insight into which group of K features Y is most dependent on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' To derive the result of BP-FI, all dependencies of Y 37 on a subset S ⊆ Ωfeat are determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' If only K variables are selected, it is natural to choose S∗ K ∈ arg max S⊆Ωfeat:|S|=K {Dep (Y |S)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' These values are stored as an intermediate step in BP-FI, thus S∗ K can be derived quickly thereafter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Larger outcome space leads to higher FI We have proven that a larger outcome space can never lead to a decrease in FI for BP-FI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' This means, that features with more possible outcomes are more likely to attain a higher FI, depending on the distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' There is a difference between a feature that has many possible outcomes that are almost never attained, and a feature where many possible outcomes are regularly observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' We do not find this property undesirable, as some articles suggest [53, 61], as we would argue that a feature can contain more information by storing the information in additional outcomes, which would lead to an non-decreasing FI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='2 Experimental design choices Regression To avoid binning issues, we only considered classification mod- els and datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' There are many more regression FI methods, that should be considered in a similar fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' However, to draw clear and accurate con- clusions, it is first necessary to understand how binning affects the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Sometimes counterintuitive results can occur due to binning, that are not necessarily wrong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' In such a case, it is crucial that the FI method is not depreciated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Runtime In the experiments, it could happen that an FI method had no result, due to an excessive runtime or incompatible FI scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' The maximum runtime for each algorithm was set to one hour per dataset on an i7-12700K processor with 4 algorithms running simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' The maximum runtime was necessary due to the sheer number of FI methods and datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Run- ning four algorithms in parallel could unfairly penalize the runtime, as the processor is sometimes limited by other algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' In some occurrences, other parallel processes were already finished, which could potentially lower the runtime of an algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' There is a potential risk here, that accurate (but slow) FI methods are not showing up in the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' However, our syn- thetic datasets are relatively small with respect to the number of samples and the number of features, and we argue that one hour should be reasonable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' 38 Depending on the use case, sometimes a long time can be used to deter- mine an FI value, whereas in other cases it could be essential to determine it rather quickly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Especially for larger datasets, it could even be unfeasible to run some FI methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' BP-FI uses Shapley values, which are exponen- tially harder to compute when the number of features grow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Approximation algorithms should be developed to faster estimate the true BP-FI outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Quick approximations could be useful if the runtime is much faster and the approximation is decent enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Already, multiple papers have suggested approaches to approximate Shapley values faster [1, 10, 24, 31, 54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' These approaches save time, but at what cost?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' A study could be done to find the best FI method given a dataset and an allowed running time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Stochasticity methods One factor we did not incorporate, is the stochas- ticity of some FI methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Some methods do not predict the same FI values, when it is repeatedly used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' As example, 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' rf predicted for Dataset 3 (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='1, 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='7, 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='9, 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='2, 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='7) rounded to the first decimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Running the method again gives a different result: (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='4, 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='0, 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='4, 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='6, 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='1), as this method uses a stochastic random forest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' In principle, it is undesirable that an FI method is stochastic, as we believe that there should be a unique assignment of FI given a dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Due to the number of FI methods and datasets, we did not repeat and averaged each FI method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' This would however give a better view on the performance of stochastic FI methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Parameter tuning All FI methods were used with default parameter val- ues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Different parameter values could lead to more or less failed tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' How- ever, the ideal parameter setting is not known beforehand, making it nec- essary to search a wide range of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' This was not the focus of our research, but future research could try to understand and learn which parameter values should be chosen for a given dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Ranking FI methods In Table 6, the 20 FI methods that passed the most tests were highlighted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' However, it is important to stress that not every test is equally difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Depending on the user, some properties could be more or less relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' It is e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=', much harder to accurately predict the specific values for 11 datasets (Test 18), than to always predict non-negatively (Test 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Every test is weighed equally, but this does not necessarily represent the difficulty of passing each test accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' However, we note that 175.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' fechner corr is the only FI method that passed Test 18, that ended up outside the top 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' We stress that we focussed on finding out if FI methods adhere to the properties, not necessarily finding the best and most fair ranking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' 39 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='3 Additional matters Global vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' local BP-FI is designed to determine the FI globally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' How- ever, another important research area focusses on local explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' These explanations should provide information about why a specific sample has a certain target value instead of a different value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' They provide the necessary interpretability that is increasingly demanded for practical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' This could give insights for questions like: ‘If my income would be higher, could I get a bigger loan?’' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=', ‘Does race play a role in this prediction?’' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=', and ‘For this automated machine learning decision, what were the critical factors?’' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Many local FI methods have been proposed, and some even use Shapley values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' A structured review should be made about all proposed local methods, simi- lar to our approach for global FI methods to find which local FI methods actually produce accurate explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' BP-FI can be modified to provide local explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' For example, we can make the characteristic function localized in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Let YS,z be Y restricted to the event that Xi = zi for i /∈ S, let us similarly define XS,z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Then, we can define a localized characteristic function by: vz(S) := Dep (YS,z|XS,z) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' (8) When dealing with continuous data, assuming equality could be too strict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' In this case, a precision vector parameter ǫ can be used, where we define YS,z,ǫ to be Y restricted to the event that |Xi − zi| ≤ ǫi for i /∈ S, and in the same way we define XS,z,ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' We then get the following localized characteristic function: vz,ǫ(S) := Dep (YS,z,ǫ|XS,z,ǫ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Additionally, there are at least two possible ways how BP-FI can be adapted to be used for local explanations if some distance function d(i, j) and param- eter δ are available to determine if sample j is close enough to i to be consid- ered ‘local’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' We can (I) discard all samples where d(i, j) > δ and/or (II) gen- erate samples, such that d(i, j) ≤ δ for all generated samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Then, we can use BP-FI on the remaining samples and/or the generated samples, which would give local FI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Note that there should still be enough samples, as we have previously discussed that too few samples could lead to different FI outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' However, there are many more ways how BP-FI can be modified to be used for local explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Model-specific FI BP-FI is in principle model-agnostic, as the FI is deter- mined of the dataset, not the FI for a prediction model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' However, BP-FI can 40 still provide insights for any specific model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' By replacing the target variable with the predicted outcomes of the model, we can apply BP-FI to this new dataset, which gives insight into which features are useful in the prediction model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Additionally, one can compare these FI results with the original FI (before replacing the target variable with the predicted outcomes) to see in what way the model changed the FI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Additional properties In this research, we have proven properties of BP- FI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' However, an in-depth study could lead to finding more useful properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' This holds both for BP-FI as well as the dependency function it is based on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Applying isomorphisms e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=', does not change the dependency function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Therefore, the BP-FI is also stable under isomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Understanding what properties BP-FI has is a double-edged sword.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Finding useful properties shows the power of BP-FI and finding undesirable behavior could lead to a future improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Additional datasets Ground truths are often unknown for FI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' In this re- search, we have given two kinds of datasets where the desirable outcomes are natural.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' It would however, be useful to create a larger collection of datasets both for global and local FI with an exact ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' We recognize that this could be a tall order, but we believe that it is essential to further improve FI methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Human labeling In some articles [35, 46], humans are used to evaluate explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' An intriguing question to investigate is if humans are good at predicting FI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' The BP-FI can be used as baseline to validate the values that are given by the participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Are humans able to identify the correct order of FI?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Even more difficult, can they predict close to the actual FI values?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' 6 Summary We started by introducing a novel FI method named Berkelmans-Pries FI (BP-FI), which combines Shapley values and the Berkelmans-Pries depen- dency function [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' In Section 3, we proved many useful properties of BP-FI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' We discussed which FI methods already exist and introduced datasets to evaluate if these methods adhere to the same properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='3, we explain how the properties are tested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' The results show that BP-FI is able to pass many more tests than any other FI method from a large collection of FI methods (468), which is a significant step forwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Most methods have not previously been tested to give exact results due to missing ground truths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' 41 In this research, we provide several specific datasets, where the desired FI can be derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' From the tests, it follows that previous methods are not able to accurately predict the desired FI values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' In Section 5, we extensively discussed the shortcomings of this paper, and the challenges for further re- search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' There are many challenging research opportunities that should be explored to further improve interpretability and explainability of datasets and machine learning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' A Datasets In this appendix, we discuss how the datasets are generated that are used in the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' We use fixed draw instead of uniformly random to draw each dataset exactly according to its distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' This is done to remove stochasticity from the dataset in order to get precise and interpretable re- sults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' An example of the difference between fixed draw and uniformly ran- dom can be seen in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' The datasets consist of 1,000 samples, except for Datasets 6 and 7 which contains 2,000 samples to ensure null-independence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' The datasets are designed to be computationally inexpensive, whilst still be- ing able to test many properties (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Below, we outline the formulas that are used to generate the datasets and give the corresponding FI values of our novel method BP-FI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Dataset 1: Binary system Feature variable(s): Xi ∼ U ({0, 1}) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' for i ∈ {1, 2, 3} Target variable: Y := �3 i=1 2i−1 · Xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Order: (X1, X2, X3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' BP-FI: (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='333, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='333, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='333).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Dataset 2: Binary system with clone Feature variable(s): Xi ∼ U ({0, 1}) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' for i ∈ {1, 2, 3} and Xclone 1 := X1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Target variable: Y := �3 i=1 2i−1 · Xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Order: (Xclone 1 , X1, X2, X3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' BP-FI: (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='202, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='202, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='298, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='298).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Dataset 3: Binary system with clone and one fully informative variable Feature variable(s): Xi ∼ U ({0, 1}) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' for i ∈ {1, 2, 3} and Xclone 1 := X1 and Xfull 4 := Y 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Target variable: Y := �3 i=1 2i−1 · Xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Order: (Xclone 1 , X1, X2, X3, Xfull 4 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' BP-FI: (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='148, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='148, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='183, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='183, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='338).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' 42 Dataset 4: Binary system with clone and two fully informative variables Feature variable(s): Xi ∼ U ({0, 1}) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' for i ∈ {1, 2, 3} and Xclone 1 := X1 and Xfull 4 := Y 2, Xfull 5 := Y 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Target variable: Y := �3 i=1 2i−1 · Xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Order: (Xclone 1 , X1, X2, X3, Xfull 4 , Xfull 5 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' BP-FI: (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='117, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='117, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='136, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='136, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='248, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='248).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Dataset 5: Binary system with clone and two fully informative variables different order Feature variable(s): Xi ∼ U ({0, 1}) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' for i ∈ {1, 2, 3} and Xclone 1 := X1 and Xfull 4 := Y 2, Xfull 5 := Y 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Target variable: Y := �3 i=1 2i−1 · Xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Order: (X3, Xfull 4 , Xfull 5 , Xclone 1 , X1, X2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' BP-FI: (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='136, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='248, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='248, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='117, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='117, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='136).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Dataset 6: Null-independent system Feature variable(s): Xnull-indep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' i ∼ U ({0, 1}) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' for i ∈ {1, 2, 3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Target variable: Y ∼ U ({0, 1}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Order: (Xnull-indep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' 1 , Xnull-indep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' 2 , Xnull-indep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' 3 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' BP-FI: (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='000, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='000, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Dataset 7: Null-independent system with constant variable Feature variable(s): Xnull-indep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' i ∼ U ({0, 1}) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' for i ∈ {1, 2, 3} and Xconst, null-indep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' 4 := 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Target variable: Y ∼ U ({0, 1}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Order: (Xnull-indep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' 1 , Xnull-indep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' 2 , Xnull-indep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' 3 , Xconst, null-indep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' 4 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' BP-FI: (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='000, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='000, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='000, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' 43 Dataset 8: Uniform system increasing bins Feature variable(s): Let Li := {0, 1/(i−1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' , 1} be an equally spaced set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Define: Xbins=10 1 := arg max x1∈L10 {Y ≥ x1}, Xbins=50 2 := arg max x2∈L50 {Y ≥ x2}, Xbins=1,000, full 3 := arg max x3∈L1,000 {Y ≥ x3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Target variable: Y ∼ U (L1,000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Order: (Xbins=10 1 , Xbins=50 2 , Xbins=1,000, full 3 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' BP-FI: (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='297, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='342, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='361).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Dataset 9: Uniform system increasing bins more variables Feature variable(s): Let Li := {0, 1/(i−1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' , 1} be an equally spaced set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Define: Xbins=10 1 := arg max x1∈L10 {Y ≥ x1}, Xbins=20 2 := arg max x2∈L20 {Y ≥ x2}, Xbins=50 3 := arg max x3∈L50 {Y ≥ x3}, Xbins=100 4 := arg max x4∈L100 {Y ≥ x4}, Xbins=1,000, full 5 := arg max x5∈L1,000 {Y ≥ x5}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Target variable: Y ∼ U (L1,000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Order: (Xbins=10 1 , Xbins=20 2 , Xbins=50 3 , Xbins=100 4 , Xbins=1,000, full 5 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' BP-FI: (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='179, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='193, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='204, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='208, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='216).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' 44 Dataset 10: Uniform system increasing bins with clone differ- ent order Feature variable(s): Let Li := {0, 1/(i−1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' , 1} be an equally spaced set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Define: Xbins=10 1 := arg max x1∈L10 {Y ≥ x1}, Xbins=50 2 := arg max x2∈L50 {Y ≥ x2}, Xbins=1,000, full 3 := arg max x3∈L1,000 {Y ≥ x3}, Xclone, full 3 := Xbins=1,000, full 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Target variable: Y ∼ U (L1,000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Order: (Xbins=1,000, full 3 , Xbins=50 2 , Xbins=10 1 , Xclone, full 3 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' BP-FI: (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='262, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='253, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='223, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='262).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Dataset 11: Dependent system: 1x fully informative variable Feature variable(s): Xfull 1 , Xnull-indep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' 2 , Xnull-indep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' 3 ∼ U ({1, 2}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Target variable: Y := Xfull 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Order: (Xfull 1 , Xnull-indep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' 2 , Xnull-indep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' 3 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' BP-FI: (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='000, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='000, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Dataset 12: Dependent system: 2x fully informative variable Feature variable(s): Xfull 1 , Xnull-indep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' 3 ∼ U ({1, 2}) and Xfull 2 := Y 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Target variable: Y := Xfull 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Order: (Xfull 1 , Xfull 2 , Xnull-indep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' 3 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' BP-FI: (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='500, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='500, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Dataset 13: Dependent system: 3x fully informative variable Feature variable(s): Xfull 1 ∼ U ({1, 2}) and Xfull 2 := Y 2, Xfull 3 := Y 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Target variable: Y := Xfull 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Order: (Xfull 1 , Xfull 2 , Xfull 3 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' BP-FI: (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='333, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='333, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='333).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Dataset 14: XOR dataset Feature variable(s): X1, X2 ∼ U ({1, 2}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Target variable: Y := X1 · (1 − X2) + X2 · (1 − X1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Order: (X1, X2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' BP-FI: (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='500, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='500).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' 45 Dataset 15: XOR dataset one variable Feature variable(s): Xnull-indep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' 1 ∼ U ({1, 2}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Target variable: Y := Xnull-indep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' 1 (1 − X2) + X2 · (1 − Xnull-indep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' 1 ) with X2 ∼ U ({1, 2}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Order: (Xnull-indep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' 1 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' BP-FI: (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Dataset 16: XOR dataset with clone Feature variable(s): X1, X2 ∼ U ({1, 2}) and Xclone 1 := X1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Target variable: Y := X1 · (1 − X2) + X2 · (1 − X1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Order: (Xclone 1 , X1, X2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' BP-FI: (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='167, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='167, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='667).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Dataset 17: XOR dataset with null independent Feature variable(s): X1, X2 ∼ U ({1, 2}) and Xnull-indep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' 3 ∼ U ({0, 3}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Target variable: Y := X1 · (1 − X2) + X2 · (1 − X1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Order: (X1, X2, Xnull-indep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' 3 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' BP-FI: (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='500, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='500, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Dataset 18-28: Probability datasets Feature variable(s): Xi = Zi + S with Zi ∼ U ({0, 2}) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' for i = 1, 2 and P(S = 1) = p, P(S = 2) = 1 − p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Target variable: Y = ⌊XS/2⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Order: (X1, X2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' BP-FI: (p, 1 − p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' B Tests This appendix gives an overview of the tests that are used for each FI method to evaluate if they adhere to the properties given in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Most tests are straightforward, but additional explanations are given in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Test 1: Efficiency sum BP-FI Evaluates: Property 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Explanation: We evaluate if the sum of all FI is equal to the sum of the Berkelmans-Pries dependency function of Y on all features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' When an FI value of NaN or infinite is assigned, the sum is automatically not equal to the sum for BP-FI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' 46 Test 2: Efficiency stable Evaluates: Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Explanation: Whenever a variable is added to a dataset, we examine if the sum of all FI changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' If a variable does not give any additional information compared to the other variables, the sum of all FI should stay the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Test 3: Symmetry Evaluates: Property 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Explanation: In some datasets, there are symmetrical variables (see Property 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' We determine for all symmetrical variables if they receive identical FI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Test 4: Range (lower) Evaluates: Property 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Explanation: We examine for all FI outcomes if they are greater or equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Test 5: Range (upper) Evaluates: Property 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Explanation: We examine for all FI outcomes if they are smaller or equal to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Test 6: Bounds BP-FI (lower) Evaluates: Property 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Explanation: We evaluate if the bounds given in Property 4 also hold for other FI methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Every FI(X) with X ∈ Ωfeat can be lower bounded for BP-FI by Dep(Y |X) Nvars ≤ FI(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Test 7: Bounds BP-FI (upper) Evaluates: Property 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Explanation: We evaluate if the bounds given in Property 4 also hold for other FI methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Every FI(X) with X ∈ Ωfeat can be upper bounded for BP-FI by X ≤ Dep (Y |Ωfeat) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Test 8: Null-independent implies zero FI Evaluates: Property 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Explanation: In some datasets, there are null-independent variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' In these cases, we investigate if they also receive zero FI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Test 9: Zero FI implies null-independent Evaluates: Property 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Explanation: When a variable gets zero FI, it should hold that such a feature is null-independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' 47 Test 10: One fully informative, two null-independent Evaluates: Property 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Explanation: feature importance: appendix: datasets) consists of a fully dependent target variable Y := Xfull 1 and two null- independent variables Xnull-indep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' 2 , Xnull-indep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' We test if FI(Xfull 1 ) = 1 and FI(Xnull-indep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' 2 ) = FI(Xnull-indep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' 3 ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Test 11: Fully informative variable in argmax FI Evaluates: Property 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Explanation: Whenever a fully informative feature exists in a dataset, there should not be a feature that attains a higher FI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Test 12: Limiting the outcome space Evaluates: Property 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Explanation: To evaluate if applying a measurable function f to a RV X could increase the FI, we examine the datasets where the same RV is binned using different bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' The binning can be viewed as applying a function f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Whenever less bins are used, the FI should not increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Test 13: Adding features can increase FI Evaluates: Property 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Explanation: Whenever a feature is added to a dataset, we examine if this ever increases the FI of an original variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' If the FI never increases, we consider this a fail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Test 14: Adding features can decrease FI Evaluates: Property 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Explanation: Whenever a feature is added to a dataset, we examine if this ever decreases the FI of an original variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' If the FI never decreases, we consider this a fail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Test 15: Cloning does not increase FI Evaluates: Property 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Explanation: We evaluate if adding a clone to a dataset increase the FI of the original variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Test 16: Order does not change FI Evaluates: Property 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Explanation: We check if the order of the variables changes the assigned FI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' 48 Test 17: Outcome XOR Evaluates: Property 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Explanation: This test evaluates the specific outcome of two datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' For Dataset 14 the desired outcome is (1/2, 1/2) and (1/2, 1/2, 0) for Dataset 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' An FI method fails this test when one of the FI values falls outside the ǫ-bound of the desired outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Test 18: Outcome probability datasets Evaluates: Property 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Explanation: This test evaluates the specific outcomes of all probabil- ity datasets (Datasets 18 to 28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' The desired outcome for probability p is (p, 1 − p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' An FI method fails this test when one of the FI values falls outside the ǫ-bound of the desired outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' References [1] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Aas, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Jullum, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Løland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' “Explaining individual predictions when features are dependent: More accurate approximations to Shapley values”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' In: Artificial Intelligence 298 (2021), page 103502.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' issn: 0004- 3702.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' doi: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='artint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='103502.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' url: https:// www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='sciencedirect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} 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Springer International Publishing, 2019, pages 655–670.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' isbn: 978-3-030-10925-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' [10] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Castro, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Gómez, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Tejada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' “Polynomial calculation of the Shapley value based on sampling”.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='1145/3429445.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' url: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content='1145/3429445.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' [62] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Zien, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Krämer, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Sonnenburg, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Rätsch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' “The feature impor- tance ranking measure”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' In: Machine Learning and Knowledge Discov- ery in Databases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Edited by W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Buntine, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Grobelnik, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Mladenić, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Shawe-Taylor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' Berlin, Heidelberg: Springer Berlin Heidelberg, 2009, pages 694–709.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' isbn: 978-3-642-04174-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} +page_content=' 55' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQf0wtX/content/2301.04740v1.pdf'} diff --git a/WNE3T4oBgHgl3EQfFQlM/content/tmp_files/2301.04303v1.pdf.txt b/WNE3T4oBgHgl3EQfFQlM/content/tmp_files/2301.04303v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a30cbf1d6c3144b62b63a4115263f5572f446a46 --- /dev/null +++ b/WNE3T4oBgHgl3EQfFQlM/content/tmp_files/2301.04303v1.pdf.txt @@ -0,0 +1,949 @@ +Dependence of simulated radiation damage on crystal structure +and atomic misfit in metals +J. C. Stimac +Department of Chemical Engineering, +University of California, Davis, CA, 95616, USA. and +Lawrence Livermore National Laboratory, Livermore, CA, 94550, USA. +C. Serrao and J. K. Mason∗ +Department of Materials Science and Engineering, +University of California, Davis, CA, 95616, USA. +1 +arXiv:2301.04303v1 [cond-mat.mtrl-sci] 11 Jan 2023 + +Abstract +This study investigates radiation damage in three metals in the low temperature and high radiant +flux regime using molecular dynamics and a Frenkel pair accumulation method to simulate up to +2.0 displacements per atom. The metals considered include Fe, equiatomic CrCoNi, and a fictitious +metal with identical bulk properties to the CrCoNi composed of a single atom type referred to as +an A-atom. CrCoNi is found to sustain higher concentrations of dislocations than either the Fe or +A-atom systems and more stacking faults than the A-atom system. The results suggest that the +concentration of vacancies and interstitials are substantially higher for the CrCoNi than the A-atom +system, perhaps reflecting that the recombination radius is smaller in CrCoNi due to the roughened +potential energy landscape. A model that partitions the major contributions from defects to the +stored energy is described, and serves to highlight a general need for higher fidelity approaches to +point defect identification. +I. +INTRODUCTION +Structural components in nuclear fission reactors need to be engineered to withstand +decades of exposure to radiation and elevated temperatures [1, 2]. Materials in the next +generation of fission reactors and future fusion reactors will be subject to substantially higher +radiation dosages and temperatures and likely highly corrosive environments as well [3, 4]. +Irradiation by exposure to high-energy particles displaces atoms and damages the crystalline +structure of pure metals and alloys [5, 6], resulting in a variety of microstructural changes. +Specifically with respect to the degrading effects of radiation, there are five considerations for +structural metals in reactor environments: radiation hardening (low temperature), radiation- +induced segregation and precipitation, void swelling, radiation-induced creep, and helium +embrittlement (high temperature) [7, 8]. The three intermediate temperature effects are +usually the most relevant in practice, and are often observed simultaneously since they are +all strongly associated with the underlying ability of point defects generated by radiation to +migrate through the lattice [9, 10]. +Radiation-induced segregation of substitutional solutes is generally attributed to the in- +verse Kirkendall effect where, as vacancies migrate to and annihilate on sinks, the different +∗ jkmason@ucdavis.edu +2 + +atomic species have different migration rates in the opposite direction [11]. The segregation +extent is governed by differences in diffusive mobility, with undersized solutes generally be- +ing enriched and oversized solutes being depleted in the vicinity of the sinks [10, 12]. Void +swelling is a serious engineering concern that entails an increase in the volume of the mate- +rial by as much as 1% per dpa [13] from the nucleation and growth of voids in the bulk. This +process is driven by an imbalance in the concentrations of vacancies and interstitials, with +annihilation of high-mobility interstitials on dislocations and other sinks leaving behind a +relative excess of vacancies that precipitate as voids [9, 14]. While a variety of mechanisms +have been proposed for irradiation creep, the two main mechanisms are believed to be the +climb of favorably-oriented dislocations by the stress-induced preferred absorption of point +defects and the glide of dislocations enabled by climb over obstacles [9, 15]. +Multi-principal component alloys (MPEAs) [16, 17] (often called high-entropy alloys or +compositionally complex alloys) consist of a few to several atom types in solid solution with +nominally equi-atomic concentrations. This emerging class of metals could perform well as +structural materials in irradiated environments, with initial evidence showing exceptional +mechanical properties [18, 19] and higher resistance to radiation damage [20, 21] than tra- +ditional metals. One experimental and computational study of the effects of radiation on a +CrCoNi MPEA showed decreases in the relative disorder, the number of defects, and large +defect clusters compared to irradiated samples of pure Ni and a NiFe binary alloy [20]. The +authors attributed these effects to decreasing dislocation mobility with increasing number of +atomic species. Lu et al. [22] studied void swelling in several Ni-based alloys including pure +Ni, NiFe, CrCoNi and two quinary alloys with and without prior nanoindentation, and found +that nanoindentation improved resistance to swelling by increasing the density of defects like +dislocations, stacking faults, and twin boundaries that promote vacancy annihilation. Cu- +riously, Veli¸sa et al. [23] found that CrCoNi showed superior irradiation resistance relative +to NiCr only at temperatures near and below 300 K though. The reason for this is not well +established, but is believed to be related to the specifics of the chemical short range order +(SRO) that developed in the two alloys. +The most direct way to simulate radiation damage at the atomic level uses molecular +dynamics (MD) simulations of collision cascades [5, 6]. This involves assigning a large initial +velocity to the primary knock-on atom (PKA) to mimic a passing neutron or other high- +energy particle transferring kinetic energy to the lattice. The PKA recoils, displacing many +3 + +of the surrounding atoms from their lattice sites and converting the initial kinetic energy into +a thermal spike with sufficient energy to facilitate the regeneration of the crystalline lattice +and the recombination of many, but not all, of the generated point defects. The resulting +interstitials and vacancies are respectively distributed on the periphery and the interior of +the affected zone, increase the point defect concentration in the material, and are directly +responsible for the most visible degrading effects of radiation. +Mass conservation requires that the interstitials and vacancies generated by a single +collision cascade occur as Frenkel pairs. The number of Frenkel pairs generated in this way +divided by the number of atoms in the material is known as the displacements per atom +(dpa), and is the standardized measure of the extent of radiation damage in crystalline +materials [24]. Early work by Kinchin and Pease modeled atoms as hard-spheres that exhibit +elastic collisions during collision cascades and laid the theoretical foundation for radiation +effects in crystalline materials [25]. If the energy transferred to an atom exceeded a material- +specific threshold value, then the atom was said to have been displaced from its lattice site. +While energies below the threshold could still displace the atom, it would return to its +lattice site after the initial perturbation. The theory developed by Kinchin and Pease was +further developed in the work of Norgett, Robinson, and Torrens (hereafter referred to as +the NRT model) who added additional terms to account for energy lost to ionization and for +the affects of inelastic collisions [26]. Several more recent studies concluded that the NRT +model overestimates the number of defects generated by a collision cascade and neglects the +mixing from atomic replacements though, spurring a number of proposed refinements [5, 6]. +These are significant for the reason that accurate damage models that can reliably predict +dpa are essential to reliably compare radiation damage resistance among various materials. +Full-scale atomic simulations of collision cascades have been performed for several decades +now, and are useful to uncover the evolution of radiation-induced primary damage [27, 28]. +However, the use of MD for this application is subject to several limitations. As Ref. [6] +points out, interatomic potentials cannot capture the effects of deviations from the Born- +Oppenheimer approximation when excited electronic states are induced. The other main +limitations are the time and length scales that can reasonably be achieved with modern +computational resources. MD simulations are usually no longer than a few nanoseconds, +limiting the overall radiation dose that can reasonably be achieved by full cascade simulations +without the events overlapping in time; Refs. [29, 30] further discuss these limitations. As +4 + +a result of the high computational cost to reach appreciable dpa, the doses investigated in +MD simulations have been historically been less than 1.0 dpa. Although these are useful to +understand defect creation at low doses, structural materials in a nuclear reactor core can +experience as much as 80 dpa over a 40 year service life [8]. +A variety of strategies have been used to circumvent the computational limitations im- +posed by cascade simulations, one of which involves the direct insertion of a high density of +Frenkel pairs [31–33]. Such Frenkel pair accumulation (FPA) techniques forgo the dynamics +of time-resolved high-energy atomic collisions stemming from the primary knock-on event. +Instead, they intermittently introduce Frenkel pairs by randomly displacing atoms from their +lattice positions, usually followed by some form of equilibration and time integration. These +methods benefit from a clearly defined radiation dose, based on the the number of displaced +atoms, and dose rate, based on the ratio of displaced atoms to the simulated time. For +example, Chartier et al. [34] used a FPA procedure to model irradiation of UO2 and found +the steady-state dislocation density to be in good agreement with experiments. A major +limitation of FPA procedures though is the absence of any effects related to the thermal +spike [30], particularly providing the thermal energy necessary for diffusion and clustering +of defects; this includes the recombination of point defects when vacancies and interstitials +collide. Analysis of FPA simulations should therefore be done with careful consideration of +this limitation to avoid the potential for nonphysical extrapolation. +Recent work by Derlet and Dudarev [29] introduced a variant of the FPA method that fur- +ther streamlines the process of sampling irradiated microstructures. Known as the creation- +relaxation algorithm (CRA), this differs from the preceding FPA methods in that there are +no time-integrated dynamics. CRA simulations randomly select atoms and displace them +with random directions and magnitudes, just as other FPA methods do, but always follow +this by potential energy minimization. The entire simulation involves repeating this process +for a specified number of displacements, with the canonical dpa equal to the number of +displaced atoms divided by the total number of atoms in the system. Dudarev and Derlet +applied the CRA to BCC Fe systems of a variety of sizes and reported good agreement +between full cascade simulations [35, 36] and the CRA for interstitial density as a function +of dpa, at least up to a linear rescaling of both the independent and dependent variables. +The need for rescaling is likely related to the CRA effectively being performed at zero Kelvin +[37]; the thermally-driven diffusion of point defects that is prevalent at high temperatures +5 + +and substantially contributes to microstructure evolution of irradiated materials is negligible +in such conditions. That said, the CRA can be viewed as simulating radiation damage in +conditions where thermally-driven diffusion is active but negligible compared to other mass +transport mechanisms. +The CRA has since been used to investigate radiation effects in materials other than +BCC Fe. One such study used the CRA to simulate irradiation of a NiFe system doped +with carbon and evaluated the ability of carbon interstitials to decrease radiation damage +[38]. Several others applied the CRA to tungsten to investigate the relationships between +dpa and specific physical parameters at relatively high doses (above 1.0 dpa), either alone +or with the assistance of other computational or experimental methods. Parameters that +were investigated include thermal conductivity [39], hydrogen embrittlement and tritium +concentration [40], and radiation induced structural evolution [41]. +This paper investigates the microstuctures of highly irradiated Fe, equi-atomic CrCoNi, +and a fictitious metal with identical bulk properties to the CrCoNi composed of a single +atom type referred to as an A-atom. The main motivation for including the Fe and A-atom +systems is to establish points of comparison for the investigation of the reported radiation +resistance of CrCoNi. The CRA is used to simulate the irradiation of all three systems up +to a final dose of 2.0 dpa. The details of the implemented CRA, as well as an analysis of +the experimental dose rates and temperatures for which it is likely relevant, are described in +Sec. II. The same section also outlines our methods for identifying material defects including +dislocations, vacancies, interstitials, and stacking faults, and a model for the energy stored +in those defects. The results and a discussion of the simulations are included in Sec. III, and +Sec. IV draws conclusions to inform further research in this area. +II. +METHODS +A. +CRA simulations +All molecular dynamics simulations were performed using the LAMMPS software [42]. +Orthorombic simulation cells were used with periodic boundary conditions for all cell faces, +and the number of atoms remained constant. No temperature or time-steps were defined for +any simulation. After constructing the initial space-filling single crystals, the volume of the +6 + +simulation cell was relaxed using a potential energy minimization and fixed thereafter. The +present study examined three material systems: BCC Fe, equi-atomic FCC CrCoNi, and +FCC A-atom designed to reproduce the bulk properties of CrCoNi using a single fictitious +atom type [43]. Comparison with an A-atom model more directly allows identification of +the effects caused by chemical short range order (SRO) and lattice distortion (LD) which +are widely implicated in the enhanced physical properties of MPEAs [19]. +The BCC Fe simulation consisted of a simulation volume of 40 units cells along each +of the three dimensions, and with two atoms in each unit cell contained a total of 128 000 +atoms. This simulation used the Mendelev-II embedded atom method (EAM) potential [44]. +The FCC CrCoNi simulation consisted of 32 unit cells along each of the three dimensions, +and with four atoms in each unit cell contained a total of 131 072 atoms. All three atomic +species were represented with equal concentrations, and were initially distributed uniformly +at random throughout the simulation cell. The purpose of reducing the number of unit cells +for the FCC systems was to make the number of atoms as close as possible to that in the Fe +simulation. The equiatomic CrCoNi simulation utilized the EAM potential developed by Li +et al. [45]. The FCC A-atom simulation used the EAM potential developed by Jian et al. +[46] but was otherwise identical to the CrCoNi simulation. +This work circumvented the computational limitations associated with full collision cas- +cade simulations by using the creation relaxation algorithm (CRA) [29] to simulate the +effects of irradiation by the repeated introduction of Frenkel pairs [34, 47]; the basic CRA is +described in Alg. 1. All three experiments used the CRA to simulate radiation damage up +to 2.0 dpa, requiring a total of 256 000 atomic displacements in the Fe system and 262 144 +displacements in both the CrCoNi and A-atom systems. The potential energy minimization +conducted after each atomic displacement consisted of two steps. First, the position of ev- +ery atom was randomly perturbed by a small Gaussian-distributed displacement. Second, a +standard gradient-based energy minimization was applied to the atomic positions to reduce +the system energy. The purpose of the small perturbations in the first step was to help +the system escape shallow local energy minima as would naturally occur in a thermal sys- +tem. The perturbation magnitude was fixed after systematically exploring the relationship +between the perturbation magnitude and the relaxed potential energy of a smaller BCC +Fe system of 2000 atoms. This involved displacing 100 atoms and performing energy min- +imizations without perturbations between displacements. After all the displacements, the +7 + +Algorithm 1 Creation Relaxation Algorithm +1: n := 0 +▷ number of displaced atoms +2: while n/N < φdpa do +▷ N is total number of atoms +3: +Randomly select an atom i uniformly over all atoms +4: +Move atom i to a randomly selected point within the simulation cell +5: +while potential energy increase exceeds a threshold do +6: +Move atom i to a randomly selected point within the simulation cell +7: +end while +8: +Relax the system using energy minimization +9: +n ← n + 1 +10: end while +atoms were subjected to repeated cycles of perturbations and energy minimization, with the +resulting potential energy profiles for a given standard deviation of the perturbation magni- +tude shown in Fig. 1 as a function of cycle number. Standard deviations of 0.005 and 0.01 +angstroms were found to rapidly result in a relatively stable potential energy minimum, with +larger displacements often introducing additional defects and smaller displacements unable +to reliably allow the system to escape shallow potential energy minima. A perturbation +magnitude of 0.01 angstroms was chosen for the subsequent simulations for the reason that +it reached the potential energy minimum in the fewest number of cycles. +Since the intention was for the perturbations to make the system more closely resemble +a thermal one, a natural question is what would be the corresponding temperature for a +given perturbation magnitude. The equipartition theorem implies that the potential energy +per atom relative to that in an ideal crystal should be (3/2)kBT for a thermal system; +equating this with the average potential energy increase per atom resulting from a single +perturbation gives an equivalent temperature of ∼30 K for all three systems. This implies +that the additional energy supplied by the perturbations is consistent with the assumption +that any thermally-driven diffusion of point defects is insignificant compared to alternative +mass transport mechanisms. +In addition to the three simulations described above, another CRA simulation of +equiatomic CrCoNi with 64 unit cells along each dimension and a total of 1 048 576 atoms +8 + +FIG. 1. The potential energy per atom of a highly defected BCC Fe system as a function of mini- +mization cycles, measured with respect to the energy of an isolated atom. Perturbation magnitudes +of 0.005 and 0.01 angstroms allowed the system to quickly reach a deep potential energy minimum. +was conducted to 3.0 dpa to investigate the effect of system size. With energy minimization +being by far the most computationally intensive step in the CRA, this simulation performed +25 atomic displacements (step 4 of Alg. 1) per energy minimization to reduce the simulation +run time; the larger volume of this simulation decreases the probability that multiple atomic +displacements occur in a given region without an intervening relaxation event. The main +results of this simulations are presented in Appendix A. +B. +Applicable temperatures and dose rates +As mentioned in Sec. I, the microstructures that develop when using the CRA are a +direct product of relaxations of the atomic-level stress fields and do not necessarily represent +microstructures that develop in conditions with appreciable thermally-driven diffusion. We +characterize the regime of physical conditions for which the CRA is applicable by a simple +argument that compares the rates of atomic transport by thermally-driven diffusion and +ballistic displacements from collision cascades. The main criterion for applicability is that +the rate of ballistic displacements per atom K0 be much greater than the interstitial hopping +rate γdiff (assumed to be the fastest thermally-driven diffusion event). In such conditions, any +effects of thermally-driven diffusion should be negligible compared to those resulting from +ballistic displacements and the subsequent stress relaxation. The interstitial hopping rate, as +9 + +FIG. 2. Contour plot of the applicability measure A = log10(K0/γdiff) as a function of the logarithm +of the dose rate and temperature. The calculations used the diffusivity and activation energy for +interstitial diffusion in CrCoNi given by Ref. [49]. +described by transition state theory, obeys the Arrhenius relation γdiff = ν0exp[−Ea/(kBT)] +where ν0 is the high-temperature limit for the site-hopping frequency, Ea is the activation +energy for interstitial diffusion, kB is Boltzmann’s constant, and T is the temperature. The +random walk diffusion model suggests that ν0 be approximated by 6D0/λ2 where D0 is the +diffusivity prefactor and λ is the distance separating two interstitial sites [48]. +Using the values for the prefactor and activation energy for interstitial diffusion for Cr- +CoNi given by Ref. [49] leaves only the temperature and K0 as independent variables. A +measure of applicability A = log10(K0/γdiff) is defined such that an increase of A by one +indicates an order of magnitude increase in the ratio of K0 to γdiff. Figure 2 gives a contour +plot of A for CrCoNi and indicates that applicability increases at lower temperatures and +higher dose rates; moreover, the dependence on the temperature is much more significant +than the dose rate, a consequence of the exponential scaling in the Arrhenius relation for the +diffusivity. For the dose rates considered, the temperatures for which A ≥ 2 are all in the +cryogenic range. This low temperature constraint is entirely consistent with the equivalent +temperature resulting from the perturbations in the atomic positions described in Sec. II A. +10 + +C. +Identifying defects +It is difficult to precisely identify crystalline defects in a material that has suffered exten- +sive radiation damage. Certainly there is still an underlying lattice, but conventional defect +models assume that the defect is isolated, or equivalently, that the surrounding atoms occupy +well-defined lattice positions. This is not true for the high defect concentrations that can +occur in irradiated materials, and the identification of reference lattice sites is further com- +plicated in MPEAs like CrCoNi where the reference lattice is already perturbed by atomic +size differences. +The numbers and types of dislocations in the simulations were determined using OVITO’s +dislocation extraction algorithm (DXA) [50]; the algorithm requires a trial circuit length and +a value for circuit stretchability which were set to 14 and 9 respectively. Surprisingly, the +DXA appeared to be relatively robust to extensive disruption of the crystalline lattice from +the CRA, with very few isolated dislocation segments appearing in the networks reported +below in Sec. III B. +The number of vacancies were determined using OVITO’s Wigner-Seitz (WS) analysis. +This constructs Voronoi cells around the lattice sites in an initial crystal structure and checks +the atom occupancy of each Voronoi cell in subsequent time steps. A Voronoi cell that +does not contain any atoms is regarded as indicating the presence of a vacancy, though this +approach does not precisely define the vacancy location. The WS analysis was compared with +two other approaches to estimate vacancy concentration in the BCC Fe systems, the first of +which was the BCC defect analysis (BDA) [51]. The second approach involved evaluating the +relaxed volume of the simulation box. Inserting a single Frenkel pair produces an interstitial +and a lattice site occupied by a vacancy, resulting in the expansion of a crystal subject +to zero traction boundary conditions. Since the simulations were conducted at constant +volume, the insertion of Frenkel pairs in the CRA instead elevated the system pressure. +The equivalent volume change was evaluated by relaxing the simulation cell subject to a +zero traction boundary condition, and enabled the number of vacancies to be estimated by +assuming that the volume change from inserting a Frenkel pair remained constant throughout +the simulation. +Figure 3 compares the three methods and shows that the WS analysis +agrees much better with with the vacancy concentration estimated from the relaxed volume +than from the BDA. The factor by which the WS and relaxed volume method differ can +11 + +FIG. 3. Three different estimates for the number of vacancies in BCC Fe as a function of dpa. The +BDA estimate is likely inaccurate due to the difficulty of identifying crystal lattice sites in highly +defected materials. +be interpreted as the reduction in volume change per effective Frenkel pair insertion with +increasing dpa and is related to point defect interactions. The poor performance of the BDA +is likely a consequence of extensive radiation damage making it difficult to identify vacancies +based on features of local atomic environments without a clear underlying crystal lattice. +Interstitials are the defect type that is most difficult to identify in our simulations. The +multiplicity of local atomic configurations that can occur for non-isolated interstitials means +that template-based approaches would be difficult or even infeasible to implement, and +examining the expected number of atoms contained within a surface that passes only through +crystalline material (the analogue of a Burgers circuit) often fails because of the obstructions +to constructing such surfaces. In the absence of a canonical alternative, our approach involves +constructing a distribution of atomic volumes as estimated by the Voronoi polyhedra. The +compressive stresses around an interstitial reduce the volumes of the interstitial and of the +surrounding atoms in a characteristic way when the interstitial is in an otherwise perfect +crystal; Fig. 4 shows an example of this phenomenon for BCC Fe. The distribution of atomic +volumes in our simulations is decomposed with a K-means algorithm into a superposition of +peaks, one for each characteristic atomic volume surrounding an isolated interstitial. Part +of the utility of K-means is that the boundaries that define the locations the peaks can be +dynamically updated from one time step to the next; this is necessary to account for the +shifting and broadening of the peaks as the structure reaches higher defect concentrations. +12 + +10.5 +11.0 +11.5 +Atomic volume [A3] +0 +1 +2 +3 +4 +5 +6 +Count +10.5 +11.0 +11.5 +Atomic volume [A3] +0 +200 +400 +600 +800 +1000 +Count +FIG. 4. +Atomic volume distribution as given by Voronoi tessellations in BCC Fe for (left) an +isolated interstitial and (right) at 0.01 dpa. All histograms were evaluated using 200 equally-sized +bins within the domain of the distribution. The simulation cells for the isolated interstitial and +the 0.01 dpa distributions contained 54 000 and 128 000 atoms, respectively. +Comparing the numbers of atoms assigned to each peak with the corresponding numbers +of atoms for an isolated interstitial gives an estimate for the number of interstitials in our +simulations, though the accuracy is expected to decrease as the peaks broaden and overlap +with increasing radiation damage. +Stacking fault (SF) densities in the FCC CrCoNi and A-atom systems are relatively simple +to evaluate by comparison. Since the local atomic structure around atoms belonging to a SF +appears to be HCP, all that is necessary for a reasonable estimate is to count the number +atoms classified as HPC by, e.g., OVITO’s common neighbor analysis, and to convert this +to an equivalent SF area using the geometry of the crystal structure. +The Warren-Cowley parameters are used to quantify the type and degree of chemical +short range order (SRO), and are defined as [52, 53] +αij = 1 − pij/cj +(1) +where cj is the overall fraction of atomic type j, and pij is the probability that an atom in +the first nearest-neighbor shell of an atom with type i is of type j. Positive values of αij +means there is a repulsion between species i and j while negative values indicate attraction. +13 + +D. +Energy model +This section develops a model to partition the energetic contributions of the various types +of material defects to the overall potential energy change as a function of dpa. The potential +energy stored in defects is defined as Estored = Epe − Ecoh − Eelas, where Epe is the overall +potential energy, Ecoh is the cohesive energy of a reference material, and Eelas is the elastic +strain energy. This stored energy is modeled as a sum over defect contributions: +Efit = Edis + Evcy + Eint − Esro +(2) +which includes terms for the contributions of dislocations, vacancies, interstitials, and chemi- +cal short range order, respectively, and where the sign of Esro reflects the fact that increasing +SRO decreases the system energy. The contribution of stacking faults to Efit was found to +be negligible. +The energy of a well-developed dislocation network in an elastically isotropic material is +well-described by the equation [54]: +Edis = χµb2 +4π ρ ln +� +1 +rc√ρ +� +(3) +where µ is the shear modulus, b is the Burgers vector, rc is a cutoff radius of the dislocation +core stress field, and ρ is the dislocation density. The value of rc is approximated as the +lattice parameter a0. χ is a parameter that accounts for the overall dislocation character, +and depending on the material and the development of the network is expected to be in +the interval 1.0 ≤ χ ≤ 1/(1 − ν) where ν is Poisson’s ratio. The elastic constants that are +necessary to evaluate Eq. 3 were found in Refs. [44, 46], and the material parameters used +in the model are included in Table I. +The energetic contributions of the point defects are modeled as: +Evcy = βevcyNvcy +(4) +Eint = γeintNint +(5) +where evcy and eint are the formation energies of an isolated vacancy or interstitial as found by +inserting a single vacancy or interstitial into an otherwise perfect crystal. More specifically, +after minimizing the potential energy and measuring the potential energy change ∆E relative +14 + +TABLE I. Material properties used as parameters to evaluate the dislocation energy. The values +for the CrCoNi and A-atom materials can be found in Refs. [44, 46]. +µ [GPa] b [A] a0 [A] +ν +Fe +49.5 +2.49 +2.87 +0.373 +CrCoNi +37.0 +2.47 +3.50 +0.414 +A-atom +43.0 +2.47 +3.50 +0.403 +TABLE II. Isolated point defect formation energies. +evcy [eV] eint [eV] +Fe +1.71 +4.01 +CrCoNi +1.49 +2.45 +A-atom +1.62 +3.85 +to the perfect crystal, the isolated point defect formation energies are defined as: +evcy = ∆Eremove + Ecoh +(6) +eint = ∆Eadd − Ecoh. +(7) +The values of these formation energies are reported in Table II. Nvcy and Nint are the numbers +of vacancies and interstitials, and β and γ are fitting parameters to account for deviations +from the isolated point defect energies that occur as the density of point defects increases +with dpa. +Finally, Esro is the energy associated with the chemical short range order relative to a +random solid solution, and was evaluated for the CrCoNi MPEA system using molecular +statics calculations. Starting with an initial configuration, the types of all the atoms were +randomly reassigned to one of the three constituent elements and the potential energy of the +structure was minimized. This procedure was repeated five times for each configuration to +account for statistical variations, and the the average potential energy change relative to the +initial structure was included in the model for each dpa for which the fitting was conducted. +15 + +Overall, the energy contribution of stacking faults is considered negligible, with an average +value over all dpa of only −0.0463 meV/atom for the CrCoNi system. +The resulting energy model contains the three adjustable parameters χ, β, and γ. Fit- +ting the model to each of the three systems involved first constructing the bounds on the +dislocation energy that would be realized by the minimum and maximum allowed values for +χ. The dislocation energy was then fixed at the lower bound, and a least-squares technique +was used to find the values of β and γ that minimized the difference between Estored and +Efit over the entire interval up to 2.0 dpa. Repeating this procedure with the dislocation +energy fixed at the upper bound allowed the construction of corresponding intervals for the +predicted contributions of the vacancy and interstitial terms. As discussed below in Sec. +III E, the largest contribution to the model error is likely the estimated numbers of point +defects Nvcy and Nint. +III. +RESULTS AND DISCUSSION +A. +Total energy and pressure +The successive generation of Frenkel pairs considerably increases the potential energy of +the simulated systems, particularly in the absence of thermally-driven point defect migration +and recombination. As displayed in Fig. 5, the energies of all three systems roughly reach +steady states by 0.5 dpa, suggesting the activation of a recovery mechanism that offsets the +energy increase of additional Frenkel pairs. The convergence of the potential energy does +not indicate that the microstructure has reached a steady state though; a slow but continual +increase in pressure past 0.5 dpa that is most visible for the Fe system in Fig. 6 implies that +at least some features of the microstructure continue to evolve up to much higher dpa. +The supplied energy that persists through the structural relaxations can be partitioned +into local material defects and an overall elastic energy—there is no kinetic energy in the +absence of atomic velocities. The increasing elastic energy is a direct consequence of the in- +troduction of Frenkel pairs; consider that displacing an internal atom to an external surface +of a crystal increases the crystal volume by one atomic volume. The simulations are per- +formed at fixed volume though, meaning that the boundary conditions can only be satisfied +by subjecting the crystal to an increasing pressure to maintain a net zero volumetric strain. +16 + +0.0 +0.5 +1.0 +1.5 +2.0 +dpa +0 +20 +40 +60 +80 +E [meV/atom] +Fe +A-atom +CrCoNi +FIG. 5. The change in the potential energy of the simulation cell as a function of dpa for all three +material systems. +0.0 +0.5 +1.0 +1.5 +2.0 +dpa +0 +1 +2 +3 +4 + Pressure [GPa] +Fe +A-atom +CrCoNi +FIG. 6. The pressure of the simulation cell as a function of dpa for all three material systems. +The elastic energy is defined as the elastic work that would need to be performed on the +relaxed system in a zero pressure configuration to return it to the required volume. This +can be calculated from the material’s elastic constants and the volume of the system in the +relaxed configuration, with the later evaluated by allowing the simulation box to expand +while minimizing the potential energy. The percent of the supplied energy that resides as +elastic energy is negligible in all situations, being 3%, 0.2% and 0.3% for the Fe, CrCoNi +and A-atom systems, respectively. The overwhelming majority of the supplied energy there- +fore resides in the form of defects, specifically dislocations, interstitials, and vacancies (the +contribution of stacking faults was found to be negligible in Sec. II D). +17 + +B. +Dislocations +While the dislocation networks that developed in the three systems are clearly distinct, +the dislocation densities of all three initially rapidly increased before falling back to steady +state values. Figure 7 shows the dislocation networks in the Fe (left), CrCoNi (middle), and +A-atom (right) systems at 0.5 dpa (top) and 2.0 dpa (bottom) as found by the DXA, with +the network density visibly lower at higher dpa for the Fe and A-atom systems. A more +quantitative analysis of the distribution of dislocation types as a function of dpa is provided +in Fig. 8 where the dislocation density for the Fe system is visibly lower than that for both +the FCC systems at all dpa. This can be explained by the main dislocation production +mechanism in irradiated materials; isolated self-interstitials precipitate as interstitial disks, +forming dislocations loops that evolve and eventually develop into a larger network. In Fe +this is known to produce more mobile 1/2⟨111⟩ and less mobile ⟨100⟩ loops, with the pop- +ulation of the former being greater at low temperatures [55, 56]. The 1/2⟨111⟩ dislocations +begin as isolated loops, but appear to be mobile enough to migrate and react once the dislo- +cation density and internal stresses reach critical values around 0.8 dpa, reducing the overall +dislocation density as a dislocation network is formed. +The expectation was that the dislocation networks in the CrCoNi and A-atom systems +would begin as isolated extrinsic Frank loops resulting from the precipitation of interstitials +on {111} planes. It is well established experimentally that such sessile Frank loops eventu- +ally unfault to form a glissle dislocation network containing both perfect dislocations and +Shockley partials, though the precise mechanism by which this occurs continues to be a sub- +ject of study [57–59]. What was unexpected in Fig. 8 is that this unfaulting process should +apparently begin at the outset, with the population of Shockley partials visibly exceeding +that of any other dislocation type in Fig. 8 even for very low dpa. A similar behavior is +observed for the larger CrCoNi simulation described in Appendix A, indicating that this is +not merely a finite size effect, and Fig. 9 even suggests that Shockley partial loops could be +nucleating directly at the earliest stages of radiation damage. While there is a measurable +population of extrinsic Frank loops that slowly decreases up to around 1.0 dpa, the rapid +growth of the population of Shockley partials well before this point strongly suggests that +there is some other mechanism by which Shockley partials are being generated. Understand- +ing the details of the dislocation formation mechanisms at the earliest stages of radiation +18 + +FIG. 7. Dislocation networks at 0.5 dpa (top) and 2.0 dpa (bottom) for Fe (left), CrCoNi (middle), +and A-atom (right) systems where color indicates dislocation types. For the Fe system green are +1/2⟨111⟩ and purple are ⟨100⟩ dislocations. For the CrCoNi and A-atom systems green are 1/6 ⟨112⟩ +Shockley partials, purple are 1/6 ⟨110⟩ stair-rod, yellow are 1/3 ⟨100⟩ Hirth, light blue are 1/3 ⟨111⟩ +Frank, and dark blue are 1/2 ⟨110⟩ perfect dislocations. +m +m +m +FIG. 8. The dislocation density and dislocation types for Fe (left), CrCoNi (middle), and A-atom +(right) as a function of dpa. +damage would require a dedicated study that is relegated to future work though. +The generally increasing populations of sessile stair-rod dislocations with dpa in the +CrCoNi and A-atom systems is an expected result of the maturation of a glissile dislocation +network mainly composed of Shockely partials. Finally, it is significant that the dislocation +network in the A-atom system continues to undergo substantial change even up to 1.6 dpa, +19 + +FIG. 9. The dislocation network for a CrCoNi system containing approximately one million atoms +at 0.007 (left), 0.01 (middle) and 0.02 (right) dpa. Green are 1/6 ⟨112⟩ Shockley partials, purple +are 1/6 ⟨110⟩ stair-rods, yellow are 1/3 ⟨100⟩ Hirth, light blue are 1/3 ⟨111⟩ Frank, and dark blue +are 1/2 ⟨110⟩ perfect dislocations. +well beyond the 0.5 dpa at which the potential energy converged in Fig. 5. The invariance +of the potential energy even as the dislocation density is reduced by nearly half requires a +corresponding increase in the population of other defect types, a point that will be significant +in the following. +C. +Stacking faults +As expected, stacking faults (SFs) were only observed in the FCC CrCoNi and A-atom +systems. The SF configurations at 2.0 dpa are shown in Fig. 10, and the SF density as +a function of dpa is reported in Fig. 11. +The very high SF densities observed in these +systems are consistent with the low reported values for their SF energies [46] and with the +high density of partial dislocations in Fig. 8. It is interesting that the SF density for the +A-atom system was higher than for the CrCoNi system up to around 0.8 dpa, but by 2.0 +dpa the situation had reversed with the SF density in the A-atom system falling by almost a +factor of three. This roughly correlates with the decrease in the density of Shockley partials +in Fig. 8, though the correspondence is not exact. If the decrease in the SF density is in +fact driven by the maturation of the dislocation network, then an important question is +why a similar decrease in the density of Shockley partials and SFs was not observed in the +CrCoNi system. Any explanation should likely involve the chemical SRO since this is (by +design) the main difference between the CrCoNi and A-atom systems. Possible mechanisms +include atomic rearrangements following the formation of the SF lowering the energy of the +20 + +FIG. 10. Stacking faults as identified by locally HCP coordinated atoms in CrCoNi (left) and +A-atom (right) systems at 2.0 dpa. +m +FIG. 11. The stacking fault density as a function of dpa for the CrCoNi and A-atom systems. +SF relative to the pristine state and increasing the unfaulting barrier, or the fluctuations +in the local stacking fault energy decreasing the difficulty of cross slip and increasing the +complexity of the dislocation network. Either of these mechanisms would help to explain +why the dislocation density for the CrCoNi system does not fall as quickly with increasing +dpa as for the A-atom system. +The elevated SF density observed here could have significant implications for the mechan- +ical strength of irradiated CrCoNi since SFs can act as barriers to dislocation motion that +increase plastic strength. Recent work by Richie et al. used in situ transmission electron +microscopy to investigate CrCoNi under cryogenic conditions and noted that a high density +of SFs and extensive cross slip likely contributed to the superior mechanical properties of +the alloy [60]. SFs have also been suggested to decrease radiation-induced void swelling of +CrCoNi by alleviating internal stress build up [22]. +21 + +FIG. 12. The Warren-Cowley SRO parameters as a function of dpa for the CrCoNi system. +D. +Short range ordering +The CrCoNi system, initialized as a random solid solution, did develop a minor degree of +SRO during the CRA simulation. While not visually apparent this ordering was statistically +significant as indicated by the WC SRO parameters reported in Fig. 12. All of the WC +parameters stabilized by 0.5 dpa, with Cr displaying an increased likelihood to neighbor +Co and both Cr and Co less likely to neighbor other atoms of the same species. +It is +interesting that this degree of order developed despite the randomizing effects of atomic +displacements in the CRA, and is significant that the order generally agrees with what has +been found experimentally [61] and with more accurate DFT-based Monte Carlo simulations +[62]. Finally, despite the magnitude of the SRO being relatively small, the contribution +toward lowering the potential energy of the system is still substantial; as will be discussed +in Sec. III E below, the magnitude of the Esro can be as much as 30% of the magnitude of +Efit. +E. +Point defects and energy balance +The results discussed up to this point have included dislocations and stacking faults, +i.e., defects for which there are standard identification techniques and about which we have +general confidence. The same is not true for point defects, for the identification of which +there are arguably no methods in the literature (including the ones used here) that work +reliably at high defect concentrations. +This section not only reports the nominal point +22 + +defect concentrations as calculated using the methods described in Sec. II, but highlights +the inconsistency of these estimates with respect to the stored energy. This is particularly +concerning since the point defect balance equations that describe the evolution of point +defect concentrations are the foundations of our understanding of damage development in +irradiated materials [2, 9], and it is unclear whether it is currently possible to evaluate the +independent variables in these equations either by experiment or simulation. Our sincere +hope is that this will be identified as an area requiring the further attention of the research +community in the future. +One expectation is that the concentrations of vacancies and intertsitials should be equal +at low dpa and increase linearly with the number of displacements. This is because both +are initially produced in equal amounts by Frenkel pair insertions and the defect density is +not high enough for there to be appreciable recombination. While this is satisfied for the +point defect concentrations reported in Fig. 13, the vacancy and interstitial concentrations +quickly diverge with increasing dpa just as in the CRA simulations of Ref. [29]. This is +conventionally believed to indicate that the more mobile interstitials are clustering to form +dislocation loops or are annihilating on previously-existing loops once a threshold density is +reached, with the less mobile vacancies remaining in the bulk. Although our simulations only +considered single crystal systems, interstitials can also migrate to and annihilate on other +types of sinks (e.g., grain boundaries, phase boundaries, voids) in more general materials. +The nominal vacancy concentrations of the Fe and CrCoNi systems were comparable at all +dpa, and were consistently ∼50% greater than that of the A-atom system. It is suspicious +that the vacancy concentrations of the two FCC systems as found by the WS method +continue to increase all the way to 2.0 dpa though, considering that the pressures of these +systems have already converged by 0.5 dpa in Fig. 6; along with the results of the energy +fitting reported below, this suggests that the WS method increasingly overestimates the +vacancy concentrations of the FCC systems at higher dpa. Confusingly, the same does not +seem to be true for the Fe system for which the vacancy concentration, pressure, and energy +model results are all consistent. The reason for this discrepancy is unknown, but highlights +the need for more robust ways to identify vacancies (or a more precise definition of what +constitutes a vacancy) in highly damaged structures. +The results of fitting the energy model Efit of Sec. II D to Estored for all three structures +are reported in Fig. 14, with the values of the fitting parameters β and γ given in Table +23 + +FIG. 13. Estimated vacancy and interstitial concentrations as measured by the WS and K-means +methods, respectively, as a function of dpa for all three material systems. +III. The model works remarkably well for the Fe system, with Efit reproducing all of the +general trends of Estored and many of the smaller features as well. The fitted values of β and +γ for the Fe system are also physically reasonable, being slightly below one as is necessary +for the clustering of point defects to be energetically favorable. This is not the case for the +CrCoNi and A-atom systems though; both of these systems exhibit a Efit that continually +increases with dpa rather than converging around 0.5 dpa, and γ values that are well above +the physically reasonable bound of 1.0. The second observation in particular suggests that +the interstitial concentration is severely underestimated, with the model compensating for +an unreasonably low value of Nint in Eq. 5 by elevating the value of γ. Supposing from the +Fe system that γ should be ∼0.9, the interstitial concentration in the FCC systems appears +to be underestimated by a factor of two to three at high dpa. While it is true that the +steady-state interstitial concentrations should not necessarily be comparable in BCC and +FCC systems, Fig. 13 is also consistent with the interstitial concentrations in FCC systems +being systematically underestimated by the same factor. The source of the error is likely +that the difference in atomic volumes of interstitials and atoms of the crystalline lattice is +less pronounced in FCC than BCC systems, with interstitials in BCC Fe often adopting +split-dumbbell configurations. +If the vacancy concentrations are relatively accurate and the errors in the interstitial +concentrations are of a similar magnitude for the CrCoNi and A-atom systems, then the +gap between the vacancy and interstitial concentrations would be significantly larger for the +24 + +sro +FIG. 14. Fit of the stored energy model described in Sec. II D for the Fe (left), CrCoNi (center), +and A-atom (right) systems. The dotted trend lines are bounds derived from the estimated bounds +on the dislocation network energy. +TABLE III. Parameters fit by the energy model that account for deviations of energy from isolated +defects. β is the coefficient for vacancies and γ is for the interstitials. +β +γ +Fe +0.879 0.840 +CrCoNi 0.454 2.90 +A-atom 0.970 1.49 +CrCoNi than for the A-atom system throughout the simulations, even at very low dpa. If +this is true, then that would have important implications for the effects of SRO and lattice +distortions on the development of radiation damage. Based on a conventional understanding +of the point defect balance equations [2, 9], prior observations of CrCoNi’s low-temperature +radiation resistance [23] imply that the gap between the vacancy and interstitial concentra- +tions should be smaller for CrCoNi, encouraging point defect recombination and suppressing +vacancy precipitation. While this is superficially inconsistent with our results, observe that +a higher sustained vacancy concentration does not necessarily lead to a higher susceptibility +to voids. Specifically, there is evidence that SRO and lattice distortions can increase the +magnitude of energetic well depths for vacancies [63], effectively decreasing the vacancy- +vacancy and vacancy-void capture radii and allowing higher vacancy concentrations to be +sustained as compared to the A-atom system. Given the magnitude of the apparent errors +in the estimated vacancy and interstitial concentrations though, there is not strong evidence +25 + +for this conclusion at present. +There are other qualifications relating to the CRA’s limitations that should be made +regarding the observed differences in point defect concentrations across the three material +systems. One important factor absent in these simulations is the thermal spike associated +with a collision cascade. MPEAs have been found to have a shorter mean-free electron path +and a lower thermal conductivity than traditional alloys, and it has been proposed that this +could prolong the thermal spike and increase the extent of defect recombination [64, 65]. It is +also worth considering the potential effects of nanotwinning, which has been experimentally +observed to be an important cryogenic strengthening mechanism for CrCoNi [60]. While the +CRA did not induce any visible nanotwinning, this could be related to the relative sizes of +the simulation cell and the critical nucleation event. Nanotwins would not only affect the +development of the dislocation network, but have shown the ability to capture or promote +the transport of point defects to sinks in irradiated Cu [66]. +IV. +CONCLUSION +CRA simulations were conducted to investigate differences in the development of irra- +diated microstructures in BCC Fe, FCC CrCoNi, and FCC A-atom systems in the low +temperature and high dose rate regime up to 2.0 dpa. The CrCoNi system developed the +highest overall dislocation density and exhibited the ability to maintain that dislocation +density even as the dislocation networks in the Fe and A-atom systems matured and sim- +plified. The higher stacking fault density in CrCoNi is likely related to the consistently +higher density of partial dislocations relative to the other systems. The successive insertion +of Frenkel pairs entailed by the CRA mildly increased the degree of chemical short range +order in CrCoNi relative to the initial random solid solution in a way that is consistent with +other modeling and experimental studies. A model was developed for the energy stored in +the material defects, and strongly suggests that the interstitial concentrations for the FCC +systems as estimated from the distribution of Voronoi cell volumes are lower than the actual +effective concentrations by a factor of two to three. It is also possible that the Wigner-Sietz +method of identifying vacancies slightly overestimates the vacancy concentration for the +FCC systems at high defect concentrations. The same methods of point defect identifica- +tion seem to be much more reliable for the BCC Fe system though, with the energy model +26 + +closely following the simulation results. Given the overwhelming importance of point defects +to the development of radiation damage and the intense interest in CrCoNi and other FCC +MPEAs for nuclear applications, our results reveal a critical need to either develop more +robust ways to measure point defect concentrations in heavily-damaged FCC materials, or +perhaps to reevaluate what is meant by a point defect in such materials. +ACKNOWLEDGMENTS +JCS gratefully acknowledges partial support by the Nuclear Regulatory Commission +(Award 31310019M0009) through the Advancing Scientific Careers to Enhance Nuclear Tech- +nologies (ASCENT) program at UC Davis. This work was partially performed under the +auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory +under Contract DE-AC52-07NA27344. +AUTHOR CONTRIBUTIONS +CS and JCS performed all the simulations and analysis. JKM conceptualized the project +and developed the methodology. All authors collaborated to writing, reviewing, and editing. +COMPETING INTERESTS +The authors declare no competing interests. +Appendix A: Convergence with system size +As described in Sec. II A, one simulation of equiatomic CrCoNi containing a total of +1 048 576 atoms was conducted to 3.0 dpa to investigate the effect of system size. The +behavior of this system is compared in Fig. 15 (top row) with that of the smaller simulation +containing a total of 131 072 atoms (bottom row). There is remarkable agreement between +the two simulations, the main difference being that the plots for the larger simulation are +smoother due to the larger number of atoms helping to average out the fluctuations. 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Zhang, Nature +communications 6, 1 (2015). +32 + diff --git a/WNE3T4oBgHgl3EQfFQlM/content/tmp_files/load_file.txt b/WNE3T4oBgHgl3EQfFQlM/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9f1d81185958d0e57d198e4e53813a52ff890c26 --- /dev/null +++ b/WNE3T4oBgHgl3EQfFQlM/content/tmp_files/load_file.txt @@ -0,0 +1,962 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf,len=961 +page_content='Dependence of simulated radiation damage on crystal structure and atomic misfit in metals J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' Stimac Department of Chemical Engineering, University of California, Davis, CA, 95616, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' and Lawrence Livermore National Laboratory, Livermore, CA, 94550, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' Serrao and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' Mason∗ Department of Materials Science and Engineering, University of California, Davis, CA, 95616, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='04303v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='mtrl-sci] 11 Jan 2023 Abstract This study investigates radiation damage in three metals in the low temperature and high radiant flux regime using molecular dynamics and a Frenkel pair accumulation method to simulate up to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='0 displacements per atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' The metals considered include Fe, equiatomic CrCoNi, and a fictitious metal with identical bulk properties to the CrCoNi composed of a single atom type referred to as an A-atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' CrCoNi is found to sustain higher concentrations of dislocations than either the Fe or A-atom systems and more stacking faults than the A-atom system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' The results suggest that the concentration of vacancies and interstitials are substantially higher for the CrCoNi than the A-atom system, perhaps reflecting that the recombination radius is smaller in CrCoNi due to the roughened potential energy landscape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' A model that partitions the major contributions from defects to the stored energy is described, and serves to highlight a general need for higher fidelity approaches to point defect identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' INTRODUCTION Structural components in nuclear fission reactors need to be engineered to withstand decades of exposure to radiation and elevated temperatures [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' Materials in the next generation of fission reactors and future fusion reactors will be subject to substantially higher radiation dosages and temperatures and likely highly corrosive environments as well [3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' Irradiation by exposure to high-energy particles displaces atoms and damages the crystalline structure of pure metals and alloys [5, 6], resulting in a variety of microstructural changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' Specifically with respect to the degrading effects of radiation, there are five considerations for structural metals in reactor environments: radiation hardening (low temperature), radiation- induced segregation and precipitation, void swelling, radiation-induced creep, and helium embrittlement (high temperature) [7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' The three intermediate temperature effects are usually the most relevant in practice, and are often observed simultaneously since they are all strongly associated with the underlying ability of point defects generated by radiation to migrate through the lattice [9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' Radiation-induced segregation of substitutional solutes is generally attributed to the in- verse Kirkendall effect where, as vacancies migrate to and annihilate on sinks, the different ∗ jkmason@ucdavis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='edu 2 atomic species have different migration rates in the opposite direction [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' The segregation extent is governed by differences in diffusive mobility, with undersized solutes generally be- ing enriched and oversized solutes being depleted in the vicinity of the sinks [10, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' Void swelling is a serious engineering concern that entails an increase in the volume of the mate- rial by as much as 1% per dpa [13] from the nucleation and growth of voids in the bulk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' This process is driven by an imbalance in the concentrations of vacancies and interstitials, with annihilation of high-mobility interstitials on dislocations and other sinks leaving behind a relative excess of vacancies that precipitate as voids [9, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' While a variety of mechanisms have been proposed for irradiation creep, the two main mechanisms are believed to be the climb of favorably-oriented dislocations by the stress-induced preferred absorption of point defects and the glide of dislocations enabled by climb over obstacles [9, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' Multi-principal component alloys (MPEAs) [16, 17] (often called high-entropy alloys or compositionally complex alloys) consist of a few to several atom types in solid solution with nominally equi-atomic concentrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' This emerging class of metals could perform well as structural materials in irradiated environments, with initial evidence showing exceptional mechanical properties [18, 19] and higher resistance to radiation damage [20, 21] than tra- ditional metals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' One experimental and computational study of the effects of radiation on a CrCoNi MPEA showed decreases in the relative disorder, the number of defects, and large defect clusters compared to irradiated samples of pure Ni and a NiFe binary alloy [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' The authors attributed these effects to decreasing dislocation mobility with increasing number of atomic species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' [22] studied void swelling in several Ni-based alloys including pure Ni, NiFe, CrCoNi and two quinary alloys with and without prior nanoindentation, and found that nanoindentation improved resistance to swelling by increasing the density of defects like dislocations, stacking faults, and twin boundaries that promote vacancy annihilation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' Cu- riously, Veli¸sa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' [23] found that CrCoNi showed superior irradiation resistance relative to NiCr only at temperatures near and below 300 K though.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' The reason for this is not well established, but is believed to be related to the specifics of the chemical short range order (SRO) that developed in the two alloys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' The most direct way to simulate radiation damage at the atomic level uses molecular dynamics (MD) simulations of collision cascades [5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' This involves assigning a large initial velocity to the primary knock-on atom (PKA) to mimic a passing neutron or other high- energy particle transferring kinetic energy to the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' The PKA recoils, displacing many 3 of the surrounding atoms from their lattice sites and converting the initial kinetic energy into a thermal spike with sufficient energy to facilitate the regeneration of the crystalline lattice and the recombination of many, but not all, of the generated point defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' The resulting interstitials and vacancies are respectively distributed on the periphery and the interior of the affected zone, increase the point defect concentration in the material, and are directly responsible for the most visible degrading effects of radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' Mass conservation requires that the interstitials and vacancies generated by a single collision cascade occur as Frenkel pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' The number of Frenkel pairs generated in this way divided by the number of atoms in the material is known as the displacements per atom (dpa), and is the standardized measure of the extent of radiation damage in crystalline materials [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' Early work by Kinchin and Pease modeled atoms as hard-spheres that exhibit elastic collisions during collision cascades and laid the theoretical foundation for radiation effects in crystalline materials [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' If the energy transferred to an atom exceeded a material- specific threshold value, then the atom was said to have been displaced from its lattice site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' While energies below the threshold could still displace the atom, it would return to its lattice site after the initial perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' The theory developed by Kinchin and Pease was further developed in the work of Norgett, Robinson, and Torrens (hereafter referred to as the NRT model) who added additional terms to account for energy lost to ionization and for the affects of inelastic collisions [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' Several more recent studies concluded that the NRT model overestimates the number of defects generated by a collision cascade and neglects the mixing from atomic replacements though, spurring a number of proposed refinements [5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' These are significant for the reason that accurate damage models that can reliably predict dpa are essential to reliably compare radiation damage resistance among various materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' Full-scale atomic simulations of collision cascades have been performed for several decades now, and are useful to uncover the evolution of radiation-induced primary damage [27, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' However, the use of MD for this application is subject to several limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' As Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' [6] points out, interatomic potentials cannot capture the effects of deviations from the Born- Oppenheimer approximation when excited electronic states are induced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' The other main limitations are the time and length scales that can reasonably be achieved with modern computational resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' MD simulations are usually no longer than a few nanoseconds, limiting the overall radiation dose that can reasonably be achieved by full cascade simulations without the events overlapping in time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' [29, 30] further discuss these limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' As 4 a result of the high computational cost to reach appreciable dpa, the doses investigated in MD simulations have been historically been less than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='0 dpa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' Although these are useful to understand defect creation at low doses, structural materials in a nuclear reactor core can experience as much as 80 dpa over a 40 year service life [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' A variety of strategies have been used to circumvent the computational limitations im- posed by cascade simulations, one of which involves the direct insertion of a high density of Frenkel pairs [31–33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' Such Frenkel pair accumulation (FPA) techniques forgo the dynamics of time-resolved high-energy atomic collisions stemming from the primary knock-on event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' Instead, they intermittently introduce Frenkel pairs by randomly displacing atoms from their lattice positions, usually followed by some form of equilibration and time integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' These methods benefit from a clearly defined radiation dose, based on the the number of displaced atoms, and dose rate, based on the ratio of displaced atoms to the simulated time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' For example, Chartier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' [34] used a FPA procedure to model irradiation of UO2 and found the steady-state dislocation density to be in good agreement with experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' A major limitation of FPA procedures though is the absence of any effects related to the thermal spike [30], particularly providing the thermal energy necessary for diffusion and clustering of defects;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' this includes the recombination of point defects when vacancies and interstitials collide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' Analysis of FPA simulations should therefore be done with careful consideration of this limitation to avoid the potential for nonphysical extrapolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' Recent work by Derlet and Dudarev [29] introduced a variant of the FPA method that fur- ther streamlines the process of sampling irradiated microstructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' Known as the creation- relaxation algorithm (CRA), this differs from the preceding FPA methods in that there are no time-integrated dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' CRA simulations randomly select atoms and displace them with random directions and magnitudes, just as other FPA methods do, but always follow this by potential energy minimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' The entire simulation involves repeating this process for a specified number of displacements, with the canonical dpa equal to the number of displaced atoms divided by the total number of atoms in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' Dudarev and Derlet applied the CRA to BCC Fe systems of a variety of sizes and reported good agreement between full cascade simulations [35, 36] and the CRA for interstitial density as a function of dpa, at least up to a linear rescaling of both the independent and dependent variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' The need for rescaling is likely related to the CRA effectively being performed at zero Kelvin [37];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' the thermally-driven diffusion of point defects that is prevalent at high temperatures 5 and substantially contributes to microstructure evolution of irradiated materials is negligible in such conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' That said, the CRA can be viewed as simulating radiation damage in conditions where thermally-driven diffusion is active but negligible compared to other mass transport mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' The CRA has since been used to investigate radiation effects in materials other than BCC Fe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' One such study used the CRA to simulate irradiation of a NiFe system doped with carbon and evaluated the ability of carbon interstitials to decrease radiation damage [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' Several others applied the CRA to tungsten to investigate the relationships between dpa and specific physical parameters at relatively high doses (above 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='0 dpa), either alone or with the assistance of other computational or experimental methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' Parameters that were investigated include thermal conductivity [39], hydrogen embrittlement and tritium concentration [40], and radiation induced structural evolution [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' This paper investigates the microstuctures of highly irradiated Fe, equi-atomic CrCoNi, and a fictitious metal with identical bulk properties to the CrCoNi composed of a single atom type referred to as an A-atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' The main motivation for including the Fe and A-atom systems is to establish points of comparison for the investigation of the reported radiation resistance of CrCoNi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' The CRA is used to simulate the irradiation of all three systems up to a final dose of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='0 dpa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' The details of the implemented CRA, as well as an analysis of the experimental dose rates and temperatures for which it is likely relevant, are described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' The same section also outlines our methods for identifying material defects including dislocations, vacancies, interstitials, and stacking faults, and a model for the energy stored in those defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' The results and a discussion of the simulations are included in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' III, and Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' IV draws conclusions to inform further research in this area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' METHODS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' CRA simulations All molecular dynamics simulations were performed using the LAMMPS software [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' Orthorombic simulation cells were used with periodic boundary conditions for all cell faces, and the number of atoms remained constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' No temperature or time-steps were defined for any simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' After constructing the initial space-filling single crystals, the volume of the 6 simulation cell was relaxed using a potential energy minimization and fixed thereafter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' The present study examined three material systems: BCC Fe, equi-atomic FCC CrCoNi, and FCC A-atom designed to reproduce the bulk properties of CrCoNi using a single fictitious atom type [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' Comparison with an A-atom model more directly allows identification of the effects caused by chemical short range order (SRO) and lattice distortion (LD) which are widely implicated in the enhanced physical properties of MPEAs [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' The BCC Fe simulation consisted of a simulation volume of 40 units cells along each of the three dimensions, and with two atoms in each unit cell contained a total of 128 000 atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' This simulation used the Mendelev-II embedded atom method (EAM) potential [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' The FCC CrCoNi simulation consisted of 32 unit cells along each of the three dimensions, and with four atoms in each unit cell contained a total of 131 072 atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' All three atomic species were represented with equal concentrations, and were initially distributed uniformly at random throughout the simulation cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' The purpose of reducing the number of unit cells for the FCC systems was to make the number of atoms as close as possible to that in the Fe simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' The equiatomic CrCoNi simulation utilized the EAM potential developed by Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' The FCC A-atom simulation used the EAM potential developed by Jian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' [46] but was otherwise identical to the CrCoNi simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' This work circumvented the computational limitations associated with full collision cas- cade simulations by using the creation relaxation algorithm (CRA) [29] to simulate the effects of irradiation by the repeated introduction of Frenkel pairs [34, 47];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' the basic CRA is described in Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' All three experiments used the CRA to simulate radiation damage up to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='0 dpa, requiring a total of 256 000 atomic displacements in the Fe system and 262 144 displacements in both the CrCoNi and A-atom systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' The potential energy minimization conducted after each atomic displacement consisted of two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' First, the position of ev- ery atom was randomly perturbed by a small Gaussian-distributed displacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' Second, a standard gradient-based energy minimization was applied to the atomic positions to reduce the system energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' The purpose of the small perturbations in the first step was to help the system escape shallow local energy minima as would naturally occur in a thermal sys- tem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' The perturbation magnitude was fixed after systematically exploring the relationship between the perturbation magnitude and the relaxed potential energy of a smaller BCC Fe system of 2000 atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' This involved displacing 100 atoms and performing energy min- imizations without perturbations between displacements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' After all the displacements,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='Algorithm 1 Creation Relaxation Algorithm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='1: n := 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='▷ number of displaced atoms ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='2: while n/N < φdpa do ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='▷ N is total number of atoms ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='3: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='Randomly select an atom i uniformly over all atoms ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='4: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='Move atom i to a randomly selected point within the simulation cell ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='5: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='while potential energy increase exceeds a threshold do ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='6: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='Move atom i to a randomly selected point within the simulation cell ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='7: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='end while ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='8: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='Relax the system using energy minimization ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='9: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='n ← n + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='10: end while ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='atoms were subjected to repeated cycles of perturbations and energy minimization,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' with the resulting potential energy profiles for a given standard deviation of the perturbation magni- tude shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' 1 as a function of cycle number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' Standard deviations of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='005 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='01 angstroms were found to rapidly result in a relatively stable potential energy minimum, with larger displacements often introducing additional defects and smaller displacements unable to reliably allow the system to escape shallow potential energy minima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' A perturbation magnitude of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='01 angstroms was chosen for the subsequent simulations for the reason that it reached the potential energy minimum in the fewest number of cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' Since the intention was for the perturbations to make the system more closely resemble a thermal one, a natural question is what would be the corresponding temperature for a given perturbation magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' The equipartition theorem implies that the potential energy per atom relative to that in an ideal crystal should be (3/2)kBT for a thermal system;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' equating this with the average potential energy increase per atom resulting from a single perturbation gives an equivalent temperature of ∼30 K for all three systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' This implies that the additional energy supplied by the perturbations is consistent with the assumption that any thermally-driven diffusion of point defects is insignificant compared to alternative mass transport mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' In addition to the three simulations described above, another CRA simulation of equiatomic CrCoNi with 64 unit cells along each dimension and a total of 1 048 576 atoms 8 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' The potential energy per atom of a highly defected BCC Fe system as a function of mini- mization cycles, measured with respect to the energy of an isolated atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' Perturbation magnitudes of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='005 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='01 angstroms allowed the system to quickly reach a deep potential energy minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' was conducted to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='0 dpa to investigate the effect of system size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' With energy minimization being by far the most computationally intensive step in the CRA, this simulation performed 25 atomic displacements (step 4 of Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' 1) per energy minimization to reduce the simulation run time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' the larger volume of this simulation decreases the probability that multiple atomic displacements occur in a given region without an intervening relaxation event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' The main results of this simulations are presented in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' Applicable temperatures and dose rates As mentioned in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' I, the microstructures that develop when using the CRA are a direct product of relaxations of the atomic-level stress fields and do not necessarily represent microstructures that develop in conditions with appreciable thermally-driven diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' We characterize the regime of physical conditions for which the CRA is applicable by a simple argument that compares the rates of atomic transport by thermally-driven diffusion and ballistic displacements from collision cascades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' The main criterion for applicability is that the rate of ballistic displacements per atom K0 be much greater than the interstitial hopping rate γdiff (assumed to be the fastest thermally-driven diffusion event).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' In such conditions, any effects of thermally-driven diffusion should be negligible compared to those resulting from ballistic displacements and the subsequent stress relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' The interstitial hopping rate, as 9 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' Contour plot of the applicability measure A = log10(K0/γdiff) as a function of the logarithm of the dose rate and temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' The calculations used the diffusivity and activation energy for interstitial diffusion in CrCoNi given by Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' described by transition state theory, obeys the Arrhenius relation γdiff = ν0exp[−Ea/(kBT)] where ν0 is the high-temperature limit for the site-hopping frequency, Ea is the activation energy for interstitial diffusion, kB is Boltzmann’s constant, and T is the temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' The random walk diffusion model suggests that ν0 be approximated by 6D0/λ2 where D0 is the diffusivity prefactor and λ is the distance separating two interstitial sites [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' Using the values for the prefactor and activation energy for interstitial diffusion for Cr- CoNi given by Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' [49] leaves only the temperature and K0 as independent variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' A measure of applicability A = log10(K0/γdiff) is defined such that an increase of A by one indicates an order of magnitude increase in the ratio of K0 to γdiff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' Figure 2 gives a contour plot of A for CrCoNi and indicates that applicability increases at lower temperatures and higher dose rates;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' moreover, the dependence on the temperature is much more significant than the dose rate, a consequence of the exponential scaling in the Arrhenius relation for the diffusivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' For the dose rates considered, the temperatures for which A ≥ 2 are all in the cryogenic range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' This low temperature constraint is entirely consistent with the equivalent temperature resulting from the perturbations in the atomic positions described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' II A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' 10 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' Identifying defects It is difficult to precisely identify crystalline defects in a material that has suffered exten- sive radiation damage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' Certainly there is still an underlying lattice, but conventional defect models assume that the defect is isolated, or equivalently, that the surrounding atoms occupy well-defined lattice positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' This is not true for the high defect concentrations that can occur in irradiated materials, and the identification of reference lattice sites is further com- plicated in MPEAs like CrCoNi where the reference lattice is already perturbed by atomic size differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' The numbers and types of dislocations in the simulations were determined using OVITO’s dislocation extraction algorithm (DXA) [50];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' the algorithm requires a trial circuit length and a value for circuit stretchability which were set to 14 and 9 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' Surprisingly, the DXA appeared to be relatively robust to extensive disruption of the crystalline lattice from the CRA, with very few isolated dislocation segments appearing in the networks reported below in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' III B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' The number of vacancies were determined using OVITO’s Wigner-Seitz (WS) analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' This constructs Voronoi cells around the lattice sites in an initial crystal structure and checks the atom occupancy of each Voronoi cell in subsequent time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' A Voronoi cell that does not contain any atoms is regarded as indicating the presence of a vacancy, though this approach does not precisely define the vacancy location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' The WS analysis was compared with two other approaches to estimate vacancy concentration in the BCC Fe systems, the first of which was the BCC defect analysis (BDA) [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' The second approach involved evaluating the relaxed volume of the simulation box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' Inserting a single Frenkel pair produces an interstitial and a lattice site occupied by a vacancy, resulting in the expansion of a crystal subject to zero traction boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' Since the simulations were conducted at constant volume, the insertion of Frenkel pairs in the CRA instead elevated the system pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' The equivalent volume change was evaluated by relaxing the simulation cell subject to a zero traction boundary condition, and enabled the number of vacancies to be estimated by assuming that the volume change from inserting a Frenkel pair remained constant throughout the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' Figure 3 compares the three methods and shows that the WS analysis agrees much better with with the vacancy concentration estimated from the relaxed volume than from the BDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' The factor by which the WS and relaxed volume method differ can 11 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' Three different estimates for the number of vacancies in BCC Fe as a function of dpa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' The BDA estimate is likely inaccurate due to the difficulty of identifying crystal lattice sites in highly defected materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' be interpreted as the reduction in volume change per effective Frenkel pair insertion with increasing dpa and is related to point defect interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' The poor performance of the BDA is likely a consequence of extensive radiation damage making it difficult to identify vacancies based on features of local atomic environments without a clear underlying crystal lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' Interstitials are the defect type that is most difficult to identify in our simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' The multiplicity of local atomic configurations that can occur for non-isolated interstitials means that template-based approaches would be difficult or even infeasible to implement, and examining the expected number of atoms contained within a surface that passes only through crystalline material (the analogue of a Burgers circuit) often fails because of the obstructions to constructing such surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' In the absence of a canonical alternative, our approach involves constructing a distribution of atomic volumes as estimated by the Voronoi polyhedra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' The compressive stresses around an interstitial reduce the volumes of the interstitial and of the surrounding atoms in a characteristic way when the interstitial is in an otherwise perfect crystal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' 4 shows an example of this phenomenon for BCC Fe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' The distribution of atomic volumes in our simulations is decomposed with a K-means algorithm into a superposition of peaks, one for each characteristic atomic volume surrounding an isolated interstitial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' Part of the utility of K-means is that the boundaries that define the locations the peaks can be dynamically updated from one time step to the next;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' this is necessary to account for the shifting and broadening of the peaks as the structure reaches higher defect concentrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' 12 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='5 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='0 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='5 Atomic volume [A3] 0 1 2 3 4 5 6 Count 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='5 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='0 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='5 Atomic volume [A3] 0 200 400 600 800 1000 Count FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' Atomic volume distribution as given by Voronoi tessellations in BCC Fe for (left) an isolated interstitial and (right) at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='01 dpa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' All histograms were evaluated using 200 equally-sized bins within the domain of the distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' The simulation cells for the isolated interstitial and the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='01 dpa distributions contained 54 000 and 128 000 atoms, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' Comparing the numbers of atoms assigned to each peak with the corresponding numbers of atoms for an isolated interstitial gives an estimate for the number of interstitials in our simulations, though the accuracy is expected to decrease as the peaks broaden and overlap with increasing radiation damage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' Stacking fault (SF) densities in the FCC CrCoNi and A-atom systems are relatively simple to evaluate by comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' Since the local atomic structure around atoms belonging to a SF appears to be HCP, all that is necessary for a reasonable estimate is to count the number atoms classified as HPC by, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=', OVITO’s common neighbor analysis, and to convert this to an equivalent SF area using the geometry of the crystal structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' The Warren-Cowley parameters are used to quantify the type and degree of chemical short range order (SRO), and are defined as [52, 53] αij = 1 − pij/cj (1) where cj is the overall fraction of atomic type j, and pij is the probability that an atom in the first nearest-neighbor shell of an atom with type i is of type j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' Positive values of αij means there is a repulsion between species i and j while negative values indicate attraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' 13 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' Energy model This section develops a model to partition the energetic contributions of the various types of material defects to the overall potential energy change as a function of dpa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' The potential energy stored in defects is defined as Estored = Epe − Ecoh − Eelas, where Epe is the overall potential energy, Ecoh is the cohesive energy of a reference material, and Eelas is the elastic strain energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' This stored energy is modeled as a sum over defect contributions: Efit = Edis + Evcy + Eint − Esro (2) which includes terms for the contributions of dislocations, vacancies, interstitials, and chemi- cal short range order, respectively, and where the sign of Esro reflects the fact that increasing SRO decreases the system energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' The contribution of stacking faults to Efit was found to be negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' The energy of a well-developed dislocation network in an elastically isotropic material is well-described by the equation [54]: Edis = χµb2 4π ρ ln � 1 rc√ρ � (3) where µ is the shear modulus, b is the Burgers vector, rc is a cutoff radius of the dislocation core stress field, and ρ is the dislocation density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' The value of rc is approximated as the lattice parameter a0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' χ is a parameter that accounts for the overall dislocation character, and depending on the material and the development of the network is expected to be in the interval 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='0 ≤ χ ≤ 1/(1 − ν) where ν is Poisson’s ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' The elastic constants that are necessary to evaluate Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' 3 were found in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' [44, 46], and the material parameters used in the model are included in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' The energetic contributions of the point defects are modeled as: Evcy = βevcyNvcy (4) Eint = γeintNint (5) where evcy and eint are the formation energies of an isolated vacancy or interstitial as found by inserting a single vacancy or interstitial into an otherwise perfect crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' More specifically, after minimizing the potential energy and measuring the potential energy change ∆E relative 14 TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' Material properties used as parameters to evaluate the dislocation energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' The values for the CrCoNi and A-atom materials can be found in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' [44, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' µ [GPa] b [A] a0 [A] ν Fe 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='49 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='87 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='373 CrCoNi 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='47 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='414 A-atom 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='47 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='403 TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' Isolated point defect formation energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' evcy [eV] eint [eV] Fe 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='71 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='01 CrCoNi 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='49 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='45 A-atom 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='62 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='85 to the perfect crystal, the isolated point defect formation energies are defined as: evcy = ∆Eremove + Ecoh (6) eint = ∆Eadd − Ecoh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' (7) The values of these formation energies are reported in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' Nvcy and Nint are the numbers of vacancies and interstitials, and β and γ are fitting parameters to account for deviations from the isolated point defect energies that occur as the density of point defects increases with dpa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' Finally, Esro is the energy associated with the chemical short range order relative to a random solid solution, and was evaluated for the CrCoNi MPEA system using molecular statics calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' Starting with an initial configuration, the types of all the atoms were randomly reassigned to one of the three constituent elements and the potential energy of the structure was minimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' This procedure was repeated five times for each configuration to account for statistical variations, and the the average potential energy change relative to the initial structure was included in the model for each dpa for which the fitting was conducted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' 15 Overall, the energy contribution of stacking faults is considered negligible, with an average value over all dpa of only −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='0463 meV/atom for the CrCoNi system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' The resulting energy model contains the three adjustable parameters χ, β, and γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' Fit- ting the model to each of the three systems involved first constructing the bounds on the dislocation energy that would be realized by the minimum and maximum allowed values for χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' The dislocation energy was then fixed at the lower bound, and a least-squares technique was used to find the values of β and γ that minimized the difference between Estored and Efit over the entire interval up to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='0 dpa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' Repeating this procedure with the dislocation energy fixed at the upper bound allowed the construction of corresponding intervals for the predicted contributions of the vacancy and interstitial terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' As discussed below in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' III E, the largest contribution to the model error is likely the estimated numbers of point defects Nvcy and Nint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' RESULTS AND DISCUSSION A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' Total energy and pressure The successive generation of Frenkel pairs considerably increases the potential energy of the simulated systems, particularly in the absence of thermally-driven point defect migration and recombination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' As displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' 5, the energies of all three systems roughly reach steady states by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='5 dpa, suggesting the activation of a recovery mechanism that offsets the energy increase of additional Frenkel pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' The convergence of the potential energy does not indicate that the microstructure has reached a steady state though;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' a slow but continual increase in pressure past 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='5 dpa that is most visible for the Fe system in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' 6 implies that at least some features of the microstructure continue to evolve up to much higher dpa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' The supplied energy that persists through the structural relaxations can be partitioned into local material defects and an overall elastic energy—there is no kinetic energy in the absence of atomic velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' The increasing elastic energy is a direct consequence of the in- troduction of Frenkel pairs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' consider that displacing an internal atom to an external surface of a crystal increases the crystal volume by one atomic volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' The simulations are per- formed at fixed volume though, meaning that the boundary conditions can only be satisfied by subjecting the crystal to an increasing pressure to maintain a net zero volumetric strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' 16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='0 dpa 0 20 40 60 80 E [meV/atom] Fe A-atom CrCoNi FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' The change in the potential energy of the simulation cell as a function of dpa for all three material systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='0 dpa 0 1 2 3 4 Pressure [GPa] Fe A-atom CrCoNi FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' The pressure of the simulation cell as a function of dpa for all three material systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' The elastic energy is defined as the elastic work that would need to be performed on the relaxed system in a zero pressure configuration to return it to the required volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' This can be calculated from the material’s elastic constants and the volume of the system in the relaxed configuration, with the later evaluated by allowing the simulation box to expand while minimizing the potential energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' The percent of the supplied energy that resides as elastic energy is negligible in all situations, being 3%, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='2% and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='3% for the Fe, CrCoNi and A-atom systems, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' The overwhelming majority of the supplied energy there- fore resides in the form of defects, specifically dislocations, interstitials, and vacancies (the contribution of stacking faults was found to be negligible in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' II D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' 17 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' Dislocations While the dislocation networks that developed in the three systems are clearly distinct, the dislocation densities of all three initially rapidly increased before falling back to steady state values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' Figure 7 shows the dislocation networks in the Fe (left), CrCoNi (middle), and A-atom (right) systems at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='5 dpa (top) and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='0 dpa (bottom) as found by the DXA, with the network density visibly lower at higher dpa for the Fe and A-atom systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' A more quantitative analysis of the distribution of dislocation types as a function of dpa is provided in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' 8 where the dislocation density for the Fe system is visibly lower than that for both the FCC systems at all dpa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' This can be explained by the main dislocation production mechanism in irradiated materials;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' isolated self-interstitials precipitate as interstitial disks, forming dislocations loops that evolve and eventually develop into a larger network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' In Fe this is known to produce more mobile 1/2⟨111⟩ and less mobile ⟨100⟩ loops, with the pop- ulation of the former being greater at low temperatures [55, 56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' The 1/2⟨111⟩ dislocations begin as isolated loops, but appear to be mobile enough to migrate and react once the dislo- cation density and internal stresses reach critical values around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='8 dpa, reducing the overall dislocation density as a dislocation network is formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' The expectation was that the dislocation networks in the CrCoNi and A-atom systems would begin as isolated extrinsic Frank loops resulting from the precipitation of interstitials on {111} planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' It is well established experimentally that such sessile Frank loops eventu- ally unfault to form a glissle dislocation network containing both perfect dislocations and Shockley partials, though the precise mechanism by which this occurs continues to be a sub- ject of study [57–59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' What was unexpected in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' 8 is that this unfaulting process should apparently begin at the outset, with the population of Shockley partials visibly exceeding that of any other dislocation type in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' 8 even for very low dpa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' A similar behavior is observed for the larger CrCoNi simulation described in Appendix A, indicating that this is not merely a finite size effect, and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' 9 even suggests that Shockley partial loops could be nucleating directly at the earliest stages of radiation damage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' While there is a measurable population of extrinsic Frank loops that slowly decreases up to around 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='0 dpa, the rapid growth of the population of Shockley partials well before this point strongly suggests that there is some other mechanism by which Shockley partials are being generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' Understand- ing the details of the dislocation formation mechanisms at the earliest stages of radiation 18 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' Dislocation networks at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='5 dpa (top) and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='0 dpa (bottom) for Fe (left), CrCoNi (middle), and A-atom (right) systems where color indicates dislocation types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' For the Fe system green are 1/2⟨111⟩ and purple are ⟨100⟩ dislocations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' For the CrCoNi and A-atom systems green are 1/6 ⟨112⟩ Shockley partials, purple are 1/6 ⟨110⟩ stair-rod, yellow are 1/3 ⟨100⟩ Hirth, light blue are 1/3 ⟨111⟩ Frank, and dark blue are 1/2 ⟨110⟩ perfect dislocations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' m m m FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' The dislocation density and dislocation types for Fe (left), CrCoNi (middle), and A-atom (right) as a function of dpa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' damage would require a dedicated study that is relegated to future work though.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' The generally increasing populations of sessile stair-rod dislocations with dpa in the CrCoNi and A-atom systems is an expected result of the maturation of a glissile dislocation network mainly composed of Shockely partials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' Finally, it is significant that the dislocation network in the A-atom system continues to undergo substantial change even up to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='6 dpa, 19 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' The dislocation network for a CrCoNi system containing approximately one million atoms at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='007 (left), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='01 (middle) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='02 (right) dpa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' Green are 1/6 ⟨112⟩ Shockley partials, purple are 1/6 ⟨110⟩ stair-rods, yellow are 1/3 ⟨100⟩ Hirth, light blue are 1/3 ⟨111⟩ Frank, and dark blue are 1/2 ⟨110⟩ perfect dislocations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' well beyond the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='5 dpa at which the potential energy converged in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' The invariance of the potential energy even as the dislocation density is reduced by nearly half requires a corresponding increase in the population of other defect types, a point that will be significant in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' Stacking faults As expected, stacking faults (SFs) were only observed in the FCC CrCoNi and A-atom systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' The SF configurations at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='0 dpa are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' 10, and the SF density as a function of dpa is reported in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' The very high SF densities observed in these systems are consistent with the low reported values for their SF energies [46] and with the high density of partial dislocations in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' It is interesting that the SF density for the A-atom system was higher than for the CrCoNi system up to around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='8 dpa, but by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='0 dpa the situation had reversed with the SF density in the A-atom system falling by almost a factor of three.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' This roughly correlates with the decrease in the density of Shockley partials in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' 8, though the correspondence is not exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' If the decrease in the SF density is in fact driven by the maturation of the dislocation network, then an important question is why a similar decrease in the density of Shockley partials and SFs was not observed in the CrCoNi system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' Any explanation should likely involve the chemical SRO since this is (by design) the main difference between the CrCoNi and A-atom systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' Possible mechanisms include atomic rearrangements following the formation of the SF lowering the energy of the 20 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' Stacking faults as identified by locally HCP coordinated atoms in CrCoNi (left) and A-atom (right) systems at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='0 dpa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' m FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' The stacking fault density as a function of dpa for the CrCoNi and A-atom systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' SF relative to the pristine state and increasing the unfaulting barrier, or the fluctuations in the local stacking fault energy decreasing the difficulty of cross slip and increasing the complexity of the dislocation network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' Either of these mechanisms would help to explain why the dislocation density for the CrCoNi system does not fall as quickly with increasing dpa as for the A-atom system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' The elevated SF density observed here could have significant implications for the mechan- ical strength of irradiated CrCoNi since SFs can act as barriers to dislocation motion that increase plastic strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' Recent work by Richie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' used in situ transmission electron microscopy to investigate CrCoNi under cryogenic conditions and noted that a high density of SFs and extensive cross slip likely contributed to the superior mechanical properties of the alloy [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' SFs have also been suggested to decrease radiation-induced void swelling of CrCoNi by alleviating internal stress build up [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' 21 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' The Warren-Cowley SRO parameters as a function of dpa for the CrCoNi system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' Short range ordering The CrCoNi system, initialized as a random solid solution, did develop a minor degree of SRO during the CRA simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' While not visually apparent this ordering was statistically significant as indicated by the WC SRO parameters reported in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' All of the WC parameters stabilized by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='5 dpa, with Cr displaying an increased likelihood to neighbor Co and both Cr and Co less likely to neighbor other atoms of the same species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' It is interesting that this degree of order developed despite the randomizing effects of atomic displacements in the CRA, and is significant that the order generally agrees with what has been found experimentally [61] and with more accurate DFT-based Monte Carlo simulations [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' Finally, despite the magnitude of the SRO being relatively small, the contribution toward lowering the potential energy of the system is still substantial;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' as will be discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' III E below, the magnitude of the Esro can be as much as 30% of the magnitude of Efit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' Point defects and energy balance The results discussed up to this point have included dislocations and stacking faults, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=', defects for which there are standard identification techniques and about which we have general confidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' The same is not true for point defects, for the identification of which there are arguably no methods in the literature (including the ones used here) that work reliably at high defect concentrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' This section not only reports the nominal point 22 defect concentrations as calculated using the methods described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' II, but highlights the inconsistency of these estimates with respect to the stored energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' This is particularly concerning since the point defect balance equations that describe the evolution of point defect concentrations are the foundations of our understanding of damage development in irradiated materials [2, 9], and it is unclear whether it is currently possible to evaluate the independent variables in these equations either by experiment or simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' Our sincere hope is that this will be identified as an area requiring the further attention of the research community in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' One expectation is that the concentrations of vacancies and intertsitials should be equal at low dpa and increase linearly with the number of displacements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' This is because both are initially produced in equal amounts by Frenkel pair insertions and the defect density is not high enough for there to be appreciable recombination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' While this is satisfied for the point defect concentrations reported in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' 13, the vacancy and interstitial concentrations quickly diverge with increasing dpa just as in the CRA simulations of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' This is conventionally believed to indicate that the more mobile interstitials are clustering to form dislocation loops or are annihilating on previously-existing loops once a threshold density is reached, with the less mobile vacancies remaining in the bulk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' Although our simulations only considered single crystal systems, interstitials can also migrate to and annihilate on other types of sinks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=', grain boundaries, phase boundaries, voids) in more general materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' The nominal vacancy concentrations of the Fe and CrCoNi systems were comparable at all dpa, and were consistently ∼50% greater than that of the A-atom system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' It is suspicious that the vacancy concentrations of the two FCC systems as found by the WS method continue to increase all the way to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='0 dpa though, considering that the pressures of these systems have already converged by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='5 dpa in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' along with the results of the energy fitting reported below, this suggests that the WS method increasingly overestimates the vacancy concentrations of the FCC systems at higher dpa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' Confusingly, the same does not seem to be true for the Fe system for which the vacancy concentration, pressure, and energy model results are all consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' The reason for this discrepancy is unknown, but highlights the need for more robust ways to identify vacancies (or a more precise definition of what constitutes a vacancy) in highly damaged structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' The results of fitting the energy model Efit of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' II D to Estored for all three structures are reported in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' 14, with the values of the fitting parameters β and γ given in Table 23 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' Estimated vacancy and interstitial concentrations as measured by the WS and K-means methods, respectively, as a function of dpa for all three material systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' The model works remarkably well for the Fe system, with Efit reproducing all of the general trends of Estored and many of the smaller features as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' The fitted values of β and γ for the Fe system are also physically reasonable, being slightly below one as is necessary for the clustering of point defects to be energetically favorable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' This is not the case for the CrCoNi and A-atom systems though;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' both of these systems exhibit a Efit that continually increases with dpa rather than converging around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='5 dpa, and γ values that are well above the physically reasonable bound of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' The second observation in particular suggests that the interstitial concentration is severely underestimated, with the model compensating for an unreasonably low value of Nint in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' 5 by elevating the value of γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' Supposing from the Fe system that γ should be ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='9, the interstitial concentration in the FCC systems appears to be underestimated by a factor of two to three at high dpa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' While it is true that the steady-state interstitial concentrations should not necessarily be comparable in BCC and FCC systems, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' 13 is also consistent with the interstitial concentrations in FCC systems being systematically underestimated by the same factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' The source of the error is likely that the difference in atomic volumes of interstitials and atoms of the crystalline lattice is less pronounced in FCC than BCC systems, with interstitials in BCC Fe often adopting split-dumbbell configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' If the vacancy concentrations are relatively accurate and the errors in the interstitial concentrations are of a similar magnitude for the CrCoNi and A-atom systems, then the gap between the vacancy and interstitial concentrations would be significantly larger for the 24 sro FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' Fit of the stored energy model described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' II D for the Fe (left), CrCoNi (center), and A-atom (right) systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' The dotted trend lines are bounds derived from the estimated bounds on the dislocation network energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' TABLE III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' Parameters fit by the energy model that account for deviations of energy from isolated defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' β is the coefficient for vacancies and γ is for the interstitials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' β γ Fe 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='879 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='840 CrCoNi 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='454 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='90 A-atom 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='970 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='49 CrCoNi than for the A-atom system throughout the simulations, even at very low dpa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' If this is true, then that would have important implications for the effects of SRO and lattice distortions on the development of radiation damage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' Based on a conventional understanding of the point defect balance equations [2, 9], prior observations of CrCoNi’s low-temperature radiation resistance [23] imply that the gap between the vacancy and interstitial concentra- tions should be smaller for CrCoNi, encouraging point defect recombination and suppressing vacancy precipitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' While this is superficially inconsistent with our results, observe that a higher sustained vacancy concentration does not necessarily lead to a higher susceptibility to voids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' Specifically, there is evidence that SRO and lattice distortions can increase the magnitude of energetic well depths for vacancies [63], effectively decreasing the vacancy- vacancy and vacancy-void capture radii and allowing higher vacancy concentrations to be sustained as compared to the A-atom system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' Given the magnitude of the apparent errors in the estimated vacancy and interstitial concentrations though, there is not strong evidence 25 for this conclusion at present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' There are other qualifications relating to the CRA’s limitations that should be made regarding the observed differences in point defect concentrations across the three material systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' One important factor absent in these simulations is the thermal spike associated with a collision cascade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' MPEAs have been found to have a shorter mean-free electron path and a lower thermal conductivity than traditional alloys, and it has been proposed that this could prolong the thermal spike and increase the extent of defect recombination [64, 65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' It is also worth considering the potential effects of nanotwinning, which has been experimentally observed to be an important cryogenic strengthening mechanism for CrCoNi [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' While the CRA did not induce any visible nanotwinning, this could be related to the relative sizes of the simulation cell and the critical nucleation event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' Nanotwins would not only affect the development of the dislocation network, but have shown the ability to capture or promote the transport of point defects to sinks in irradiated Cu [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' CONCLUSION CRA simulations were conducted to investigate differences in the development of irra- diated microstructures in BCC Fe, FCC CrCoNi, and FCC A-atom systems in the low temperature and high dose rate regime up to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='0 dpa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' The CrCoNi system developed the highest overall dislocation density and exhibited the ability to maintain that dislocation density even as the dislocation networks in the Fe and A-atom systems matured and sim- plified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' The higher stacking fault density in CrCoNi is likely related to the consistently higher density of partial dislocations relative to the other systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' The successive insertion of Frenkel pairs entailed by the CRA mildly increased the degree of chemical short range order in CrCoNi relative to the initial random solid solution in a way that is consistent with other modeling and experimental studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' A model was developed for the energy stored in the material defects, and strongly suggests that the interstitial concentrations for the FCC systems as estimated from the distribution of Voronoi cell volumes are lower than the actual effective concentrations by a factor of two to three.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' It is also possible that the Wigner-Sietz method of identifying vacancies slightly overestimates the vacancy concentration for the FCC systems at high defect concentrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' The same methods of point defect identifica- tion seem to be much more reliable for the BCC Fe system though, with the energy model 26 closely following the simulation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' Given the overwhelming importance of point defects to the development of radiation damage and the intense interest in CrCoNi and other FCC MPEAs for nuclear applications, our results reveal a critical need to either develop more robust ways to measure point defect concentrations in heavily-damaged FCC materials, or perhaps to reevaluate what is meant by a point defect in such materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' ACKNOWLEDGMENTS JCS gratefully acknowledges partial support by the Nuclear Regulatory Commission (Award 31310019M0009) through the Advancing Scientific Careers to Enhance Nuclear Tech- nologies (ASCENT) program at UC Davis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' This work was partially performed under the auspices of the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' AUTHOR CONTRIBUTIONS CS and JCS performed all the simulations and analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' JKM conceptualized the project and developed the methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' All authors collaborated to writing, reviewing, and editing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' COMPETING INTERESTS The authors declare no competing interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' Appendix A: Convergence with system size As described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' II A, one simulation of equiatomic CrCoNi containing a total of 1 048 576 atoms was conducted to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='0 dpa to investigate the effect of system size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' The behavior of this system is compared in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' 15 (top row) with that of the smaller simulation containing a total of 131 072 atoms (bottom row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' There is remarkable agreement between the two simulations, the main difference being that the plots for the larger simulation are smoother due to the larger number of atoms helping to average out the fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' Notice particularly that the pressure change and potential energy changes reach similar asymptotic values at similar dpa, that the dislocations in both are overwhelmingly Shockley partials, 27 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' Comparison of two simulations of equiatomic CrCoNi of different system sizes, with one million (one hundred thousand) atoms on the top (bottom) row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' Included in this comparison are the change in pressure (left), change in potential energy (middle), and the dislocation density (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' and that the overall dislocation density is relatively stable from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content='25 dpa to the end of the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' [1] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' J.' metadata={'source': 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and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' Zhang, Nature communications 6, 1 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} +page_content=' 32' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE3T4oBgHgl3EQfFQlM/content/2301.04303v1.pdf'} diff --git a/WdE3T4oBgHgl3EQfFQnF/content/tmp_files/2301.04304v1.pdf.txt b/WdE3T4oBgHgl3EQfFQnF/content/tmp_files/2301.04304v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..d9225a27998404646b16cc4c25e03d6cb94f45d5 --- /dev/null +++ b/WdE3T4oBgHgl3EQfFQnF/content/tmp_files/2301.04304v1.pdf.txt @@ -0,0 +1,2458 @@ +3D Bosons and W1+∞ algebra +Wang Na†∗, Wu Ke‡ +†School of Mathematics and Statistics, Henan University, Kaifeng, 475001, China +‡School of Mathematical Sciences, Capital Normal University, Beijing 100048, China +Abstract +In this paper, we consider 3D Young diagrams with at most N layers in z- +axis direction, which can be constructed by N 2D Young diagrams on slice z = j, +j = 1, 2, · · · , N from the Yang-Baxter equation. Use 2D Bosons {aj,m, m ∈ Z} +associated to 2D Young diagrams on the slice z = j, we constructed 3D Bosons. +Then we show the 3D Boson representation of W1+∞ algebra, and the Littlewood- +Richardson rule for 3-Jack polynomials from the actions of 3D Bosons on 3D Young +diagrams. +Keywords: +Affine Yangian, 3D Young diagrams, 3D Bosons, 3-Jack polynomials, +Littlewood-Richardson rule. +1 +Introduction +The Schur functions defined on 2D Young diagrams are an attractive research object, +which were used to determine irreducible characters of highest weight representations +of the classical groups, and the Littlewood-Richarson rule for Schur functions show the +relations between the representation spaces[1, 2, 3]. There are many structures, such as +2D Bosons and Boson-Fermion correspondence, defined on Schur functions or 2D Young +diagrams. These structures have many applications in mathematical physics. In [4], +the group in the Kyoto school uses Schur functions in a remarkable way to understand +the KP and KdV hierarchies. In [5, 6], Tsilevich and Su�lkowski, respectively, give the +realization of the phase model in the algebra of Schur functions and build the relations +between the q-boson model and Hall-Littlewood functions. In [7], we build the relations +between the statistical models, such as phase model, and KP hierarchy by using 2D +Young diagrams and Schur functions. In [8], the authors show that the states in the β- +deformed Hurwitz-Kontsevich matrix model can be represented as the Jack polynomials. +3D Young diagrams (plane partitions) are a generalization of 2D Young diagrams, +which arose naturally in crystal melting model[9, 10]. 3D Young diagrams also have +many applications in many fields of mathematics and physics, such as statistical mod- +els, number theory, representations of some algebras (Ding-Iohara-Miki algebras, affine +Yangian, etc). In this paper, we consider 3D Bosons and the Littlewood-Richardson +rule for 3-Jack polynomials on 3D Young diagrams which parallel to 2D Bosons and +the Littlewood-Richardson rule for Schur functions or Jack polynomials on 2D Young +diagrams. +Let aj,n be the 2D Bosons associated to 2D Young diagrams which are on the slice +z = j of 3D Young diagrams with the relation +[aj,n, ai,m] = − +1 +h1h2 +δi,jnδn+m,0, +∗Corresponding author: wangna@henu.edu.cn +1 +arXiv:2301.04304v1 [math-ph] 11 Jan 2023 + +where h1, h2 are the parameters in the affine Yangian of gl(1). 3D Bosons bn,j can be +represented by these 2D Bosons. We treat 3D Young diagrams which have one layer in +z-axis direction as 2D Young diagrams. Since we require 3D Bosons become 2D Bosons +when N = 1, which means 3D Bosons bn,1 become 2D Bosons and bn,j≥2 become zero, we +know that ⟨0|bn,j≥2b−n,j≥2|0⟩ must contain the factor 1 + h1h2ψ0, where ψ0 = − +N +h1h2 is +the generator in affine Yangian of gl(1). Since all the results we constructed on 3D Young +diagrams are symmetric about three coordinate axes, which means they are symmetric +about the parameters h1, h2, h3, then ⟨0|bn,j≥2b−n,j≥2|0⟩ must contain the factor +(1 + h1h2ψ0)(1 + h1h3ψ0)(1 + h2h3ψ0). +The Littlewood-Richardson rule for Schur functions are well known, for example, +S(2)S(1,1) = S(3,1) + S(2,1,1). +They show the relations between the irreducible representation spaces of the general lin- +ear groups or permutation groups. In this paper, we calculate the Littlewood-Richardson +rule for 3-Jack polynomials, we will find that it is more complicated than that for Schur +functions, but it will become that for Schur functions in the special case h1 = 1, h2 = +−1, N = 1. We believe that the Littlewood-Richardson rule should have applications in +representation theory which we will consider later. +The paper is organized as follows. In section 2, we recall the definition of affine +Yangian of gl(1) and its representation on 3D Young diagrams. In section 3, we recall +the definition of the W1+∞ algebra, then we construct the fields in W1+∞ algebra from +the Miura transformation. The Virasoro field V2(z) become that in [11] when h1 = +h, h2 = −h−1. The spin 3 field V3(z) is given new. In section 4, we construct 3D Boson +fields and give the 3D Boson representation of W1+∞ algebra. In section 5, we give the +Littlewood-Richardson rule for 3-Jack polynomials. In section 6, we show the actions +of 3D Bosons on 3D Young diagrams and the relations between 3D Bosons and the +generators of affine Yangian of gl(1). +2 +Affine Yangian of gl(1) +In this section, we recall the definition of affine Yangian of gl(1) and its representation +on 3D Young diagrams. The affine Yangian Y of gl(1) is an associative algebra with +generators ej, fj and ψj, j = 0, 1, · · · and the following relations[12, 13] +[ψj, ψk] = 0, +(1) +[ej+3, ek] − 3 [ej+2, ek+1] + 3 [ej+1, ek+2] − [ej, ek+3] ++σ2 [ej+1, ek] − σ2 [ej, ek+1] − σ3 {ej, ek} = 0, +(2) +[fj+3, fk] − 3 [fj+2, fk+1] + 3 [fj+1, fk+2] − [fj, fk+3] ++σ2 [fj+1, fk] − σ2 [fj, fk+1] + σ3 {fj, fk} = 0, +(3) +[ej, fk] = ψj+k, +(4) +[ψj+3, ek] − 3 [ψj+2, ek+1] + 3 [ψj+1, ek+2] − [ψj, ek+3] ++σ2 [ψj+1, ek] − σ2 [ψj, ek+1] − σ3 {ψj, ek} = 0, +(5) +[ψj+3, fk] − 3 [ψj+2, fk+1] + 3 [ψj+1, fk+2] − [ψj, fk+3] ++σ2 [ψj+1, fk] − σ2 [ψj, fk+1] + σ3 {ψj, fk} = 0, +(6) +together with boundary conditions +[ψ0, ej] = 0, [ψ1, ej] = 0, [ψ2, ej] = 2ej, +(7) +[ψ0, fj] = 0, [ψ1, fj] = 0, [ψ2, fj] = −2fj, +(8) +2 + +and a generalization of Serre relations +Sym(j1,j2,j3) [ej1, [ej2, ej3+1]] = 0, +(9) +Sym(j1,j2,j3) [fj1, [fj2, fj3+1]] = 0, +(10) +where Sym is the complete symmetrization over all indicated indices which include 6 +terms. +The notations σ2, σ3 in the definition of affine Yangian are functions of three complex +numbers h1, h2 and h3: +σ1 += +h1 + h2 + h3 = 0, +(11) +σ2 += +h1h2 + h1h3 + h2h3, +(12) +σ3 += +h1h2h3. +(13) +The affine yangian Y has a representation on the plane partitions. A plane partition +π is a 2D Young diagram in the first quadrant of plane xOy filled with non-negative +integers that form nonincreasing rows and columns [14, 9]. The number in the position +(i, j) is denoted by πi,j +� +� +π1,1 +π1,2 +· · · +π2,1 +π2,2 +· · · +· · · +· · · +· · · +� +� . +The integers πi,j satisfy +πi,j ≥ πi+1,j, +πi,j ≥ πi,j+1, +lim +i→∞ πi,j = lim +j→∞ πi,j = 0 +for all integers i, j ≥ 0. Piling πi,j cubes over position (i, j) gives a 3D Young diagram. +3D Young diagrams arose naturally in the melting crystal model[9, 10]. +We always +identify 3D Young diagrams with plane partitions as explained above. For example, the +3D Young diagram +can also be denoted by the plane partition (1, 1). +As in our paper [15], we use the following notations. For a 3D Young diagram π, the +notation 2 ∈ π+ means that this box is not in π and can be added to π. Here “can be +added” means that when this box is added, it is still a 3D Young diagram. The notation +2 ∈ π− means that this box is in π and can be removed from π. Here “can be removed” +means that when this box is removed, it is still a 3D Young diagram. For a box 2, we +let +h2 = h1y2 + h2x2 + h3z2, +(14) +where (x2, y2, z2) is the coordinate of box 2 in coordinate system O − xyz. Here we +use the order y2, x2, z2 to match that in paper [12]. +Following [12, 13], we introduce the generating functions: +e(u) += +∞ +� +j=0 +ej +uj+1 , +f(u) += +∞ +� +j=0 +fj +uj+1 , +(15) +ψ(u) += +1 + σ3 +∞ +� +j=0 +ψj +uj+1 , +where u is a parameter. Introduce +ψ0(u) = u + σ3ψ0 +u +(16) +3 + +and +ϕ(u) = (u + h1)(u + h2)(u + h3) +(u − h1)(u − h2)(u − h3). +(17) +For a 3D Young diagram π, define ψπ(u) by +ψπ(u) = ψ0(u) +� +2∈π +ϕ(u − h2). +(18) +In the following, we recall the representation of the affine Yangian on 3D Young diagrams +as in paper [12] by making a slight change. The representation of affine Yangian on 3D +Young diagrams is given by +ψ(u)|π⟩ += +ψπ(u)|π⟩, +(19) +e(u)|π⟩ += +� +2∈π+ +E(π → π + 2) +u − h2 +|π + 2⟩, +(20) +f(u)|π⟩ += +� +2∈π− +F(π → π − 2) +u − h2 +|π − 2⟩ +(21) +where |π⟩ means the state characterized by the 3D Young diagram π and the coefficients +E(π → π + □) = −F(π + □ → π) = +� +1 +σ3 +resu→h□ ψπ(u) +(22) +Equations (20) and (21) mean generators ej, fj acting on the 3D Young diagram π by +ej|π⟩ = +� +□∈π+ +hj +□E(π → π + □)|π + □⟩, +(23) +fj|π⟩ = +� +hj +□F(π → π − □)|π − □⟩. +(24) +In the following of this paper, we consider 3D Young diagrams which have at most +N layers in the z-axis direction, and slice the 3D Young diagrams into a series of 2D +Young diagrams by the plane z = n for n = 1, 2, · · · , N. Then the symmetry of the +affine Yangian of gl(1) about the coordinate axes are broken. For example, ψ0 = − +N +h1h2 . +3 +W1+∞ algebra +We begin this section by the definition of W1+∞ algebra. The W1+∞ algebra contains +the Heisenberg algebra, the Virasoro algebra as subalgebras[12]. The generators are Vj,m +for j ∈ Z+, m ∈ Z. The relations are +[V1,m, V1,n] += +mδm+n,0c1, +(25) +[V2,m, V2,n] += +(m − n)V2,m+n + m3 − m +12 +δm+n,0c2, +(26) +[V2,m, V1,n] += +−nV1,m+n, +(27) +and generally +[Vj,m, Vk,n] = +� +0≤l≤j+k−2 +j+k−leven +Cl +jkNl +jk(m, n)Vl,m+n, +(28) +4 + +where the coefficients Nl +jk(m, n) are +N0 +jk(m, n) += +� m + j − 1 +j + k − 1 +� +δm+n,0, +(29) +Nl +jk(m, n) += +j+k−l−1 +� +s=0 +(−1)s +(j + k − l − 1)!(2l)j+k−l−1 +� j + k − l − 1 +s +� +× +[j + m − 1]j+k−l−1−s[j − m − 1]s[k + n − 1]s[k − n − 1]j+k−l−1−s, +and the structure constants Cl +jk are +C0 +jk += +(j − 1)!2(2j − 1)! +4j−1(2j − 1)!!(2j − 3)!!δjkcj, +(30) +Cl +jk += +1 +2 × 4j+k−l−2 (2l)j+k−l−1 × 4F3 +� 1 +2, 1 +2, − 1 +2(j + k − l − 2), − 1 +2(j + k − l − 1) +3 +2 − j, 3 +2 − k, 1 +2 + l +; 1 +� +, +with +(a)n += +a(a + 1) · · · (a + n − 1), +(31) +[a]n += +a(a − 1) · · · (a − n + 1), +(32) +mFn +� a1, · · · , am +b1, · · · , bn ; z +� += +∞ +� +k=0 +(a1)k · · · (am)k +(b1)k · · · (bn)k +zk +k! . +(33) +Note that here we allow that the central charges can be different. +We consider Jj(z) +Jj(z) = +� +n∈Z +aj,nz−n−1 +(34) +with the relation +[aj,n, ak,m] = − +1 +h1h2 +δj,knδn+m,0. +(35) +Define +J(z) = +� +n∈Z +anz−n−1 = J1(z) + J2(z) + · · · + J3(z), +(36) +then the bosons an satisfy +[an, am] = − N +h1h2 +nδn+m,0 = ψ0nδn+m,0. +(37) +In the following two subsections, we consider the Boson aj,n representation of W1+∞ +algebra. +3.1 +Miura transformation and the W1+∞ algebra Vn(z) +Let +α0 = − h3 +h1h2 +, +and define the operator Uk(z) as in [11] by +: (α0∂ + J1(z))(α0∂ + J2(z)) · · · (α0∂ + JN(z)) := +N +� +k=0 +Uk(z)(α0∂)N−k. +(38) +5 + +The fields Uk(z) generate an algebra, which is W1+N. The fields Vn(z) can be realized +by Uk(z). We list the concrete expressions of the first few Uk(z) as in [16] +U0 += +1, +(39) +U1 += +N +� +j=1 +Jj, +(40) +U2 += +� +j 0. +(78) +11 + +Define +|0⟩ = +N +� +j=1 +|0⟩j, +(79) +3D Bosons are the operators acting on |0⟩. For n > 0, define +Pn,k = b−n,k|0⟩. +(80) +We calculate Pn,k in order to obtain its property. When k = 1, +Pn,1 = +N +� +j=1 +aj,−n|0⟩ = +N +� +j=1 +pj,n. +(81) +When k = 2, +b−1,2 = −2h1h2(1 − 1 +N ) +N +� +j=1 +� +l>0 +aj,−l−1aj,l + 2h1h2 +N +� +j0 +(aj,−l−1ak,l + ak,−l−1aj,l),(82) +then P1,2 = 0. +b−2,2 += +−h1h2(1 − 1 +N ) +N +� +j=1 +(a2 +j,−1 + 2 +� +l>0 +aj,−l−2aj,l) ++2h1h2 +N +� +j0 +(aj,−l−2ak,l + ak,−l−2aj,l) +� +−h3 +N +� +j=1 +(N + 1 − 2j)aj,−2, +(83) +then +P2,2 = −h1h2(1 − 1 +N ) +N +� +j=1 +p2 +j,1 + 2h1h2 +N +� +j0,k+l=n +pj,kpj,l + 2h1h2 +N +� +j0,l+q=n +pj,lpk,q +−h3 +N +� +j=1 +(N + 1 − 2j)(n − 1)pj,n. +(85) +12 + +When k = 3, +P1,3 += +0, +(86) +P2,3 += +0, +(87) +P3,3 += +6h2 +1h2 +2 +� +�− +� +j 0. +14 + +Since we have known the representation of affine Yangian of gl(1) on 3D Young +diagrams, in the following, we use the generators of affine Yangian of gl(1) to represent +3D Bosons. We know that [18] +ψ2 += +−2h1h2 +N +� +j=1 +� +k>0 +aj,−kaj,k, +(94) +ψ3 += +3h2 +1h2 +2 +N +� +i=1 +� +j,k>0 +(ai,−j−kai,jai,k + ai,−jai,−kai,j+k) ++6σ3 +� +i10 +kai1,−kai2,k + (−4N + 6j − 3)σ3 +N +� +j=1 +� +k>0 +aj,−kaj,k ++3σ3 +N +� +j=1 +� +k>0 +kaj,−kaj,k, +(95) +and +e0 += +N +� +j=1 +aj,−1, +(96) +e1 += +−h1h2 +N +� +j=1 +� +k>0 +aj,−k−1aj,k. +(97) +Compare with the expression of bn,1, we obtain that +b−n,1 += +1 +(n − 1)!adn−1 +e1 +e0, +(98) +bn,1 += +− +1 +(n − 1)!adn−1 +f1 +f0 +(99) +for n > 0, since +[aj,−k−1aj,k, ai,−l] = − +1 +h1h2 +δi,jδk,llaj,−l−1. +The expressions in (98) and (99) equals that in [18]. +Let ψ0 = 1 without loss the generality. For Boson field B2(z), we have +b−1,2 += +e1 − 2 +� +n≥1 +b−n−1,1bn,1, +b1,2 += +−f1 − 2 +� +n≥1 +b−n,1bn+1,1, +which also explains b−1,2|0⟩ = 0 since e1|0⟩ = 0. For n ≥ 1, +b−(n+1),2 += +1 +(n − 1)!adn−1 +e1 +([e2, e0] − σ3[e1, e0]) − +� +i+j=−(n+1) +: bi,1bj,1 : +(100) +bn+1,2 += +− +1 +(n − 1)!adn−1 +f1 +([f2, f0] − σ3[f1, f0]) − +� +i+j=(n+1) +: bi,1bj,1 :, +(101) +which equal that in [20]. +15 + +As operators, +˜J += +1 +(h1 − h2)(h1 − h3) +� +(1 + h2h3)b2 +−1,1 + (1 + h2h3)h1b−2,1 + b−2,2 +� += +1 +(h1 − h2)(h1 − h3) +� +�h2h3e2 +0 + h1[e1, e0] + [e2, e0] − 2 +� +n≥1 +b−n−2,1bn,1 +� +� , +and +˜J += +1 +(h2 − h1)(h2 − h3) +� +�h1h3e2 +0 + h2[e1, e0] + [e2, e0] − 2 +� +n≥1 +b−n−2,1bn,1 +� +� . +From +ej · ˜Jπ += +� +2∈π+ +hj +2 ˜Jπ+2, +(102) +fj · ˜Jπ += +− +� +2∈π− +hj +2F 2(π → π − 2) ˜Jπ+2, +(103) +we have +b1,1 · ˜J += +2(1 + h1h3) +(h2 − h1)(h2 − h3) +˜J +, +b2,1 · ˜J += +2(1 + h1h3)h2 +(h2 − h1)(h2 − h3), +and +b−3,1 · ˜J += +1 +2 (e1e1e0 − 2e1e0e1 + e0e1e1) · ˜J += +h2 +1 ˜J ++ h2 +2 ˜J ++ h2 +3 ˜J ++h2 +1 ˜J +h1,2h1,h2 ++ 3h1h2 − 4h2 +1 +2 +˜J +h1,h2,2h1 ++ 2h2 +1 − 3h1h2 +2 +˜J +h2,h1,2h1 ++h2 +1 ˜J +h1,2h1,h3 ++ 3h1h3 − 4h2 +1 +2 +˜J +h1,h3,2h1 ++ 2h2 +1 − 3h1h3 +2 +˜J +h3,h1,2h1 ++h2 +2 ˜J +h2,2h2,h1 ++ 3h1h2 − 4h2 +2 +2 +˜J +h2,h1,2h2 ++ 2h2 +2 − 3h1h2 +2 +˜J +h1,h2,2h2 ++h2 +2 ˜J +h2,2h2,h3 ++ 3h2h3 − 4h2 +2 +2 +˜J +h2,h3,2h2 ++ 2h2 +2 − 3h2h3 +2 +˜J +h3,h2,2h2 ++h2 +3 ˜J +h3,2h3,h2 ++ 3h2h3 − 4h2 +3 +2 +˜J +h3,h2,2h3 ++ 2h2 +3 − 3h2h3 +2 +˜J +h2,h3,2h3 ++h2 +3 ˜J +h3,2h3,h1 ++ 3h1h3 − 4h2 +3 +2 +˜J +h3,h1,2h3 ++ 2h2 +3 − 3h1h3 +2 +˜J +h1,h3,2h3 ++h2 +3 − 2h2 +1 +2 +˜J +h1,h3,h1+h3 ++ h2 +1 − 2h2 +3 +2 +˜J +h3,h1,h1+h3 +16 + ++h2 +2 − 2h2 +1 +2 +˜J +h1,h2,h1+h2 ++ h2 +1 − 2h2 +2 +2 +˜J +h2,h1,h1+h2 ++h2 +3 − 2h2 +2 +2 +˜J +h2,h3,h2+h3 ++ h2 +2 − 2h2 +3 +2 +˜J +h3,h2,h2+h3 ++−h2 +2 − 2h1h3 +2 +� +˜J +h3,h2,h1 ++ ˜J +h1,h2,h3 +� ++−h2 +1 − 2h2h3 +2 +� +˜J +h2,h1,h3 ++ ˜J +h3,h1,h2 +� ++−h2 +3 − 2h1h2 +2 +� +˜J +h1,h3,h2 ++ ˜J +h2,h3,h1 +� +, +b−4,1 · ˜J0 += +1 +6e1e1e1 · ˜J += +h3 +1 ˜J ++ h3 +2 ˜J ++ h3 +3 ˜J ++h2 +1h2 +3 +� +˜J +h1,2h1,h2 ++ ˜J +h1,h2,2h1 ++ ˜J +h2,h1,2h1 +� ++h2 +1h3 +3 +� +˜J +h1,2h1,h3 ++ ˜J +h1,h3,2h1 ++ ˜J +h3,h1,2h1 +� ++h2 +2h1 +3 +� +˜J +h2,2h2,h1 ++ ˜J +h2,h1,2h2 ++ ˜J +h1,h2,2h2 +� ++h2 +2h3 +3 +� +� ˜J +h2,2h2,h3 ++ ˜J +h2,h3,2h2 ++ ˜J +h3,h2,2h2 +� +� ++h2 +3h2 +3 +� +� ˜J +h3,2h3,h2 ++ ˜J +h3,h2,2h3 ++ ˜J +h2,h3,2h3 +� +� ++h2 +3h1 +3 +� +˜J +h3,2h3,h1 ++ ˜J +h3,h1,2h3 ++ ˜J +h1,h3,2h3 +� ++h1h3(h1 + h3) +6 +� +˜J +h1,h3,h1+h3 ++ ˜J +h3,h1,h1+h3 +� ++h1h2(h1 + h2) +6 +� +˜J +h1,h2,h1+h2 ++ ˜J +h2,h1,h1+h2 +� ++h2h3(h2 + h3) +6 +� +˜J +h2,h3,h2+h3 ++ ˜J +h3,h2,h2+h3 +� ++h1h2h3 +6 +� +˜J +h3,h2,h1 ++ ˜J +h1,h2,h3 ++ ˜J +h2,h1,h3 ++ ˜J +h3,h1,h2 ++ ˜J +h1,h3,h2 ++ ˜J +h2,h3,h1 +� +, +17 + +we calculate +˜J +({Pn,j}) × ˜J +({Pn,j}) = ˜J +({b−n,j}) · ˜J +({Pn,j}) += +1 +(h1 − h2)(h1 − h3) +� +h2h3e2 +0 + h1[e1, e0] + [e2, e0] − 2(b−3,1b1,1 + b−4,1b2,1) +� +· ˜J += +−4h2 +1(1 + h1h3)(1 + h1h2) +(h1 − h2)(h1 − h3)(h2 − h1)(h2 − h3) +˜J ++ +−4h2 +3(1 + h1h3)(1 + h2h3) +(h1 − h2)(h1 − h3)(h2 − h1)(h2 − h3) +˜J ++ +−4h2 +2(1 + h2h3)(1 + h1h2) +(h1 − h2)(h1 − h3)(h2 − h1)(h2 − h3) +˜J ++ +−4h2 +1(1 + h1h3)(3 + h2 +2) +3(h1 − h2)(h1 − h3)(h2 − h1)(h2 − h3) +˜J +h1,2h1,h2 ++−2h2 +1(1 + h1h3)(9h1h2 − 12h2 +1 + 2h2 +1h2 +2) +3(h1 − h2)(h1 − h3)(h2 − h1)(h2 − h3) +˜J +h1,h2,2h1 ++4h4 +1h2 +2 + 4h3 +1h3 +2 − 15h3 +1h2 + 8h2 +1h2 +2 − 3h1h3 +2 − 6h4 +2 − 12h2 +1 + 18h1h2 +3(h1 − h2)(h1 − h3)(h2 − h1)(h2 − h3) +˜J +h2,h1,2h1 ++ +−4h2 +1(1 + h1h3)(3 + h2h3) +3(h1 − h2)(h1 − h3)(h2 − h1)(h2 − h3) +˜J +h1,2h1,h3 ++−2(1 + h1h3)(9h1h3 − 12h2 +1 + 2h2 +1h2h3) +3(h1 − h2)(h1 − h3)(h2 − h1)(h2 − h3) +˜J +h1,h3,2h1 ++−2(1 + h1h3)(6h2 +1 − 9h1h3 + 2h2 +1h2h3) +3(h1 − h2)(h1 − h3)(h2 − h1)(h2 − h3) +˜J +h3,h1,2h1 ++4h3 +1h3 +2 + 4h2 +1h4 +2 − 6h4 +1 + 3h3 +1h2 + 48h2 +1h2 +2 + 5h1h3 +2 − 30h4 +2 − 12h2 +2 +3(h1 − h2)(h1 − h3)(h2 − h1)(h2 − h3) +˜J +h2,2h2,h1 ++4h3 +1h3 +2 + 4h2 +1h4 +2 + 6h4 +1 + 21h3 +1h2 − 30h2 +1h2 +2 − 31h1h3 +2 + 18h4 +2 − 18h1h2 + 24h2 +2 +3(h1 − h2)(h1 − h3)(h2 − h1)(h2 − h3) +˜J +h2,2h2,h1 ++ −(2(h1h3 + 1))h2(2h1h2 +2 − 9h1 + 6h2) +3(h1 − h2)(h1 − h3)(h2 − h1)(h2 − h3) +˜J +h1,h2,2h2 ++−4h3 +1h3 +2 − 8h2 +1h4 +2 − 4h1h5 +2 + 6h3 +1h2 + 30h2 +1h2 +2 + 16h1h3 +2 − 20h4 +2 − 12h2 +2 +3(h1 − h2)(h1 − h3)(h2 − h1)(h2 − h3) +˜J +h2,2h2,h3 ++−4h3 +1h3 +2 − 8h2 +1h4 +2 − 4h1h5 +2 − 18h3 +1h2 − 66h2 +1h2 +2 − 44h1h3 +2 + 16h4 +2 + 18h1h2 + 42h2 +2 +3(h1 − h2)(h1 − h3)(h2 − h1)(h2 − h3) +˜J +h2,h3,2h2 ++ −(2(h1h3 + 1))h2(2h2 +2h3 + 6h2 − 9h3) +3(h1 − h2)(h1 − h3)(h2 − h1)(h2 − h3) +˜J +h2,h3,2h2 +18 + ++ +−(4(h1h3 + 1))h2 +3(h2 +2 + 3) +3(h1 − h2)(h1 − h3)(h2 − h1)(h2 − h3) +˜J +h3,2h3,h2 ++−(2(h1h3 + 1))h3(2h2 +2h3 + 9h2 − 12h3) +3(h1 − h2)(h1 − h3)(h2 − h1)(h2 − h3) +˜J +h3,h2,2h3 ++−2h3(2h1h2 +2h2 +3 − 6h1h2h3 + 3h1h2 +3 − h2 +2h3 + 3h2h2 +3 − 9h2 + 6h3) +3(h1 − h2)(h1 − h3)(h2 − h1)(h2 − h3) +˜J +h2,h3,2h3 ++ +−(4(h1h3 + 1))h2 +3(h1h2 + 3) +3(h1 − h2)(h1 − h3)(h2 − h1)(h2 − h3) +˜J +h3,2h3,h1 ++−(2(h1h3 + 1))h3(2h1h2h3 + 9h1 − 12h3) +3(h1 − h2)(h1 − h3)(h2 − h1)(h2 − h3) +˜J +h3,h1,2h3 ++−2(h1h3 + 1)h3(2h1h2h3 − 9h1 + 6h3) +3(h1 − h2)(h1 − h3)(h2 − h1)(h2 − h3) +˜J +h1,h3,2h3 ++−(2(h1h3 + 1))(h2 +1h3 + h1h2 +3 − 6h2 +1 + 3h2 +3) +3(h1 − h2)(h1 − h3)(h2 − h1)(h2 − h3) +˜J +h1,h3,h1+h3 ++−(2(h1h3 + 1))(h2 +1h3 + h1h2 +3 + 3h2 +1 − 6h2 +3) +3(h1 − h2)(h1 − h3)(h2 − h1)(h2 − h3) +˜J +h3,h1,h1+h3 ++−(2(h1h3 + 1))(h2 +1h2 + h1h2 +2 − 6h2 +1 + 3h2 +2) +3(h1 − h2)(h1 − h3)(h2 − h1)(h2 − h3) +˜J +h1,h2,h1+h2 ++(−2h3(h3 +1h2 + h2 +1h2 +2 + 3h3 +1 − 6h1h2 +2 − h1h2 − 3h1 + 6h2) +3(h1 − h2)(h1 − h3)(h2 − h1)(h2 − h3) +˜J +h2,h1,h1+h2 ++2h1(h3 +1h2 + 2h2 +1h2 +2 + h1h3 +2 + 3h3 +1 + 9h2 +1h2 + 3h1h2 +2 − 3h3 +2 − h1h2 − h2 +2 − 3h1 − 9h2) +3(h1 − h2)(h1 − h3)(h2 − h1)(h2 − h3) +˜J +h2,h3,h2+h3 ++−(2(h1h3 + 1))(h2 +2h3 + h2h2 +3 + 3h2 +2 − 6h2 +3) +3(h1 − h2)(h1 − h3)(h2 − h1)(h2 − h3) +˜J +h3,h2,h2+h3 ++−(2(h1h3 + 1))(h1h2h3 − 6h1h3 − 3h2 +2) +3(h1 − h2)(h1 − h3)(h2 − h1)(h2 − h3) +� +� ˜J +h1,h2,h3 ++ ˜J +h3,h2,h1 +� +� ++−(2(h1h3 + 1))(3h2 +1 − h3h2 + h3h2h1) + 3h3h2 + 3h1h3 − 6h2 +1 + 3h2 +3 +3(h1 − h2)(h1 − h3)(h2 − h1)(h2 − h3) +˜J +h2,h1,h3 ++−(2(h1h3 + 1))(h1h2h3 + 3h2 +1 − 6h2h3) +3(h1 − h2)(h1 − h3)(h2 − h1)(h2 − h3) +˜J +h3,h1,h2 ++−(2(h1h3 + 1))(h1h2h3 − 6h1h2 + 3h2 +3) +3(h1 − h2)(h1 − h3)(h2 − h1)(h2 − h3) +˜J +h1,h3,h2 ++−(2(h1h3 + 1))(−6h1h2 + 3h2 +3 + h3h2h1) + 3h3h2 − 3h1h3 + 6h2 +1 − 3h2 +3 +3(h1 − h2)(h1 − h3)(h2 − h1)(h2 − h3) +˜J +h1,h3,h2 +. +This is the Littlewood-Richardson rule for ˜J +({Pn,j}) × ˜J +({Pn,j}). We can check +that +˜J +({Pn,j}) × ˜J +({Pn,j}) = ˜J +({Pn,j}) × ˜J +({Pn,j}), +19 + +and +˜J +� +˜J ++ ˜J ++ ˜J +� += ˜J +˜J2 += ˜J2 +˜J +. +We can see that the Littlewood-Richardson rule for 3-Jack polynomials are complicated. +When h1 = 1, h2 = −1, the Littlewood-Richardson rule for 3-Jack polynomials becomes +that for Schur functions: when h1 = 1, h2 = −1, 3-Jack polynomials of 3D Young di- +agrams which have more than one layer in z-axis direction become zero, and 3-Jack +polynomials of 3D Young diagrams which have one layer in z-axis direction become the +Schur functions of the corresponding 2D Young diagrams, for example, 3-Jack polynomi- +als ˜J +h1,h2,h1+h2 +and ˜J +h2,h1,h1+h2 +all become Schur function S(2,2). Schur functions +of 2D Young diagrams are not dependent on the box growth processes of 2D Young +diagrams, while Jack polynomials of 2D Young diagrams and 3-Jack polynomials of 3D +Young diagrams are dependent on the box growth processes of 2D/3D Young diagrams. +Then we can see that the Littlewood-Richardson rule for ˜J +({Pn,j}) × ˜J +({Pn,j}) +becomes +S(2)S(1,1) = S(3,1) + S(2,1,1). +We also can check that when h1 = h, h2 = −h−1, the Littlewood-Richardson rule for +˜J +({Pn,j}) × ˜J +({Pn,j}) becomes that for 2D Jack polynomials ˜J(2) ˜J(1,1). +Data availability statement +The data that support the findings of this study are available from the corresponding +author upon reasonable request. +Declaration of interest statement +The authors declare that we have no known competing financial interests or personal +relationships that could have appeared to influence the work reported in this paper. +Acknowledgements +This research is supported by the National Natural Science Foundation of China under +Grant No. +12101184 and No. +11871350, and supported by Key Scientific Research +Project in Colleges and Universities of Henan Province No. 22B110003. +References +[1] W. Fulton and J. Harris. Representation theory, A first course, Springer-Verlag, +New York, 1991. +[2] I. G. Macdonald. Symmetric functions and Hall polynomials. Oxford Mathematical +Monographs, Clarendon Press, Oxford, 1979. +[3] H. Weyl, The classical groups; their invariants and representations. Princeton +Univ. Press, Princeton, 1946. +[4] T. Miwa, M. Jimbo, and E. Date. +Solitons: Differential equations, symmetries +and infinite dimensional algebras. Cambridge University Press, Cambridge, 2000. +20 + +[5] N. Tsilevich. Quantum inverse scattering method for the q-boson model and sym- +metric functions. Funct. Anal. Appl. 40, No. 3 (2006) 207-217. +[6] P. Su�lkowski. Deformed boson-fermion correspondence, Q-bosons, and topological +strings on the conifold. JHEP 10 (2008) 1127-1134. +[7] N. Wang. Young diagrams in an N × M box and the KP hierarchy. Nucl. phys. B +937 (2018) 478-501. +[8] R. Wang, F. Liu, C. H. Zhang, W. Z. Zhao. 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Wu. 3D Bosons, 3-Jack polynomials and affine Yangian of gl(1), +arXiv: 2212.05665. +21 + diff --git a/WdE3T4oBgHgl3EQfFQnF/content/tmp_files/load_file.txt b/WdE3T4oBgHgl3EQfFQnF/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..710660887410adb8b19de9c58a1491f5128ad0e8 --- /dev/null +++ b/WdE3T4oBgHgl3EQfFQnF/content/tmp_files/load_file.txt @@ -0,0 +1,702 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf,len=701 +page_content='3D Bosons and W1+∞ algebra Wang Na†∗, Wu Ke‡ †School of Mathematics and Statistics, Henan University, Kaifeng, 475001, China ‡School of Mathematical Sciences, Capital Normal University, Beijing 100048, China Abstract In this paper, we consider 3D Young diagrams with at most N layers in z- axis direction, which can be constructed by N 2D Young diagrams on slice z = j, j = 1, 2, · · · , N from the Yang-Baxter equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' Use 2D Bosons {aj,m, m ∈ Z} associated to 2D Young diagrams on the slice z = j, we constructed 3D Bosons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' Then we show the 3D Boson representation of W1+∞ algebra, and the Littlewood- Richardson rule for 3-Jack polynomials from the actions of 3D Bosons on 3D Young diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' Keywords: Affine Yangian, 3D Young diagrams, 3D Bosons, 3-Jack polynomials, Littlewood-Richardson rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' 1 Introduction The Schur functions defined on 2D Young diagrams are an attractive research object, which were used to determine irreducible characters of highest weight representations of the classical groups, and the Littlewood-Richarson rule for Schur functions show the relations between the representation spaces[1, 2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' There are many structures, such as 2D Bosons and Boson-Fermion correspondence, defined on Schur functions or 2D Young diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' These structures have many applications in mathematical physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' In [4], the group in the Kyoto school uses Schur functions in a remarkable way to understand the KP and KdV hierarchies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' In [5, 6], Tsilevich and Su�lkowski, respectively, give the realization of the phase model in the algebra of Schur functions and build the relations between the q-boson model and Hall-Littlewood functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' In [7], we build the relations between the statistical models, such as phase model, and KP hierarchy by using 2D Young diagrams and Schur functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' In [8], the authors show that the states in the β- deformed Hurwitz-Kontsevich matrix model can be represented as the Jack polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' 3D Young diagrams (plane partitions) are a generalization of 2D Young diagrams, which arose naturally in crystal melting model[9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' 3D Young diagrams also have many applications in many fields of mathematics and physics, such as statistical mod- els, number theory, representations of some algebras (Ding-Iohara-Miki algebras, affine Yangian, etc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' In this paper, we consider 3D Bosons and the Littlewood-Richardson rule for 3-Jack polynomials on 3D Young diagrams which parallel to 2D Bosons and the Littlewood-Richardson rule for Schur functions or Jack polynomials on 2D Young diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' Let aj,n be the 2D Bosons associated to 2D Young diagrams which are on the slice z = j of 3D Young diagrams with the relation [aj,n, ai,m] = − 1 h1h2 δi,jnδn+m,0, ∗Corresponding author: wangna@henu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content='cn 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content='04304v1 [math-ph] 11 Jan 2023 where h1, h2 are the parameters in the affine Yangian of gl(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' 3D Bosons bn,j can be represented by these 2D Bosons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' We treat 3D Young diagrams which have one layer in z-axis direction as 2D Young diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' Since we require 3D Bosons become 2D Bosons when N = 1, which means 3D Bosons bn,1 become 2D Bosons and bn,j≥2 become zero, we know that ⟨0|bn,j≥2b−n,j≥2|0⟩ must contain the factor 1 + h1h2ψ0, where ψ0 = − N h1h2 is the generator in affine Yangian of gl(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' Since all the results we constructed on 3D Young diagrams are symmetric about three coordinate axes, which means they are symmetric about the parameters h1, h2, h3, then ⟨0|bn,j≥2b−n,j≥2|0⟩ must contain the factor (1 + h1h2ψ0)(1 + h1h3ψ0)(1 + h2h3ψ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' The Littlewood-Richardson rule for Schur functions are well known, for example, S(2)S(1,1) = S(3,1) + S(2,1,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' They show the relations between the irreducible representation spaces of the general lin- ear groups or permutation groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' In this paper, we calculate the Littlewood-Richardson rule for 3-Jack polynomials, we will find that it is more complicated than that for Schur functions, but it will become that for Schur functions in the special case h1 = 1, h2 = −1, N = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' We believe that the Littlewood-Richardson rule should have applications in representation theory which we will consider later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' In section 2, we recall the definition of affine Yangian of gl(1) and its representation on 3D Young diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' In section 3, we recall the definition of the W1+∞ algebra, then we construct the fields in W1+∞ algebra from the Miura transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' The Virasoro field V2(z) become that in [11] when h1 = h, h2 = −h−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' The spin 3 field V3(z) is given new.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' In section 4, we construct 3D Boson fields and give the 3D Boson representation of W1+∞ algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' In section 5, we give the Littlewood-Richardson rule for 3-Jack polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' In section 6, we show the actions of 3D Bosons on 3D Young diagrams and the relations between 3D Bosons and the generators of affine Yangian of gl(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' 2 Affine Yangian of gl(1) In this section, we recall the definition of affine Yangian of gl(1) and its representation on 3D Young diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' The affine Yangian Y of gl(1) is an associative algebra with generators ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' fj and ψj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' j = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' · · · and the following relations[12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' 13] [ψj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' ψk] = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' (1) [ej+3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' ek] − 3 [ej+2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' ek+1] + 3 [ej+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' ek+2] − [ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' ek+3] +σ2 [ej+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' ek] − σ2 [ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' ek+1] − σ3 {ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' ek} = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' (2) [fj+3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' fk] − 3 [fj+2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' fk+1] + 3 [fj+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' fk+2] − [fj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' fk+3] +σ2 [fj+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' fk] − σ2 [fj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' fk+1] + σ3 {fj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' fk} = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' (3) [ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' fk] = ψj+k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' (4) [ψj+3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' ek] − 3 [ψj+2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' ek+1] + 3 [ψj+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' ek+2] − [ψj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' ek+3] +σ2 [ψj+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' ek] − σ2 [ψj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' ek+1] − σ3 {ψj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' ek} = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' (5) [ψj+3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' fk] − 3 [ψj+2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' fk+1] + 3 [ψj+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' fk+2] − [ψj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' fk+3] +σ2 [ψj+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' fk] − σ2 [ψj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' fk+1] + σ3 {ψj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' fk} = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' (6) together with boundary conditions [ψ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' ej] = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' [ψ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' ej] = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' [ψ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' ej] = 2ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' (7) [ψ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' fj] = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' [ψ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' fj] = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' [ψ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' fj] = −2fj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' (8) 2 and a generalization of Serre relations Sym(j1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content='j2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content='j3) [ej1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' [ej2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' ej3+1]] = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' (9) Sym(j1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content='j2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content='j3) [fj1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' [fj2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' fj3+1]] = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' (10) where Sym is the complete symmetrization over all indicated indices which include 6 terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' The notations σ2, σ3 in the definition of affine Yangian are functions of three complex numbers h1, h2 and h3: σ1 = h1 + h2 + h3 = 0, (11) σ2 = h1h2 + h1h3 + h2h3, (12) σ3 = h1h2h3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' (13) The affine yangian Y has a representation on the plane partitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' A plane partition π is a 2D Young diagram in the first quadrant of plane xOy filled with non-negative integers that form nonincreasing rows and columns [14, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' The number in the position (i, j) is denoted by πi,j � � π1,1 π1,2 · · π2,1 π2,2 · · · · · · · · � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' The integers πi,j satisfy πi,j ≥ πi+1,j, πi,j ≥ πi,j+1, lim i→∞ πi,j = lim j→∞ πi,j = 0 for all integers i, j ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' Piling πi,j cubes over position (i, j) gives a 3D Young diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' 3D Young diagrams arose naturally in the melting crystal model[9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' We always identify 3D Young diagrams with plane partitions as explained above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' For example, the 3D Young diagram can also be denoted by the plane partition (1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' As in our paper [15], we use the following notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' For a 3D Young diagram π, the notation 2 ∈ π+ means that this box is not in π and can be added to π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' Here “can be added” means that when this box is added, it is still a 3D Young diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' The notation 2 ∈ π− means that this box is in π and can be removed from π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' Here “can be removed” means that when this box is removed, it is still a 3D Young diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' For a box 2, we let h2 = h1y2 + h2x2 + h3z2, (14) where (x2, y2, z2) is the coordinate of box 2 in coordinate system O − xyz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' Here we use the order y2, x2, z2 to match that in paper [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' Following [12, 13], we introduce the generating functions: e(u) = ∞ � j=0 ej uj+1 , f(u) = ∞ � j=0 fj uj+1 , (15) ψ(u) = 1 + σ3 ∞ � j=0 ψj uj+1 , where u is a parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' Introduce ψ0(u) = u + σ3ψ0 u (16) 3 and ϕ(u) = (u + h1)(u + h2)(u + h3) (u − h1)(u − h2)(u − h3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' (17) For a 3D Young diagram π, define ψπ(u) by ψπ(u) = ψ0(u) � 2∈π ϕ(u − h2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' (18) In the following, we recall the representation of the affine Yangian on 3D Young diagrams as in paper [12] by making a slight change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' The representation of affine Yangian on 3D Young diagrams is given by ψ(u)|π⟩ = ψπ(u)|π⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' (19) e(u)|π⟩ = � 2∈π+ E(π → π + 2) u − h2 |π + 2⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' (20) f(u)|π⟩ = � 2∈π− F(π → π − 2) u − h2 |π − 2⟩ (21) where |π⟩ means the state characterized by the 3D Young diagram π and the coefficients E(π → π + □) = −F(π + □ → π) = � 1 σ3 resu→h□ ψπ(u) (22) Equations (20) and (21) mean generators ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' fj acting on the 3D Young diagram π by ej|π⟩ = � □∈π+ hj □E(π → π + □)|π + □⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' (23) fj|π⟩ = � hj □F(π → π − □)|π − □⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' (24) In the following of this paper, we consider 3D Young diagrams which have at most N layers in the z-axis direction, and slice the 3D Young diagrams into a series of 2D Young diagrams by the plane z = n for n = 1, 2, · · · , N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' Then the symmetry of the affine Yangian of gl(1) about the coordinate axes are broken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' For example, ψ0 = − N h1h2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' 3 W1+∞ algebra We begin this section by the definition of W1+∞ algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' The W1+∞ algebra contains the Heisenberg algebra, the Virasoro algebra as subalgebras[12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' The generators are Vj,m for j ∈ Z+, m ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' The relations are [V1,m, V1,n] = mδm+n,0c1, (25) [V2,m, V2,n] = (m − n)V2,m+n + m3 − m 12 δm+n,0c2, (26) [V2,m, V1,n] = −nV1,m+n, (27) and generally [Vj,m, Vk,n] = � 0≤l≤j+k−2 j+k−leven Cl jkNl jk(m, n)Vl,m+n, (28) 4 where the coefficients Nl jk(m, n) are N0 jk(m, n) = � m + j − 1 j + k − 1 � δm+n,0, (29) Nl jk(m, n) = j+k−l−1 � s=0 (−1)s (j + k − l − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' (2l)j+k−l−1 � j + k − l − 1 s � × [j + m − 1]j+k−l−1−s[j − m − 1]s[k + n − 1]s[k − n − 1]j+k−l−1−s, and the structure constants Cl jk are C0 jk = (j − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content='2(2j − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' 4j−1(2j − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content='!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' (2j − 3)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content='δjkcj, (30) Cl jk = 1 2 × 4j+k−l−2 (2l)j+k−l−1 × 4F3 � 1 2, 1 2, − 1 2(j + k − l − 2), − 1 2(j + k − l − 1) 3 2 − j, 3 2 − k, 1 2 + l ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' 1 � , with (a)n = a(a + 1) · · · (a + n − 1), (31) [a]n = a(a − 1) · · · (a − n + 1), (32) mFn � a1, · · · , am b1, · · · , bn ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' z � = ∞ � k=0 (a1)k · · · (am)k (b1)k · · · (bn)k zk k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' (33) Note that here we allow that the central charges can be different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' We consider Jj(z) Jj(z) = � n∈Z aj,nz−n−1 (34) with the relation [aj,n, ak,m] = − 1 h1h2 δj,knδn+m,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' (35) Define J(z) = � n∈Z anz−n−1 = J1(z) + J2(z) + · · · + J3(z), (36) then the bosons an satisfy [an, am] = − N h1h2 nδn+m,0 = ψ0nδn+m,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' (37) In the following two subsections, we consider the Boson aj,n representation of W1+∞ algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content='1 Miura transformation and the W1+∞ algebra Vn(z) Let α0 = − h3 h1h2 , and define the operator Uk(z) as in [11] by : (α0∂ + J1(z))(α0∂ + J2(z)) · · · (α0∂ + JN(z)) := N � k=0 Uk(z)(α0∂)N−k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' (38) 5 The fields Uk(z) generate an algebra, which is W1+N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' The fields Vn(z) can be realized by Uk(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE3T4oBgHgl3EQfFQnF/content/2301.04304v1.pdf'} +page_content=' We list the concrete expressions of the first few Uk(z) as in [16] U0 = 1, (39) U1 = N � j=1 Jj, (40) U2 = � jβ= Z−1 +γ T r +� +exp(−βHγ)A +� +, +(2) +where the partition function +Zγ = T r +� +exp(−βHγ) +� +. +(3) +For a small γ we have the expansion (till the first order in γ) +Zγ = Z0 − γ +� β +0 +dsT r +� +exp(−βH) exp(sH)H1 exp(−sH) +� +. +In terms of the partition function we can define other thermodynamic func- +tions as ,e.g.,the internal energy U +Uγ = −∂β ln Zγ. +(4) +The susceptibility to H1 can be defined as +χβ = ∂γ(Zγ)|γ=0. +(5) +We can use the expansion +exp(sH)H1 exp(−sH) = H1 + s[H, H1] + .... +in eq.(5) to see that for a small β +χβ = ∂γ(Zγ)|γ=0 = −βT r +� +exp(−βH)H1 +� +. +(6) +For H0 of the harmonic oscillator with the frequency ω ( the ground state +energy subtracted) we have +ln Z0 = − ln(1 − exp(−βω)) +(7) +and +U = −∂β ln Z0 = ω(exp(βω) − 1)−1 ≃ β−1 +(8) +for a small β (high temperature). +2 + +If the oscillators are the modes of the electromagnetic field in a cavity then +ω = |k|c where c is the velocity of light and k is the wave vector in the cavity. +In such a case in eqs.(7)-(8) we have a sum over modes. The modified ther- +modynamics [7] comes from the modified distribution of modes in cavities with +fractal geometry. +We consider Hamiltonians H such that its similarity transformation +ˆH = Ω−1HΩ +(9) +gives a generator of a diffusion process (exp(−β ˆH) is a Markov semigroup). +According to our assumption +(exp(−β ˆH)ψ)(ξ) = E[ψ(ξβ(ξ))], +(10) +where ξ are the coordinates of the oscillators, ξβ(ξ) is a Markov process starting +from ξ and the expectation value E[...] is over the paths of the process. In our +models we consider ξ = (x, X) ∈ Rn+d and assume that H1 = V (P) (a function +of momentum). Then, according to eq.(6) +T r +� +exp(−β ˆH)V (P) +� += (2π)−dE +� +δ(xβ(x) − x) exp(iP(Xβ − Y )) < Y |V (P)|X > +� +dxdXdY dP, +(11) +where +< Y |V (P)|X >= (2π)−d +� +dK exp(iK(Y − X))V (K). +(12) +So that +T r +� +exp(−β ˆH)V (P) +� += (2π)−d � +E +� +δ(xβ(x) − x) exp(iP(Xβ − X)) +� +V (P)dxdXdP. +(13) +2 +Statistical mechanics of oscillators +Let us consider in Rn+d the coordinates ξA and a diffusion operator of the form +(sum over repeated indices) +ˆH = − σ2 +2 gAB∂A∂B + ωAξA∂A + ωD +2 gCD∂CgABξAξB∂D. +(14) +By means of +Ω = exp +� +− ωA +2σ gABξAξB� +we obtain the Hamiltonian H of eq.(9) +H = − σ2 +2 gAB∂A∂B + ω2 +A +2 gABξAξB ++σ ωC +8 gCD∂CgRMξMξR∂DgABξAξB − 1 +2 +� +A ωA. +(15) +3 + +We divide the coordinates ξ = (x, X) into two classes x and X where x ∈ Rn +and X ∈ Rd. +In order to simplify the model we assume that only the X +coordinates are coupled to the random metric (or a random mass). So, (gAB) = +(1, gµν) and in gµν(x) the dependence on X is negligible (the coordinates xj +of x have Latin indices j = 1, ..., n and the coordinates of X the Greek indices +µ = n+1, .., n+d). We can imagine a random mass distribution (producing the +metric) which depends only on some coordinates. To make the model simple we +assume ωj = ν and ωµ = ω. If ωA are the modes of a massless field in a cavity +which is a rectangular box of sides LA then ωA = c 2πnA +LA +where nA are integers. +We could arrange the model so that ωj are small and the non-linear terms in +eq.(14) with ωj = ν are negligible. Then, ˆH is of the form +ˆH = − σ2 +2 ∇2 +x + νx∇x + ω +2 ∇xgµνXµXν∇x − σ2 +2 gµν(x)∂µ∂ν + ωXµ∂µ, +(16) +where +∂µ = +∂ +∂Xµ . +With +Ω = exp +� +− 1 +2νσ−1x2 − 1 +2ωσ−1gµνXµXν� +(17) +the similarity transformation (9) gives +H = − σ2 +2 ∇2 +x + 1 +2ν2x2 + σ ω +8 ∇xgµνXµXν∇xgσρXσXρ +− σ2 +2 gµν(x)∂µ∂ν + 1 +2ω2gµνXµXν − d +2ω − n +2 ν. +(18) +exp(−t ˆH) can be expressed (according to eq.(10)) by the solution of the stochas- +tic equations [12] +dxt = −νxtdt − ω +2 ∇xgµνXµXν + σdbt, +(19) +dXµ +t = −ωXµ +t dt + σeµ +a(xt)dBa +t , +(20) +where we expressed the metric g by vierbeins (tetrads) e +gµν = eµ +aeν +a. +(21) +(bt, Bt) is the Brownian motion on Rn+d ,i.e., the Gaussian process with mean +zero and the covariance +E[bj +tbl +s] = min(t, s)δjl +(and similarly for Bt). In order to take the expectation value over the metric +we need an explicit solution of eq.(20). For xt we require only some estimates +on the behaviour in t . However, for a simplicity of the arguments we neglect +the non-linear term in eq.(19) (we assume that eµ +a = δµ +a + κǫµ +a, where κ is a +4 + +small parameter, then ∇g ≃ κ). After a negligence of the non-linear term the +solution of eq.(19) with the initial condition x is +xt = exp(−νt)x + σ +� t +0 +exp(−ν(t − s))dbs. +(22) +The solution of eq.(20) reads +Xµ +t = exp(−ωt)Xµ + σ +� t +0 +exp(−ω(t − s))eµ +a(xs)dBa +s . +(23) +The kernel K of exp(−β ˆH) can be expressed by means of the Fourier transform +Kβ(x, X; y, Y ) = (2π)−d � +dPE +� +δ(xβ(x) − y) exp +� +iP(exp(−βω)X ++σ +� β +0 exp(−ω(β − s))ea(xs)dBa − Y ) +�� +. +(24) +3 +Random metric +We assume that eµ +a are Gaussian variables with the mean δµ +a +eµ +a = δµ +a + κǫµ +a +(25) +where +< ǫµ +a(x)ǫν +c(x′) >= δµν +ac G(x − x′). +(26) +We calculate the Gaussian integral in eqs.(13) and (24)(with δµν +ac = δµνδac chosen +for simplicity) +< exp(iP(exp(−ωβ)X + σ +� β +0 exp(−ω(β − s))ea(xs)dBa)) > += exp +� +iP exp(−ωβ)X − 1 +2 < (PQβ)2 > +iσPµ +� β +0 exp(−ω(β − s))dBµ +s +� +) += exp +� +− 1 +2κ2PµPν +� β +0 +� β +0 exp(−ω(β − s)) exp(−ω(β − s′))G(xs − xs′)dBµ +s dBν +s′ ++iP exp(−ωβ)X + iσPµ +� β +0 exp(−ω(β − s))dBµ +s +� +. +(27) +In eq.(27) we used the formula for Gaussian expectation value of ǫ +< exp(iPµQµ +β) >= exp(−1 +2 < (PµQµ +β)2 >) +(28) +with +PµQµ +t = Pµσ +� t +0 +exp(−ω(t − s))ǫµ +a(xs)dBa +s ≡ Ft. +(29) +This expression can be written in a different way using the Ito calculus [12][14] +� +dF 2 +t = 2 +� +FtdFt + +� +dFtdFt, +(30) +5 + +where we have (here P 2 = PµP µ) +dFtdFt = dκ2G(0)P 2dt. +(31) +Next, we consider a singular covariance G. For this purpose at the beginning +we treat ǫµ +a as a regularized random field. Then, we remove the regularization. +In order to make Hψ a well-defined random field we need the normal ordering +of H with +: gµν :=: eµ +a(x)eν +a(x) := eµ +a(x)eν +a(x) − κ2 < ǫµ +a(x)ǫν +a(x) > . +The normal ordering in the exponential of eq.(27) removes the second term +dκ2G(0)P 2t on the rhs of eq.(30) whereas the first term there (i.e., +� β +0 FsdFs) +becomes a time-ordered integral (renormalization of such expressions appear +also in QED [15][16]). Owing to the normal ordering in eq.(27) +< (PQt)2 >→< (PQt)2 > −tP 2κG(0)d +(32) +and because of the subtraction (32) we can define the action of exp(−tH) upon +a test function ψ (decaying fast in the momentum space) so that exp(−Ht)ψ is +a well-defined random field. After the averaging over the translation invariant +random field eµ +a(x) and the renormalization (32) we can write < exp(−β : H : +)ψ > in terms of Fourier transforms as +(< exp(−β : H :) > ψ)(x, X) = +E +� +δ(xβ − y) exp +� +− κ2PµPν +� β +0 exp(−ω(β − s))dBµ +s +� s +0 dBν +s′ exp(−ω(β − s′))G(xs − xs′) ++iσPµ +� β +0 exp(−ω(β − s))dBµ +s + iP exp(−ωβ)X) +�� +ψ(y, P)dPdy. +(33) +At κ = 0 we have +Z0 = T r(exp(−βH0)) = +� +dxdX < exp(−βH0) > (x, X; x, X) = +(2π)−d � +dPdxdXE +� +δ(xβ − x) exp(iP((exp(−ωβ) − 1)X + iσP +� β +0 exp(−ω(β − s))dBs)) +� += +� +K(0) +β (x, X; x, X)dxdX, +(34) +where H(0) is the Hamiltonian of uncoupled harmonic oscillators and K(0) is +the well-known Mehler kernel of the harmonic oscillator. So, at κ = 0 we obtain +the formula (7). We are interested in the κ-term as a perturbation resulting +from an interaction with a random metric. In eq.(33) we apply the identity for +Bs and bs (in the sense that both sides have the same probability law) +Bs = +√ +λB s +λ . +(35) +After rescaling +xs′ = exp(−νs′)x + +√ +λ +� +s′ +λ +0 +exp +� +− λν(s′ +λ − τ) +� +dbτ. +(36) +6 + +It is easy to see that for a small t in eq.(22) (set λ = s′ in eq.(36)) +xt = x + +√ +tqt, +(37) +where qt ≃ a + c +√ +t + ... with |a| > 0 for a small t. This scaling behaviour is +all what we need to assume about solutions of eq.(19) with a random metric g +of eq.(26). In eq.(33) we rescale the Brownian motion Bs′ and xs′ as in (36) +(with λ = s) and subsequently Bs and xs with λ = β. After such a change of +variables the integral of the κ-dependent part in the formula (33) reads +exp +� +− 1 +2κ2βPµPν +� 1 +0 exp(−βω(1 − s))dBµ +s +� s +0 exp(−βω(1 − s′))G(xs − xs′)dBν +s′ +� +, +(38) +where 0 ≤ s′ ≤ s ≤ 1, xs = x + √βqs and xs′ = x + √β˜qs′ where qs ̸= 0 and +˜qs ̸= 0 at β = 0. We consider the covariance (the upper bound on α > 0 will +be discussed at the end of this section) +G(x − x′) = |x − x′|−2α. +(39) +Then, from eq.(38) for a small β +G(xs − xs′) = β−αg(β, s, s′), +(40) +where s′ ≤ s ∈ [0, 1] and g ≃ A + Cβ (with A > 0) for a small β. Hence,eq.(33) +is of the form +E +� +exp +� +iP((exp(−ωβ) − 1)X − � β +0 exp(−ω(β − s))dB) +� +× exp +� +− κ2β1−αPµPνf µν(β) +�� +, +(41) +where |f µν(β)PµPν| is bounded from below by a constant. +We can now estimate the behaviour of the partition function in the metric +field (26) ( note that : gµν : − <: gµν :> has the covariance |x − x′|−4α if ǫµ +a has +the covariance (39)). We have +gµν = δµν + 2κǫµ +ν + κ2ǫµ +aǫν +a. +(42) +The Hamiltonian (16) is of the form H = H0 + κH1 + κ2H2.From eq.(27) we +can see that the contribution to the partition function Z is of order κ2. We +consider Hγ = H + γV (P) then in the approximation (6) +∂γZγ = T r(exp(−βH)V (P)). +(43) +The anomalous (fractional) dependence β1−α in eq.(41) of the partition function +is a characteristic of the coupling to a singular metric field. We can calculate +the κ2-derivative of the susceptibility (6) to the metric for small β (high tem- +perature) +7 + +∂κ2∂γZ|κ=γ=0 = β1−αE +� +PµPνf µν(β)δ(xβ − x) +exp(iP(exp(−ωβ) − 1)X + iσP +� β +0 exp(−ω(β − s))dBs)) +� +βV (P)dxdXdP +(44) +and +∂2 +κ2∂γZ|κ=γ=0 = β2−2αE +�� +PµPνf µν(β) +�2 +δ(xβ − x) +exp(iP(exp(−ωβ) − 1)X + iσP +� β +0 exp(−ω(β − s))dBs)) +� +βV (P)dxdXdP. +(45) +For a small κ and small β the partition function has the expansion in non-integer +powers of β +∂γZκ2 = ∂γZ0 + κ2χ1ββ1−α + κ4χ2ββ2−2α + .... +(46) +with certain constants χ1 and χ2. +We still have to estimate the integrals in eqs.(43)-(46). The integral (43) is +expressed by kernels in eq.(13). The expectation value in eq.(13) after the renor- +malization (32) involves the time-ordered stochastic integral in eq.(33) which +fails to be positive definite. Hence, if the expectation value in eq.(43) is to be +finite V (P) must decrease faster than exp(−RP 2) for any R (the derivatives in +eqs.(44)-(46) impose milder requirements on the decay of V (P) for a large P). +There is still the problem of the convergence of the stochastic integrals in the +definition of fµνP µP ν. This stochastic integral is of the form +PµPν +� β +0 +dBµ +s +� s +0 +dBν +s′G(xs − xs′). +(47) +The stochastic integrals in eq. (47) can be estimated by ordinary integrals using +the formula [13](better estimates on the multiple stochastic integrals (33) and +(47) can be obtained using the results of ref.[17]) +E +�� � +FdBs +�2k� +≤ CkE +� � +F 2kds +� +(48) +with certain constants Ck. The rhs of eq.(48) involves the Ornstein-Uhlenbeck +process (22) [18] whose transition function is expressed by the Mehler for- +mula. In calculations in eq.(48) for small β we can approximate the Ornstein- +Uhlenbeck process xs by the Brownian motion with the transition function +p(s, x) = (2πs)− n +2 exp(− |x|2 +2s ). Then, the integral in eq.(47) can be estimated +by (use eq.(48) twice with k = 1 ) +� +dsds′ +� +dxp(s − s′, x)|x|−4α < ∞ +(49) +if 2α < 1. The expansion (46) must be terminated at the k-th order if 2kα > 1 +because the rhs of the estimate (48) is infinite. +8 + +4 +The outlook +Some approximate calculations [8][9][11] indicate that the singularity of the +quantum gravitational field at small distances can be different than the canon- +ical one which in n dimensions is of the form |x − x′|−n+2 ( where n = 4 +corresponds to α = 1 +2 in eq.(39)). The fractal dimensionality of the physical +space-time has been discussed in [19][20] on the basis of the Cosmic Microwave +Background measurements. Some limits on the deviation of the observational +space-time dimension from the physical four dimensions have been obtained. +The effect of quantum gravity could be observed either at small distances or at +high energies (which in cosmology are connected with high temperatures). The +non-integer indices in the expansion of the partition function in eqs.(44)-(46) +could indicate the relevance of quantum gravity for some extremal processes in +astrophysics ( which possibly could be tested on the quantum level in gravi- +tational wave interferometers [21]). The model of a random mass distribution +which according to eqs.(14) and (23) is equivalent to a random diffusivity is +of interest in condensed matter physics [22][23]. An anomalous behaviour of +the partition function (46) or other thermodynamic functions of complex sys- +tems (e.g. molecules or crystals) could be an indication of the random mass or +random metric present in these systems. +References +[1] J.D. Farmer, E. Ott and J.A. Yorke, The dimension of chaotic attractors, +Physica D7,153(1983) +[2] E.A. Novikov, The effect of intermittency on statistical characteristics +of turbulence and scale similarity of breakdown coefficients, Phys.Fluids, +A2,814(1990) +[3] J.R. Banavar and J.F. Willemsen,Probability density for diffusion on frac- +tals, Phys.Rev.B30,6778(1984) +[4] G.V. Dunne, Heat kernels and zeta functions on fractals, +J.Phys.A45,374016(2012) +[5] S. Carlip,Dimension and dimensional reduction in quantum gravity, +Class.Quant.Grav.34,193001(2017) +[6] B.B. Mandelbrot, The Fractal Geometry of Nature, Freeman, San Fran- +cisco,1983 +[7] E. Akkermans, G.V.Dunne and A. Teplyaev, +Thermodynamics of photons on fractals, Phys.Rev. Lett.105,230407(2010) +9 + +[8] J. Ambjorn, J. Jurkiewicz and R.Loll, +The +spectral +dimension +of +the +universe +is +scale +dependent, +Phys.Rev.Lett.95,171301(2005) +[9] P. Horava,Spectral dimension of the universe in quantum gravity at a Lif- +shitz point, Phys.Rev.Lett.102,161301(2009) +[10] L.Crane and L. Smolin, Renormalization of general relativity on a back- +ground of spacetime foam +Nucl.Phys.B267,714(1986) +[11] Z.Haba, Universal regular short distance behavior from an interaction with +a scale invariant gravity, +Phys.Lett.B528,129(2002) +[12] N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion +Processes, Amsterdam, North Holland,1981 +[13] I.I. Gikhman and A.V. Skorohod, Stochastic Differential Equations, +Springer,New York,1972 +[14] B. Simon, Functional Integration and +Quantum Physics,Academic, New York,1979 +[15] V. Betz and F. Hiroshima, Gibbs measures with double stochastic integrals +on a path space, +Inf.Dim.Anal.Quant.Prob. Rel.Topics,12,135(2009) +[16] S. Albeverio and S. Kusuoka, A basic estimate for two-dimensional stochas- +tic holonomy along Brownian bridges, +J.Funct.Anal.127,132(1994) +[17] E. Carlen and P. Kree, +Lp estimates on iterated stochastic integrals, +Ann.Probab.19,354(1991) +[18] S. Chandrasekhar, Stochastic problems in physics and astronomy, +Rev.Mod.Phys.15,1 (1943) +[19] F. Caruso and V. Oguri, The cosmic microwave background spectrum and +an upper limit for fractal space dimensionality, +Astroph.Journ.694,151(2009) +[20] G. Amelino-Camelia, F. Brighenti, G. Gubitosi and G. Santos, +Thermal dimension of quantum spacetime, Phys.Lett.B76748(2017) +10 + +[21] M. Parikh, F. Wilczek and G. Zahariade, +Signatures of the quantization of gravity at gravitational wave detectors, +Phys.Rev.D104,046021(2021) +[22] R. Metzler and J. Klafter, The random walk’s guide to anomalous diffu- +sion:a fractional dynamics approach, +Phys.Rep.339,1(2000) +[23] V. Sposini, A.V.Chechkin, F.Seno, G. Pagnini +and R. Metzler, +Random diffusivity from stochastic equations:comparison of two models for +Brownian yet non-Gaussian diffusion, +New Journ.Phys.20,043044(2018) +11 + diff --git a/XNE0T4oBgHgl3EQfWAA7/content/tmp_files/load_file.txt b/XNE0T4oBgHgl3EQfWAA7/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c6f8acb00ddca4c425c5a057dd468b4badd1f68f --- /dev/null +++ b/XNE0T4oBgHgl3EQfWAA7/content/tmp_files/load_file.txt @@ -0,0 +1,274 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf,len=273 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content='02271v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content='stat-mech] 5 Jan 2023 Response of a canonical ensemble of quantum oscillators to a random metric Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' Haba Institute of Theoretical Physics, University of Wroclaw, 50-204 Wroclaw, Plac Maxa Borna 9, Poland email:zbigniew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content='haba@uwr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content='pl January 9, 2023 Abstract We calculate the susceptibility of a canonical ensemble of quantum oscillators to the singular random metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' If the covariance of the metric is |x − x′|−4α (0 < α < 1 2)then the expansion of the partition function in powers of the temperature involves non-integer indices .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' 1 Introduction It is known that the dynamics in an irregular domain can be chaotic and con- versely the chaotic motion can have a fractal attractor [1][2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' Diffusion in frac- tal domains exhibits their fractal dimension which in general is not a natural number[3][4][5][6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' Thermodynamics of a canonical ensemble of particles in an irregular domain depends on the spectral dimension of the domain [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' In such a case thermodynamic properties (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' critical indices) may be functions of a non-integer dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' For some time fractal geometry has been associated with quantum gravity [5][8][9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content='Quantum gravity can be expressed as a random geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' It is believed that quantum gravity leads to a fractal geometry by a ”foamy” behaviour of the metric at short distances [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' Such irregular shapes of random figures have been at the basis of the fractal geometry [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content='It has been suggested that irregular metric at the Planck scale can modify the short distance behaviour of quantum fields at short distances [5][10][11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' In this paper we study quantum oscillators in a random singular metric ( or random position dependent mass).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' We define the susceptibility to the metric which has an expansion in powers of the inverse temperature β if the metric is a regular random field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' We show that if the random metric is singular then the susceptibility has an expansion in non-integer powers of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' It is known that an ensemble of oscillators can serve as an approximation to field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' 1 The quantum statistical mechanics of oscillators will resemble the quantum field theory at finite temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' We consider a Hamiltonian Hγ perturbed around the free theory(harmonic oscillators) Hγ = H + γH1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' (1) The statistical expectation value of an observable A at the temperature β−1 = kBT (where kB is the Boltzman constant) is defined by < A >β= Z−1 γ T r � exp(−βHγ)A � , (2) where the partition function Zγ = T r � exp(−βHγ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' (3) For a small γ we have the expansion (till the first order in γ) Zγ = Z0 − γ � β 0 dsT r � exp(−βH) exp(sH)H1 exp(−sH) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' In terms of the partition function we can define other thermodynamic func- tions as ,e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=',the internal energy U Uγ = −∂β ln Zγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' (4) The susceptibility to H1 can be defined as χβ = ∂γ(Zγ)|γ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' (5) We can use the expansion exp(sH)H1 exp(−sH) = H1 + s[H, H1] + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content='. in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' (5) to see that for a small β χβ = ∂γ(Zγ)|γ=0 = −βT r � exp(−βH)H1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' (6) For H0 of the harmonic oscillator with the frequency ω ( the ground state energy subtracted) we have ln Z0 = − ln(1 − exp(−βω)) (7) and U = −∂β ln Z0 = ω(exp(βω) − 1)−1 ≃ β−1 (8) for a small β (high temperature).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' 2 If the oscillators are the modes of the electromagnetic field in a cavity then ω = |k|c where c is the velocity of light and k is the wave vector in the cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' In such a case in eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' (7)-(8) we have a sum over modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' The modified ther- modynamics [7] comes from the modified distribution of modes in cavities with fractal geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' We consider Hamiltonians H such that its similarity transformation ˆH = Ω−1HΩ (9) gives a generator of a diffusion process (exp(−β ˆH) is a Markov semigroup).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' According to our assumption (exp(−β ˆH)ψ)(ξ) = E[ψ(ξβ(ξ))], (10) where ξ are the coordinates of the oscillators, ξβ(ξ) is a Markov process starting from ξ and the expectation value E[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content='] is over the paths of the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' In our models we consider ξ = (x, X) ∈ Rn+d and assume that H1 = V (P) (a function of momentum).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' Then, according to eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' (6) T r � exp(−β ˆH)V (P) � = (2π)−dE � δ(xβ(x) − x) exp(iP(Xβ − Y )) < Y |V (P)|X > � dxdXdY dP, (11) where < Y |V (P)|X >= (2π)−d � dK exp(iK(Y − X))V (K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' (12) So that T r � exp(−β ˆH)V (P) � = (2π)−d � E � δ(xβ(x) − x) exp(iP(Xβ − X)) � V (P)dxdXdP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' (13) 2 Statistical mechanics of oscillators Let us consider in Rn+d the coordinates ξA and a diffusion operator of the form (sum over repeated indices) ˆH = − σ2 2 gAB∂A∂B + ωAξA∂A + ωD 2 gCD∂CgABξAξB∂D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' (14) By means of Ω = exp � − ωA 2σ gABξAξB� we obtain the Hamiltonian H of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' (9) H = − σ2 2 gAB∂A∂B + ω2 A 2 gABξAξB +σ ωC 8 gCD∂CgRMξMξR∂DgABξAξB − 1 2 � A ωA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' (15) 3 We divide the coordinates ξ = (x, X) into two classes x and X where x ∈ Rn and X ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' In order to simplify the model we assume that only the X coordinates are coupled to the random metric (or a random mass).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' So, (gAB) = (1, gµν) and in gµν(x) the dependence on X is negligible (the coordinates xj of x have Latin indices j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=', n and the coordinates of X the Greek indices µ = n+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content='., n+d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' We can imagine a random mass distribution (producing the metric) which depends only on some coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' To make the model simple we assume ωj = ν and ωµ = ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' If ωA are the modes of a massless field in a cavity which is a rectangular box of sides LA then ωA = c 2πnA LA where nA are integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' We could arrange the model so that ωj are small and the non-linear terms in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' (14) with ωj = ν are negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' Then, ˆH is of the form ˆH = − σ2 2 ∇2 x + νx∇x + ω 2 ∇xgµνXµXν∇x − σ2 2 gµν(x)∂µ∂ν + ωXµ∂µ, (16) where ∂µ = ∂ ∂Xµ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' With Ω = exp � − 1 2νσ−1x2 − 1 2ωσ−1gµνXµXν� (17) the similarity transformation (9) gives H = − σ2 2 ∇2 x + 1 2ν2x2 + σ ω 8 ∇xgµνXµXν∇xgσρXσXρ − σ2 2 gµν(x)∂µ∂ν + 1 2ω2gµνXµXν − d 2ω − n 2 ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' (18) exp(−t ˆH) can be expressed (according to eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' (10)) by the solution of the stochas- tic equations [12] dxt = −νxtdt − ω 2 ∇xgµνXµXν + σdbt, (19) dXµ t = −ωXµ t dt + σeµ a(xt)dBa t , (20) where we expressed the metric g by vierbeins (tetrads) e gµν = eµ aeν a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' (21) (bt, Bt) is the Brownian motion on Rn+d ,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=', the Gaussian process with mean zero and the covariance E[bj tbl s] = min(t, s)δjl (and similarly for Bt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' In order to take the expectation value over the metric we need an explicit solution of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content='(20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' For xt we require only some estimates on the behaviour in t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' However, for a simplicity of the arguments we neglect the non-linear term in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' (19) (we assume that eµ a = δµ a + κǫµ a, where κ is a 4 small parameter, then ∇g ≃ κ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' After a negligence of the non-linear term the solution of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' (19) with the initial condition x is xt = exp(−νt)x + σ � t 0 exp(−ν(t − s))dbs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' (22) The solution of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' (20) reads Xµ t = exp(−ωt)Xµ + σ � t 0 exp(−ω(t − s))eµ a(xs)dBa s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' (23) The kernel K of exp(−β ˆH) can be expressed by means of the Fourier transform Kβ(x, X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' y, Y ) = (2π)−d � dPE � δ(xβ(x) − y) exp � iP(exp(−βω)X +σ � β 0 exp(−ω(β − s))ea(xs)dBa − Y ) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' (24) 3 Random metric We assume that eµ a are Gaussian variables with the mean δµ a eµ a = δµ a + κǫµ a (25) where < ǫµ a(x)ǫν c(x′) >= δµν ac G(x − x′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' (26) We calculate the Gaussian integral in eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' (13) and (24)(with δµν ac = δµνδac chosen for simplicity) < exp(iP(exp(−ωβ)X + σ � β 0 exp(−ω(β − s))ea(xs)dBa)) > = exp � iP exp(−ωβ)X − 1 2 < (PQβ)2 > +iσPµ � β 0 exp(−ω(β − s))dBµ s � ) = exp � − 1 2κ2PµPν � β 0 � β 0 exp(−ω(β − s)) exp(−ω(β − s′))G(xs − xs′)dBµ s dBν s′ +iP exp(−ωβ)X + iσPµ � β 0 exp(−ω(β − s))dBµ s � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' (27) In eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' (27) we used the formula for Gaussian expectation value of ǫ < exp(iPµQµ β) >= exp(−1 2 < (PµQµ β)2 >) (28) with PµQµ t = Pµσ � t 0 exp(−ω(t − s))ǫµ a(xs)dBa s ≡ Ft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' (29) This expression can be written in a different way using the Ito calculus [12][14] � dF 2 t = 2 � FtdFt + � dFtdFt, (30) 5 where we have (here P 2 = PµP µ) dFtdFt = dκ2G(0)P 2dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' (31) Next, we consider a singular covariance G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' For this purpose at the beginning we treat ǫµ a as a regularized random field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' Then, we remove the regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' In order to make Hψ a well-defined random field we need the normal ordering of H with : gµν :=: eµ a(x)eν a(x) := eµ a(x)eν a(x) − κ2 < ǫµ a(x)ǫν a(x) > .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' The normal ordering in the exponential of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' (27) removes the second term dκ2G(0)P 2t on the rhs of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' (30) whereas the first term there (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=', � β 0 FsdFs) becomes a time-ordered integral (renormalization of such expressions appear also in QED [15][16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' Owing to the normal ordering in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' (27) < (PQt)2 >→< (PQt)2 > −tP 2κG(0)d (32) and because of the subtraction (32) we can define the action of exp(−tH) upon a test function ψ (decaying fast in the momentum space) so that exp(−Ht)ψ is a well-defined random field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' After the averaging over the translation invariant random field eµ a(x) and the renormalization (32) we can write < exp(−β : H : )ψ > in terms of Fourier transforms as (< exp(−β : H :) > ψ)(x, X) = E � δ(xβ − y) exp � − κ2PµPν � β 0 exp(−ω(β − s))dBµ s � s 0 dBν s′ exp(−ω(β − s′))G(xs − xs′) +iσPµ � β 0 exp(−ω(β − s))dBµ s + iP exp(−ωβ)X) �� ψ(y, P)dPdy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' (33) At κ = 0 we have Z0 = T r(exp(−βH0)) = � dxdX < exp(−βH0) > (x, X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' x, X) = (2π)−d � dPdxdXE � δ(xβ − x) exp(iP((exp(−ωβ) − 1)X + iσP � β 0 exp(−ω(β − s))dBs)) � = � K(0) β (x, X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' x, X)dxdX, (34) where H(0) is the Hamiltonian of uncoupled harmonic oscillators and K(0) is the well-known Mehler kernel of the harmonic oscillator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' So, at κ = 0 we obtain the formula (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' We are interested in the κ-term as a perturbation resulting from an interaction with a random metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' In eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' (33) we apply the identity for Bs and bs (in the sense that both sides have the same probability law) Bs = √ λB s λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' (35) After rescaling xs′ = exp(−νs′)x + √ λ � s′ λ 0 exp � − λν(s′ λ − τ) � dbτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' (36) 6 It is easy to see that for a small t in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' (22) (set λ = s′ in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' (36)) xt = x + √ tqt, (37) where qt ≃ a + c √ t + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' with |a| > 0 for a small t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' This scaling behaviour is all what we need to assume about solutions of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' (19) with a random metric g of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content='(26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' In eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' (33) we rescale the Brownian motion Bs′ and xs′ as in (36) (with λ = s) and subsequently Bs and xs with λ = β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' After such a change of variables the integral of the κ-dependent part in the formula (33) reads exp � − 1 2κ2βPµPν � 1 0 exp(−βω(1 − s))dBµ s � s 0 exp(−βω(1 − s′))G(xs − xs′)dBν s′ � , (38) where 0 ≤ s′ ≤ s ≤ 1, xs = x + √βqs and xs′ = x + √β˜qs′ where qs ̸= 0 and ˜qs ̸= 0 at β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' We consider the covariance (the upper bound on α > 0 will be discussed at the end of this section) G(x − x′) = |x − x′|−2α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' (39) Then, from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' (38) for a small β G(xs − xs′) = β−αg(β, s, s′), (40) where s′ ≤ s ∈ [0, 1] and g ≃ A + Cβ (with A > 0) for a small β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' Hence,eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' (33) is of the form E � exp � iP((exp(−ωβ) − 1)X − � β 0 exp(−ω(β − s))dB) � × exp � − κ2β1−αPµPνf µν(β) �� , (41) where |f µν(β)PµPν| is bounded from below by a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' We can now estimate the behaviour of the partition function in the metric field (26) ( note that : gµν : − <: gµν :> has the covariance |x − x′|−4α if ǫµ a has the covariance (39)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' We have gµν = δµν + 2κǫµ ν + κ2ǫµ aǫν a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' (42) The Hamiltonian (16) is of the form H = H0 + κH1 + κ2H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content='From eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' (27) we can see that the contribution to the partition function Z is of order κ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' We consider Hγ = H + γV (P) then in the approximation (6) ∂γZγ = T r(exp(−βH)V (P)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' (43) The anomalous (fractional) dependence β1−α in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' (41) of the partition function is a characteristic of the coupling to a singular metric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' We can calculate the κ2-derivative of the susceptibility (6) to the metric for small β (high tem- perature) 7 ∂κ2∂γZ|κ=γ=0 = β1−αE � PµPνf µν(β)δ(xβ − x) exp(iP(exp(−ωβ) − 1)X + iσP � β 0 exp(−ω(β − s))dBs)) � βV (P)dxdXdP (44) and ∂2 κ2∂γZ|κ=γ=0 = β2−2αE �� PµPνf µν(β) �2 δ(xβ − x) exp(iP(exp(−ωβ) − 1)X + iσP � β 0 exp(−ω(β − s))dBs)) � βV (P)dxdXdP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' (45) For a small κ and small β the partition function has the expansion in non-integer powers of β ∂γZκ2 = ∂γZ0 + κ2χ1ββ1−α + κ4χ2ββ2−2α + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content='. (46) with certain constants χ1 and χ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' We still have to estimate the integrals in eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content='(43)-(46).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' The integral (43) is expressed by kernels in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content='(13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' The expectation value in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' (13) after the renor- malization (32) involves the time-ordered stochastic integral in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' (33) which fails to be positive definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' Hence, if the expectation value in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' (43) is to be finite V (P) must decrease faster than exp(−RP 2) for any R (the derivatives in eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' (44)-(46) impose milder requirements on the decay of V (P) for a large P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' There is still the problem of the convergence of the stochastic integrals in the definition of fµνP µP ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' This stochastic integral is of the form PµPν � β 0 dBµ s � s 0 dBν s′G(xs − xs′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' (47) The stochastic integrals in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' (47) can be estimated by ordinary integrals using the formula [13](better estimates on the multiple stochastic integrals (33) and (47) can be obtained using the results of ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' [17]) E �� � FdBs �2k� ≤ CkE � � F 2kds � (48) with certain constants Ck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' The rhs of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' (48) involves the Ornstein-Uhlenbeck process (22) [18] whose transition function is expressed by the Mehler for- mula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' In calculations in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' (48) for small β we can approximate the Ornstein- Uhlenbeck process xs by the Brownian motion with the transition function p(s, x) = (2πs)− n 2 exp(− |x|2 2s ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' Then, the integral in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' (47) can be estimated by (use eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' (48) twice with k = 1 ) � dsds′ � dxp(s − s′, x)|x|−4α < ∞ (49) if 2α < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' The expansion (46) must be terminated at the k-th order if 2kα > 1 because the rhs of the estimate (48) is infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' 8 4 The outlook Some approximate calculations [8][9][11] indicate that the singularity of the quantum gravitational field at small distances can be different than the canon- ical one which in n dimensions is of the form |x − x′|−n+2 ( where n = 4 corresponds to α = 1 2 in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content='(39)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' The fractal dimensionality of the physical space-time has been discussed in [19][20] on the basis of the Cosmic Microwave Background measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' Some limits on the deviation of the observational space-time dimension from the physical four dimensions have been obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' The effect of quantum gravity could be observed either at small distances or at high energies (which in cosmology are connected with high temperatures).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' The non-integer indices in the expansion of the partition function in eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' (44)-(46) could indicate the relevance of quantum gravity for some extremal processes in astrophysics ( which possibly could be tested on the quantum level in gravi- tational wave interferometers [21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' The model of a random mass distribution which according to eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' (14) and (23) is equivalent to a random diffusivity is of interest in condensed matter physics [22][23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' An anomalous behaviour of the partition function (46) or other thermodynamic functions of complex sys- tems (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' molecules or crystals) could be an indication of the random mass or random metric present in these systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content=' References [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} +page_content='D.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQfWAA7/content/2301.02271v1.pdf'} diff --git a/XNFRT4oBgHgl3EQfNTdg/content/tmp_files/2301.13509v1.pdf.txt b/XNFRT4oBgHgl3EQfNTdg/content/tmp_files/2301.13509v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..276d149001c8b3e82a5635368ea3c4e66815f9af --- /dev/null +++ b/XNFRT4oBgHgl3EQfNTdg/content/tmp_files/2301.13509v1.pdf.txt @@ -0,0 +1,1649 @@ +Waves in a Stochastic Cell Motility Model +C. H. S. Hamster a,∗, P. van Heijster b, +a Biometris - Wageningen University and Research +Wageningen; The Netherlands +Email: christian.hamster@wur.nl +b Biometris - Wageningen University and Research +Wageningen; The Netherlands +Email: peter.vanheijster@wur.nl +Abstract +In Bhattacharya et al. (Science Advances, 2020), a set of chemical reactions involved in the dynamics of +actin waves in cells was studied. Both at the microscopic level, where the individual chemical reactions are +directly modelled using Gillespie-type algorithms, and on a macroscopic level where a deterministic reaction- +diffusion equation arises as the large-scale limit of the underlying chemical reactions. In this work, we derive, +and subsequently study, the related mesoscopic stochastic reaction-diffusion system, or Chemical Langevin +Equation, that arises from the same set of chemical reactions. We explain how the stochastic patterns that +arise from this equation can be used to understand the experimentally observed dynamics from Bhattacharya +et al. In particular, we argue that the mesoscopic stochastic model better captures the microscopic behaviour +than the deterministic reaction-diffusion equation, while being more amenable for mathematical analysis and +numerical simulations than the microscopic model. +Key words: Gillespie Algorithms, Cell Motility, Mesoscopic Patterns, SPDEs, Chemical Langevin +Equation. +1 +Introduction +In order to move around, an amoeboid cell can change its shape by polymerising actin to curve +the cell membrane. The actin polymerisation is controlled by signalling molecules and experiments +in Dictyostelium discoideum have shown that activation of these signalling molecules happens at +localised patches that can move along the membrane like a wave [1, 21]. In wild-type (WT) cells, +these waves move fast and die out, creating familiar-shaped pseudopods, while in cancerous cells +these waves stick to a point, creating elongated protrusions [1], see Figure 1.1. In absence of a signal, +the formation of pseudopods happens at random places on the cell membrane, resulting in random +motion. In contrast, when a cell senses a chemical signal, it can concentrate the random protrusions +at the side of the cell where the signal comes from, leading to movement in the direction of the +signal [6]. As cells are small, the difference in signal strength between the front and the back of +the cell (the gradient) is small as well. Furthermore, the cell can only use discrete points at the +membrane where the receptors are to estimate the direction of the signal [6]. Therefore, one of the +∗Corresponding author. +Preprint submitted to .... +February 1, 2023 +arXiv:2301.13509v1 [math.AP] 31 Jan 2023 + +Fig. 1.1: Stochastic simulations of the microscopic Gillespie-type model from [1]. The figures on the left show +stochastic simulations of the Ras activity for parameter values applicable to (A) wild-type cells and to (B) +genetically modified cells, where the phosphatase PTEN has been switched off. The figures on the right show +typical cell shapes corresponding to the dynamics in the left figures. This shows that mutations in the gene +that codes for PTEN lead to elongated protrusions typically associated with cancer. The dotted yellow line is +an indicator of the wave speed, i.e. the actin waves in (B) are slower and live longer than in (A). Reproduced +from [1] under creative commons license 4.0. +main questions is “How can a cell use a small gradient in the signal to concentrate the actin activity +in the front?”. This question has been studied intensively, but no complete description of all the +microscopic chemical processes involved has been given yet, see [8] for a review. +In [1], the choice is made to describe the highly complex actin dynamics with a conceptual acti- +vator u and inhibitor v that diffuse and react with each other as summarised in Table 1. The species +u and v are an abstraction of the dozens of components that regulate the actual cell movement, but +the activator u can be thought of as Ras activity [1], which plays an important role in cell growth +and differentiation [28]. In particular, u is being activated by Reaction #3 and Reaction #4, while +being inhibited by Reaction #1 and Reaction #2, with propensities as indicated in the table. In +addition, v is inhibited by Reaction #5, while Reaction #6 activates the inhibitor. +The information on the chemical reactions, in combination with the diffusion of both species, +is generally used in one of two ways. First, there is a Gillespie-type algorithm [15, 16] which can +be used to simulate the involved chemical reactions on a microscopic level. For these simulations, +(uk(tn), vk(tn)) (the solution at time tn at grid cell k) is treated as the number of molecules of type +u and v at time tn in a grid cell with finite size. For all these individual molecules the probabilities +of diffusing to other grid cells or taking part in a chemical reaction are prescribed as by Table 1. To +be precise, Reaction #1 implies that the time to the next reaction that degrades a u molecule in +grid cell k is exponentially distributed with rate parameter (a1uk(tn))−1. See the panels on the left +of Figure 1.1 for examples of these simulations. This Gillespie-type algorithm approach takes the +stochastic nature of a single cell into account. However, it is computationally very expensive and +difficult to analyse mathematically. Hence, it is hard to use this type of modelling approach to make +valuable predictions. +2 + +A +Space (x) +ime +B +Space (x) +ime(t)No. +Reaction +Propensity +u +v +1 +u → ∅ +a1u +−1 +0 +2 +u → ∅ +a2uv +−1 +0 +3 +∅ → u +a3u2/(a4 + u2) +1 +0 +4 +∅ → u +a5 +1 +0 +5 +v → ∅ +εc1v +0 +−1 +6 +∅ → v +εc2u +0 +1 +Table 1: The chemical reactions that determine the actin wave dynamics from [1]. +A second way to use the reactions in Table 1 is to derive an average large-scale limit macro- +scopic equation. Hence, we assume that u and v are densities on a continuous domain, described +by a reaction-rate equation with diffusion, also known as a Reaction-Diffusion Equation (RDE). In +particular, the RDE1 related to the chemical reactions in Table 1 is given by +∂tu = Du∂xxu − a1u − a2uv + +a3u2 +a4 + u2 + a5 , +∂tv = Dv∂xxv + ε(−c1v + c2u), +(1.1) +which is a specific version of the general RDE we will encounter in §2. This model is a variation on the +classic FitzHugh-Nagumo model for neuron spiking [12, 30]. Protrusions are formed at places with +high activator u and u is inhibited by the terms −a1u and −a2uv, see Reaction #1 and Reaction #2 +in Table 1. This implies that an increase in u or v leads to a decrease in u, unless the increase is high +enough such that activation from Reaction #3, modelled by a nonlinear Hill function a3u2/(a4+u2), +takes over and negates the inhibiting effects. Effectively, this means that a small increase in u can +lead to a much larger increase in u, that is, the system is locally activated. Once u is large and the +Hill function levels off at a fixed value a3, the amount of inhibitor v increases via the term εc2u +(related to Reaction #6), leading to a fast decay in u by the −a2uv term (related to Reaction #2). +The inhibitor v then decays via Reaction #5 to the rest state and activation can happen again. In +addition, both species diffuse with diffusion coefficient Du, respectively Dv, where it is assumed that +Du < Dv. It is important to realise that, in both approaches, the modelled actin waves happen on +the surface of the cell, and, as in [1], we only study a slice of this surface. Therefore, the spatial +domain must be thought of as an (approximate) circle. +For deterministic RDEs like (1.1), a plethora of analytical tools are available (see, for instance, +Appendix B) and numerical simulations are relatively straightforward. However, being a determin- +istic equation, this RDE does not show the same stochastic dynamics as the Gillespie simulations +and experiments. A crucial difference between the macroscopic RDE model (1.1) and the Gillespie +simulations revolves around the duration of the patterns. In the RDE, an established pattern, e.g. +a standing or travelling wave, will, if uninterrupted, remain there for a very long time, while these +patterns are destroyed quickly both in stochastic simulations and experiments. Furthermore, when +the rest state of the RDE (1.1) is stable, activation cannot come from the RDE itself, but it needs +an external signal large enough to activate the nonlinear term a3u2/(a4 + u2) related to Reaction +#3. We generally refer to the activation of these patterns as activation events. +It is important to realise that the dynamics of the different chemical processes in the cell are +inherently stochastic and at the size of a single cell chemical reactions are not well approximated +by large-scale approximations, as Figures 1.1 and 1.2 show. In other words, treating the relevant +enzymes and receptors like a continuous medium of infinitely many, infinitely small, particles is +invalid, and the stochastic nature of reactions between individual molecules becomes important. +This so-called internal noise can serve as a signal to activate the dynamics if it is large enough at a +1Note that (1.1) can also be obtained from a quasi-steady-state approximation from a more complex three- +component model introduced in [2], commonly referred to as a Signal Transduction Excitation Network. +3 + +(a) +(b) +Fig. 1.2: Comparison of the deterministic model (1.1) and its stochastic counterpart (1.2). In Figure (a) we +show a simulation of (1.1), which is excited at t = 0, resulting in two counterpropagating travelling waves. +In the stochastic simulation in Figure (b), the influence of the initial excitation quickly disappears and new +pulses appear constantly. The same parameters are used as in the simulations shown in the second row of +Figure 1.1. Observe the similarities in the shape of the pattern. In Figure (a), the waves travel around the +cell where they cancel each other, while in Figure (b) the waves cancel each other at a much shorter scale. +See §3.4 for more details. +certain point in space and time. As we noted before, the cell hence executes a random walk in the +absence of a signal2. This implies that an external signal does not necessarily activate the dynamics +at a certain point on the membrane, but rather changes the random walk of the cell into a biased +random walk in the direction of the signal. Using a more extended model than presented here, it is +shown in [2] that coupling an external signal to the stochastic dynamics of the cell indeed can lead +to movement in the direction of that signal. +Instead of studying the complex internal dynamics of the cell, it can be advantageous to perturb +the deterministic RDE (1.1). For instance, in [1], an external source of noise is applied to the +RDE (1.1), turning it into a Stochastic RDE (or Stochastic Partial Differential Equation (SPDE)). +While this approach can indeed activate the dynamics and make long-term deterministic waves +collapse, it is inherently ad hoc and not a priori based on any of the involved biologically relevant +processes. +In between the macroscopic level of the RDE and the microscopic level of the chemical reactions, +one can derive a mesoscopic SPDE, known as a Chemical Langevin Equation (CLE) [18], that also +incorporates the internal noise of the cell. In §2, we will show that the SPDE associated with the +chemical reactions as described in Table 1 plus diffusion is given by +du = +� +Du∂xxu − (a1 + a2v)u + +a3u2 +a4 + u2 + a5 +� +dt + σ +� +(a1 + a2v)u + +a3u2 +a4 + u2 + a5 dW 1 +t ++ σ∂x +� +2Duu d ˜W 1 +t , +dv = (Dv∂xxv + ε(−c1v + c2u)) dt + σ +� +ε(−c1v + c2u) dW 2 +t + σ∂x +� +2Dvv d ˜W 2 +t . +(1.2) +Here, (dW 1 +t , dW 2 +t ) and (d ˜W 1 +t , d ˜W 2 +t ) are two independent noise vectors with space-time white noise +(each component is also independent of the other) and σ is a measure for the strength of the noise. +Indeed, in the no-noise limit σ → 0 the mesoscopic SPDE (1.2) reduces to the macroscopic RDE (1.1). +In that sense, σ serves as a scale parameter. +The main advantage of the SPDE description is, on one hand, that the solutions still show +the rich dynamics of the Gillespie models, i.e. the activation and destruction of waves, but are +computationally significantly less expensive. On the other hand, since the SPDE in the no-noise +2Describing the motion of free cells is a very subtle problem and random motion does not necessarily mean Brownian +motion [27, 31]. +4 + +40 +6 +20 +4 +X +0 +2 +-20 +-40 +0 +0 +5 +10 +15 +20 +t30 +20 +4 +10 +3 +X +0 +2 +-10 +-20 +-30 +0 +10 +20 +30 +40 +50 +tlimit reduces to the deterministic RDE model (1.1), we can use well-developed Partial Differential +Equation (PDE) theory to gain insight into the dynamics of the RDE (1.1) and use this to study +the closely related SPDE, see for instance [19, 25]. To give an idea of the differences between the +deterministic and stochastic models we plot two simulations in Figure 1.2 that will be discussed +later in §3. It is clear that the simulation of the SPDE paints a much more dynamic picture than +the deterministic one, which is more in line with the inherently noisy nature of the cell’s chemical +processes. Hence, SPDEs are an invaluable tool in unravelling the dynamics of a cell. +This article is now organised as follows. In §2 we explain how to derive the SPDE (1.2) from +Table 1. Subsequently, in §3 we study both the SPDE (1.2) and the RDE (1.1) numerically in different +parameter regimes and qualitatively compare the observed dynamics to the Gillespie simulations +from [1]. In §4, we discuss the results and how they relate to the questions posed in this introduction. +2 +Derivation of the SPDE +Our starting point to derive (1.2) is the set of chemical reactions as laid out in Table 1. First, we +introduce the column vector X(t) = (u(t), v(t)))T , where T indicates that we transpose the row +vector, and the column vector R(X(t)) with the propensities of the six reactions: +R(X(t)) = +� +a1u(t), a2u(t)v(t), +a3u(t)2 +a4 + u(t)2 , a5, εc1v(t), εc2u(t) +�T +. +The associated stoichiometric matrix S, which describes the change in X(t) for each reaction, is +then given by +S = +� +−1 +−1 +1 +1 +0 +0 +0 +0 +0 +0 +−1 +1 +� +, +(2.1) +see the last two columns of Table 1. On top of these reactions, we assume that both variables also +diffuse, so for a well-mixed solution in a large container we find the classic PDE +∂tX = D∂xxX + SR(X), +(2.2) +where D is a diagonal diffusion matrix with coefficients Du and Dv on the diagonal [3]. This PDE +is identical to the RDE (1.1) and describes the dynamics of X(t), averaged over many individual +reactions. When the number of reacting molecules is large enough, and when we zoom out far enough +such that all individual molecules become effectively a density, the macroscopic PDE gives a good +approximation of the microscopic behaviour. Statistically speaking, this means that the probability +distribution of all possible states must be very sharply peaked around the average value described +by the PDE, so the deviations from the mean can be ignored. +2.1 +Motivating Example +The assumption that we can ignore deviations from the mean is not always valid. For example, in +population dynamics, we can write down birth-death models for several hundred individuals and with +this number of individuals, random deviations from the mean are actually significant. To further +exemplify, and to set the stage for the upcoming derivation, let us study such a simple discrete +birth-death process: suppose a population is at time t in state X(t). In the next timestep dt, there +are three possible outcomes: (i) the population grows by one individual with probability b(X(t))dt, +(ii) the population decreases by one individual with probability d(X(t))dt, or (iii) nothing happens +to the population with probability 1 − b(X(t))dt − d(X(t))dt. +Now, assume we have a continuous Stochastic Differential Equation (SDE) +dx(t) = f(x(t))dt + g(x(t))dβt, +(2.3) +5 + +where βt is Brownian motion, i.e. we can think of dβt as a random step with average zero and +variance dt. We now ask the question: “When is this continuous SDE a good approximation of the +described discrete birth-death process?”. Or, more precisely, “What should f(x) and g(x) be such +that (2.3) is a good approximation of the described discrete process?”. Given a solution x of the +SDE, we see that the average expected value at x(t + dt) is approximated, at lowest order in dt, by +E[x(t + dt)] = x(t) + f(x(t))dt + O(dt2). +For the described birth-death process, we have that the expectation is +E[X(t + dt)] = X(t) + [b(X(t)) − d(X(t))]dt. +Hence, the average expected jump size in population is identical for the SDE (2.3) and the birth- +death process if we take f(x) := b(x) − d(x). +Next, we compute the deviation from the mean of the SDE (2.3) +Var[x(t + dt)] = Var[g(x(t))dβt] + O(dt2) = g(x(t))2dt + O(dt2) , +while this deviation for the birth-death process is +Var[X(t + dt)] = b(x(t)) + d(x(t)) + O(dt2). +Therefore, to make these deviations coincide at first order in dt, we must take g(x) := +� +b(x) + d(x). +Hence, the process x(t) described in (2.3), which is continuous in population size and time, is a good +approximation of the discrete process X(t) when +dx(t) = (b(x(t)) − d(x(t)))dt + +� +b(x(t)) + d(x(t))dβt. +(2.4) +The stochastic process x(t) shares the average and variance with X(t) but differs in other points. +Higher order moments of x(t) and X(t) will not be identical and x(t) can become negative, even +when b and d are chosen such that this is not possible in the discrete model. +In order to link the SDE above to chemical reactions, we make the following observation. The +birth of an individual can be thought of as the chemical reaction ∅ → X with propensity b(X) and +stoichiometric value 1, while the death of an individual can be seen as the chemical reaction X → ∅ +with propensity d(X) and stoichiometric value −1. Next, we make an assumption which is called +the leap condition [3]. That is, we assume that, given a state X(t), enough reactions happen in the +interval [t, t+dt] to describe the average jump size in [t, t+dt] by a Poisson process whose parameters +depend on X(t). With this leap condition assumption, we implicitly also assume that X(t) is a good +approximation of the solution in the whole time interval [t, t + dt]. We now turn the discrete process +X(t) into a continuous process x(t) by approximating the discrete Poisson process by a continuous +Gaussian, see [24] for details. This approach results in an SDE similar to the SDE (2.4): +dx(t) = (b(x(t)) − d(x(t)))dt + +� +b(x(t))dβ1 +t − +� +d(x(t))dβ2 +t , +(2.5) +for two independent Brownian motions β1 +t and β2 +t . Although visually different from (2.4), both +SDEs have a noise term that is Gaussian with identical average and variance. Therefore, both SDEs +describe the same stochastic process and hence we can say that (2.4) and (2.5) are equivalent. +2.2 +Derivation of the CLE +We have now gained some intuition for linking more general discrete chemical reactions to continuous +S(P)DEs: if we have M different molecules in a vector X(t) with diffusion matrix D, N reactions +given by a vector R(X(t)) and a stoichiometric matrix S, then the continuous SPDE for X(t) is +given by +dX(t) = (D∂xxX(t) + SR(X(t))) dt + +1 +√ +Ω +S +� +diag(R(X(t)))dWt + +1 +√ +Ω +∂x +� +2DX(t)d ˜Wt, +(2.6) +6 + +see [3, 24]. The equation is made of two parts, a local equation that describes the kinetics as in SDE +(2.4) +dX(t) = SR(X(t))dt + +1 +√ +Ω +S +� +diag(R(X(t)))dWt +(2.7) +and a stochastic diffusion equation +dX(t) = D∂xxX(t) + +1 +√ +Ω +∂x +� +2DX(t)d ˜Wt, +(2.8) +as derived in [10]. Here, dWt and d ˜Wt are two independent vectors with space-time white noise. The +vector dWt has N components coming from the N reactions, while d ˜Wt has the dimension M of X(t). +SPDE (2.6) is known as the Chemical Langevin Equation (CLE) [18]. The vector X(t) now describes +the densities of the molecules involved, not the actual number of molecules. How well the discrete +number of molecules is approximated by a density is determined by the scale parameter Ω and is +in that sense a measure for the noisiness of the system. In the no-noise limit Ω → ∞, we recover +the classic RDE (2.2). In contrast, for small Ω the dynamics of the discrete process is dominated +by random events and the discrete process should be described in full detail by a chemical master +equation [17]. The CLE can be understood as the lowest order approximation of the chemical master +equation for large Ω, see for more details [3]. For an overview of all different paths leading from +molecular kinetics to (S)PDEs, see [26, Fig. 3.4]. It is important to realise that SPDE (2.6) does not +necessarily inherit all the statistical properties of the chemical master equation, only averages and +variances. Another potential issue is that it does not necessarily ensures positivity of the solutions. +Just as (2.4) and (2.5) are identical, we can rewrite (2.6) in the following way: +dX(t) = (D∂xxX(t) + SR(X(t)))dt + +1 +√ +Ω +� +Sdiag(R(X(t)))ST dWt + +1 +√ +Ω +∂x +� +2DX(t)d ˜Wt. (2.9) +This time, the noise vector dWt has just M components, reducing the number of random vectors +that must be generated (when M < N). The downside is that the computation of +� +Sdiag(R(X))ST +is in general numerically more expensive then the computation of S +� +diag(R(X)). However, in the +present setting, there are no connections between the two variables in the stoichiometric matrix +S (2.1) and the matrix Sdiag(R(X))ST is thus diagonal, making the computation of the square root +trivial. +Note that once we have the CLE (2.9), it can be applied to any set of chemical reactions and +can therefore have widespread use. For example, we can now return to Table 1 and apply the CLE +to these reactions, which results in +du = +� +Du∂xxu − (a1 + a2v)u + +a3u2 +a4 + u2 + a5 +� +dt + σ +� +(a1 + a2v)u + +a3u2 +a4 + u2 + a5dW 1 +t ++ σ∂x +� +2Duud ˜W 1 +t , +dv = (Dv∂xxv + ε(−c1v + c2u)) dt + σ +� +ε(−c1v + c2u)dW 2 +t + σ∂x +� +2Dvvd ˜W 2 +t . +(2.10) +For notational convenience, we replaced 1/ +√ +Ω by a small parameter σ, resulting in the SPDE (1.2) +from the introduction. In the remainder of this work, we will study the SPDE above, mainly using +numerical techniques. +Remark 1. It is important to realise that the SPDE above does not have a function-valued solution in +general. The term ∂x +� +2DX(t)d ˜Wt makes the equation ill-posed and solutions can only be understood +in terms of distributions. Therefore, it is not a priori clear if the numerical solutions shown in the +next section converge to a solution of the SPDE when the spatio-temporal discretisations dx and dt +are sent to 0. In §3.4, we will discuss the implications of omitting this term on the wave dynamics. +7 + +(a) +(b) +Fig. 3.1: (a) The green line is the v-nullcline for c1 = 0.18, while the red line is the nullcline for c1 = 0.35. The +blue line is the u-nullcline, independent of c1. The u-axis is plotted logarithmically to better highlight the shape +of the nullcline for small u. Note how the background state moved around the fold. (b) Visual representation +of the evolution of the two (complex) eigenvalues of the Jacobian matrix (3.2) for c1 varying from 0.18 (dark +blue) to 0.35 (yellow), following the black arrows. The other parameters are fixed at a1 = 0.167, a2 = 16.67, +a3 = 167, a4 = 1.44, a5 = 1.47, ε = 0.52 and c2 = 3.9. +3 +Simulations +In this section, we will numerically investigate the PDE (1.1) and SPDE (2.10). We investigate three +of the main building blocks of the PDE dynamics: localised standing waves, localised travelling +waves and time-periodic solutions, together with their counterparts in the SPDE. However, before +we can investigate the dynamics, we must first establish some basic properties of the (S)PDE, like +the existence, uniqueness and stability of the background state(s). +Since we are interested in localised waves and expect the activator to be in rest otherwise, we +need for the existence of these localised waves that the spatially homogeneous background state is +stable. In contrast, for the time-periodic solutions, we expect the background state to be unstable +such that continuous excitations of the background state can happen. The possible background states +(u∗, v∗) of (1.1) are given by the positive real solutions of the u-nullcline and v-nullcline +0 = −a1u − a2uv + +a3u2 +a4 + u2 + a5 , +0 = ε(−c1v + c2u) . +(3.1) +See Figure 3.1a for a typical representation of the shape of the nullclines. Since the system parameters +are all assumed to be positive, this is equivalent to finding the positive solutions u∗ of +−a2c2 +c1 +u4 − a1u3 + +� +a3 + a5 − a2a4c2 +c1 +� +u2 − a1a4u + a5a4 = 0 , +with v∗ = c2u∗/c1. Due to the complexity of the general solution formula for quartic polynomials, +it is not feasible to write down its solutions explicitly. However, by Descartes’ rule of signs [7] we +know that there is only one positive real root if c1(a3 + a5) < a2a4c2 and one or three positive real +roots otherwise3. The stability of a background state (u∗, v∗) is then determined by the eigenvalues +of the associated Jacobian matrix +J(u∗, v∗) = +� +�−a1 − a2v∗ − +2a3a4u∗ +(a4 + (u∗)2)2 +−a2u∗ +εc2 +−εc1 +� +� . +(3.2) +Since we do not have an explicit formula for (u∗, v∗), we must compute these eigenvalues numerically. +For example, when we allow one free parameter, e.g. c1, and fix the other values, then we can compute +3Note that the origin (0, 0) is only a background state if a5 = 0. +8 + +the background states and the associated eigenvalues of the Jacobian matrix. Taking the parameter +values a1 = 0.167, a2 = 16.67, a3 = 167, a4 = 1.44, a5 = 1.47, ε = 0.52 and c2 = 3.9 from [1] and +letting c1 range from 0.18 to 0.35, such that c1(a3 + a5) < a2a4c2, results in one admissible positive +background state ranging from (u∗, v∗) ≈ (0.077, 1.669) to (u∗, v∗) ≈ (0.142, 1.586). Initially, for +the lower values of c1, the eigenvalues are real and negative, resulting in a stable background state. +Increasing the value of c1 to approximately 0.25 results in complex eigenvalues, still with negative +real parts. When we further increase the value of c1 to approximately 0.29, both eigenvalues cross the +imaginary axis, i.e. the background state undergoes a Hopf bifurcation and we expect to see time- +periodic solutions. See Figure 3.1b for a visual representation of the evolution of the eigenvalues. In +Figure 3.1a we show the nullclines for c1 = 0.18 and c1 = 0.35. The unique background state moved +along the fold in the u-nullcline and as long as the background state is in between the two folds, the +fixed point is unstable. +In the next sections, we will study localised standing and travelling waves for the same parameter +set with c1 < 0.25 and for time-periodic solutions with c1 > 0.29. The complex dynamics of pulse +adding for c1-values in the intermediate regime between these two boundary values, where the +eigenvalues of the Jacobian are stable but complex-valued, is outside the scope of this work, see for +example [4] for more information. +So far, we only looked at background states, which are spatially homogeneous. However, we are +interested in spatially nonhomogeneous patterns. By definition, a localised wave is a fixed profile +(Φu, Φv) that moves with a fixed speed c (possibly zero). Therefore, when we change the spatial +coordinate x to ξ = x − ct using the chain rule, the profile (Φu, Φv) is a stationary solution of the +following shifted Ordinary Differential Equation (ODE): +0 = Du∂ξξΦu + c∂ξΦu − (a1 + a2Φv)Φu + +a3Φ2 +u +a4 + Φ2u ++ a5, +0 = Dv∂ξξΦv + c∂ξΦv + ε(−c1Φv + c2Φu). +(3.3) +This ODE problem can be solved using numerical fixed-point algorithms. For these algorithms, +a crude starting point is needed for the profile and the value of c, which can come from a PDE +simulation. Note that this problem is translation invariant, meaning that we find a one-dimensional +family of travelling waves, all shifted versions of each other. Hence, for the solver to converge, an +extra condition to fix the location of the wave is necessary. +3.1 +Standing Waves +In this section, we will study standing waves, which means we look for solutions of (3.3) with c = 0. A +solution to this ODE is shown in Figure 3.2a. We observe that both components u and v indeed start +at and return to their background state (u∗, v∗) ≈ (0.0523, 2.0394). We observe that the activator u +changes rapidly in a small region in the spatial domain and we, therefore, call the activator u the +fast variable. On the other hand, the inhibitor v is the slow variable as it changes more gradually +over a larger spatial distance. Figure 3.2b shows the corresponding phase plane. The majority of +the spatial dynamics happens near the lower branch of the u-nullcline before it has a fast excursion +from the lower branch to the upper branch of this nullcline and, by the symmetry x �→ −x of +the ODE (3.3), it then returns back to the lower branch in a similar fashion. The fact that both +components of the standing pulse evolve on a different spatial scale allows us to mathematically +analyse this standing pulse, see Appendix B. For instance, the value ¯v at which the activator u +makes a sharp transition (approximately 3.8 in Figure 3.2b), can be approximated by the algebraic +relation (B.10). The analysis also explains why the solution trajectory in the phase plane closely +follows the lower branch of the u-nullcline for the most part of the trajectory. +By assumption, the standing wave in Figure 3.2a is a stationary solution of the PDE (1.1). This +can be confirmed by using the wave from the ODE as the initial condition for a PDE simulation +(not shown). However, we are not likely to find this single standing wave in a PDE simulation +9 + +(a) +(b) +Fig. 3.2: Figure (a) shows a localised standing wave solution to ODE (3.3), found numerically with Matlab’s +fsolve. The green curve is the u-component and the red curve is the v-component. In Figure (b), the u- +nullcline (blue) and v-nullcline (green) of (1.1) are shown together with the v-u phase plane of the standing +wave from Figure (a). The phase plane is plotted on a semi-log scale to better highlight the dynamics for +small u. We observe that the standing wave starts from the background state (indicated by an asterisk) and +initially follows the lower branch of the u-nullcline before jumping to the upper branch of the u-nullcline and +follows the same track back to the background state. The system parameters are taken from [1] and set to +Du = 0.1, Dv = 1, a1 = 0.167, a2 = 16.67, a3 = 167, a4 = 1.44, a5 = 1.47, ε = 0.52, c1 = 0.1, and c2 = 3.9. +(a) +(b) +Fig. 3.3: Simulation of the PDE (1.1), Figure (a) shows the activator u and Figure (b) the inhibitor v with +an initial condition as described in the main text. The same parameters are used as in Figure 3.2. Note that +the v-component does not return to its rest state in the region between the two pulses. +10 + +30 +4 +20 +10 +3 +X +0 +2 +-10 +1 +-20 +-30 +0 +0 +5 +10 +15 +20 +t30 +4 +20 +10 +3 +X +0 +2 +-10 +1 +-20 +-30 +0 +0 +5 +10 +15 +20 +t(a) +(b) +Fig. 3.4: Same simulation as in Figure 3.3, but on different time scales. Figure (a) shows the u-component, +zoomed in to highlight the short-time dynamics, while Figure (b) shows the long-time dynamics of u high- +lighting the pulse splitting phenomenon. Both simulations were done on a larger grid [−60, 60], so the waves +would not affect each other on the other side of the domain on this large time scale. +without a fine-tuned initial condition. As an example, we use for the simulation the initial condition +u0 = u∗ + e−x2 and v0 = v∗ + 2/ cosh2(5x) as a crude approximation of the wave. The resulting +simulation is shown in Figure 3.3. This initial condition splits in, what appears to be, two well- +separated localised standing waves4. However, the plot of the slow v-component makes clear that +this is not the case, and that the two standing waves are connected through the slow component, +i.e. the slow component is not in its rest state in between the two standing waves. For more details +on the numerics of the (S)PDE simulations, see Appendix A. +The interaction between the two standing waves in Figure 3.3 through the slow v-component +makes that the two standing waves repel each other on a very slow timescale as is made clear by +taking long integration times, see Figure 3.4b. On an infinite domain, the two standing waves slowly +drift apart forever, but on a periodic domain, we can expect them to stabilise once they are at an +equal distance on both sides. On the timescales of biological processes, this slow continuous splitting +is probably not relevant and on short timescales, the term ‘standing waves’ for the solution at later +times in Figure 3.3 is biologically justifiable. Furthermore, note that for our understanding of the +presented dynamics, it is essential to look at both components simultaneously. In other words, for +our understanding of Figure 3.3a it is essential to also look at Figure 3.3b. +We now take a closer look at the short-time dynamics presented in Figure 3.4a. In [1], this +splitting of the initial condition is described as two counter-propagating travelling waves, sometimes +called trigger waves [14]. By the formal mathematical definition, a travelling wave is a fixed profile +moving with a fixed speed, i.e. a solution of (3.3). Therefore, mathematically speaking, these do not +classify as travelling waves. Instead, what we observe here would be classified as transient dynamics +and pulse splitting. However, it is clear that at t = 0, the activity of u is around x = 0, and after +some time it moved to two different places, justifying the term ‘travelling’. If we adopt the terms +‘standing’ and ‘travelling’, it is clear from Figure 3.3a that around t = 3 a transition occurs from +travelling to standing. +Standing waves with noise +For the same parameter values as in the previous paragraph, we +now study the full SPDE (2.10). In Figure 3.5, we plot realisations of the SPDE for different noise +intensities. For low noise levels, we see two quasi-stationary waves appear, like in Figure 3.3, before +they are destroyed at different points in time by the noise. Since the noise is low, no new activation +events happen. When we increase the noise intensity, the noise is able to activate the stable back- +ground state, but the waves are also destroyed more quickly, resulting in a constant appearance and +disappearance of waves. Note the comparison between Figures 3.5c and the figures in [2], where a +4We also observe the evolution of the initial condition back to the stable background state (u∗, v∗), especially for +initial conditions with smaller amplitudes. Simulations are not shown. +11 + +5 +3 +2.5 +2 +X0 +1.5 +L +0.5 +-5 +0 +1 +2 +3 +4 +t10 +3 +5 +2.5 +2 +X +0 +1.5 +1 +-5 +0.5 +-10 +0 +200 +400 +600 +t(a) σ = 0.002 +(b) σ = 0.035 +(c) σ = 0.05 +(d) σ = 0.5 +Fig. 3.5: The u-component of the SPDE (2.10) for four different values of the noise σ. The other system +parameters and initial conditions are the same as in the previous figures. In Figure (a), we only show the +simulation of wave integrated up to T = 20 because the solution remains in the background state afterwards, +the other three figures are shown up to T = 100. +similar model is studied using Gillespie algorithms. This activation of the background state is not +possible in the deterministic PDE (1.1) without an external force. In Figures 3.5b and 3.5c, we see +that in the first instances, many patterns are generated, causing the inhibitor to increase everywhere +which blocks new activation events. After this initial phase, new activation events appear, and sig- +nificantly more for higher values of the noise as expected. When we increase the noise even further, +it becomes impossible to form patterns as every activation event is destroyed instantly. Therefore, +pattern formation happens at intermediate values of the noise. The idea that there is some ‘opti- +mal’ value of the noise resulting in complex dynamics has been observed before in, for instance, the +context of nerve impulses [13]. +In order to quantify this notion of optimality in the noise intensity we must first quantify the size +and shape of the patterns in Figures 3.5b and 3.5c. Using Matlab’s regionprops algorithm we can +automatically detect the patches with a high value for the activator u (see Appendix A for details), +giving us the possibility to compute the number of activation events and determine the width and +duration of each event, see Figure 3.6a. In Figure 3.6b we show the statistics for a range of σ values. +This figure shows that there is a clear cutoff for when activation events are likely to happen. For +values of σ < 0.035, the average number of events is lower than 1, and the number of activation +events increases sharply after this value. We observe that the width, the length and the maximum +height of the events are all higher when the number of excitation events is low, but the variability +in these values is also larger. in Figure 3.7, we look at the statistics of the events for the specific +value σ = 0.046. The value of the maximum is sharply peaked. This is something we expect, as the +maximum is mainly determined by the deterministic dynamics after the excitation. The width and +length of the events are much more spread out. Especially for the width, we see a heavy tail towards +zero. This is also expected because activation events come in two forms. Most events result in two +waves, but a small part of the events has the shape of just a single wave, which has a width of 0.87 +in the deterministic case. We checked whether or not these histograms are well approximated by a +12 + +30 +5 +20 +4 +10 +3 +X +0 +2 +-10 +1 +-20 +-30 +0 +0 +5 +10 +15 +20 +t30 +5 +20 +4 +10 +3 +X +0 +2 +-10 +1 +-20 +-30 +0 +0 +20 +40 +60 +80 +100 +t30 +5 +20 +4 +10 +3 +X +0 +-10 +-20 +-30 +0 +20 +40 +60 +80 +100 +t30 +5 +20 +4 +10 +3 +X +0 +2 +-10 +1 +-20 +-30 +0 +0 +20 +40 +60 +80 +100 +t(a) +(b) +Fig. 3.6: In Figure (a) we show a simulation similar to those in Figure 3.5, but with σ = 0.45 and a +homogeneous initial condition with (u∗, 4v∗). The red boxes are the result of the pattern finding algorithm +regionprops in Matlab; it identifies all the regions of excitations which we would also find by eye, see +Appendix A for details. In Figure (b), we used this algorithm to find the length, width and maximum of +these pulses (left axis), as well as the total number of activation events (right axis). For each value of σ, the +number of events is averaged over 100 simulations, and the length, width and maximum are averaged over +all events in the 100 simulations. We plot the average together with the standard deviation. +Gaussian distribution, but this was rejected using a Kolmogorov-Smirnov test (p ∼ 10−14). +Using the statistics on the width, length, and maximum, we can compare the solutions of +SPDE (2.10) to SPDEs with the same deterministic part but different noise terms. First, we can set +the ∂x +√ +2DXdWt term coming from the diffusion to zero. As noted in Remark 1, this term makes +the mathematical analysis of the SPDE (2.10) significantly harder. Figures 3.7d-3.7f show that the +statistics of the solutions do not change significantly when we delete this term. This indicates that +the noise coming from the reaction terms plays a more influential role in determining the shape of +the patterns. +We are now also in the position to compare the CLE approach with the more ad hoc approach of +adding additive white noise the to u-component to mimic the inherent noisiness of the system, see +Figures 3.7g-3.7i. Indeed, the properties of the patterns are significantly different when we compare +them to the full SPDE. In particular, with just white noise, the patterns are all short and narrow +and do not reflect the complicated dynamics of the underlying chemical reactions and experiments +(not shown). +3.2 +Travelling Waves +In order to find a travelling wave solution of (1.1), understood as a solution of (3.3) with c ̸= 0, we +must ensure that the dynamics starting from the initial condition does not reach the standing phase +or returns to the background state. This can be achieved by increasing the value of c1. Increasing +c1 results in a faster exponential decay of v back to the background state after an excitation, see +Table 1, preventing the inhibitor from glueing the two waves together like in Figure 3.3. Simulations +for an increased value of c1, from 0.1 to 0.25, are shown in Figure 3.8. Note that the PDE (1.1) +still only has one stable background state (u∗, v∗) ≈ (0.0833, 1.625). The initial condition splits into +two counter-propagating travelling waves, but opposite to what happened with the standing wave +before, they keep separating and move away from each other at a fixed speed until they collide and +cancel each other out due to the periodicity of the domain, see Figure 3.8. +To find a single travelling wave, we again need to properly tune the initial condition. This can be +done by selecting one of the two waves in Figure 3.8 and using it as the initial condition of the PDE +simulation (not shown). In Figure 3.9 we show the travelling wave profile and its associated phase +5For values in between, say c1 = 0.15, the numerics becomes very sensitive to the chosen discretisation, see +Appendix A. +13 + +60 +40 +C +3 +20 +X +0 +2 +-20 +1 +-40 +-60 +C +0 +0 +20 +40 +60 +80 +100 +t(a) +(b) +(c) +(d) +(e) +(f) +(g) +(h) +(i) +Fig. 3.7: Figures (a)-(c) show the histograms for the width, length and maximum of the pulses for σ = 0.046, +for the same data as in Figure 3.6b. For Figures (a) and (b), the bin width is fixed to 0.25, and for Figure (c) +to 0.1. Figures (d)-(f) show the same histograms, but in the simulations, the noise coming from the diffusion +(the last term in (2.10)) was set to 0. In Figures (g)-(i), we again show the same histograms, but with just +white noise on the u-component. In order to compare the noise levels, we did not choose the same σ value +for the three cases but chose σ values such that the average number of activation events per simulation is +approximately 50. For the Figures (d)-(f), this means σ = 0.056, and for (g)-(i) σ = 0.3. +(a) +(b) +Fig. 3.8: Simulation of the PDE (1.1), Figure (a) shows the activator u and Figure (b) the inhibitor v. We +observe the splitting of the initial condition in two counterpropagating travelling waves with a constant speed +that exist until they cancel each other out due to the periodicity of the domain. The slow inhibitor v decays +back to its rest state in between the pulses. The red dotted line has a speed of −2.10, which is close to the value +of approximately −2.17 found by solving (3.3) using a fixed point method. The parameters are Du = 0.1, +a1 = 0.167, a2 = 16.67, a3 = 167, a4 = 1.44, a5 = 1.47, Dv = 1, ε = 0.52, c1 = 0.2 and c2 = 3.9. +14 + +40 +4 +20 +3 +0 +X +2 +-20 +1 +-40 +0 +0 +5 +10 +15 +20 +25 +t40 +4 +20 +3 +0 +X +2 +-20 +1 +-40 +0 +0 +5 +10 +15 +20 +25 +t(a) +(b) +Fig. 3.9: Profile of a single travelling wave. Figure (a) shows both components u (green) and v (red) and +Figure (b) the related phase plane, plotted on a semi-log scale to highlight the dynamics for small u, as well +as the nullclines. The asterisk indicates the fixed point. This solution is obtained as the endpoint of a PDE +simulation (not shown), i.e. similar to Figure 3.8, but with just one of the two waves as initial condition. +plane. As with the standing pulse, the dynamics around the u-nullcline is essential. The solution +trajectory starts from near the background state and follows the lower branch of the u-nullcline, +jumps towards the upper branch of the nullcline and keeps following it until it falls off and returns +to the lower branch to slowly evolve back towards the stable background state. In contrast to the +standing pulse, see Figure 3.2, the travelling wave is no longer symmetric and it jumps back to the +lower branch by falling off the edge of the upper branch. These travelling wave solutions could be +analysed further using techniques similar to Appendix B. +It is important to realise that we do not expect to see travelling waves in practice as the travelling +wave gets destroyed when it collides with another wave. Therefore, in the stochastic simulations, it +might not always be clear if we are looking at a travelling wave that collapses or at the transient +dynamics towards a double pulse that subsequently gets destroyed by the noise. +Travelling waves with noise +When we now return to SPDE (2.10), there are now four regimes +for the same parameters as in the previous section. For high values of the noise, we, as before, do not +observe any patterns (not shown). For low values of the noise, we just find the travelling wave (if the +simulation is initiated by an appropriate initial condition) since the noise is not strong enough to +destroy the wave, nor to activate another pattern, on the timescales of the simulation (not shown). +The interesting dynamics happens again at the intermediate levels of the noise. As Figure 3.10a +shows, the noise activates the dynamics, resulting in many counter-propagating travelling waves. +A travelling wave is subsequently annihilated when it collides with a travelling wave coming from +the other direction. Hence, the collision dynamics of Figure 3.8 is repeated many times on smaller +spatial-temporal scales. We see in Figure 3.10 that after the annihilation of the travelling waves, the +slow inhibitor v initially remains high preventing the activation of new counterpropagating travelling +waves. Only when after a certain time the inhibitor has sufficiently decayed, do we see the activation +of new counterpropagating travelling waves by the noise. The creation and annihilation of travelling +waves happen at a shorter time scale than the decay of the inhibitor, which makes the dynamics +look synchronised, or even periodic. In Figure A.2a we plot the approximate period versus the +intensity of the noise. As expected, the period decreases with the intensity of the noise. It differs +however significantly from the true time periodic motion we will discuss in §3.3. When we increase +the noise, the quasi-periodic pattern is broken up, as the counter-propagating travelling waves are +destroyed before they collide and annihilate each other, so no synchronised patterns emerge, see +Figures 3.10cand 3.10d. These patterns become relevant when we discuss the comparison between +the CLE and the Gillespie simulations in Figure 1.1, see §3.4 +15 + +(a) +(b) +(c) +(d) +Fig. 3.10: Simulation of the SPDE (2.10) for σ = 0.02, Figures (a) and (b), and σ = 0.05, Figures (c) +and (d). The red dashed line in (a) has a slope of 2.05, close to the deterministic wave speed, but given +the short time interval the wave exists, precise estimates are difficult to obtain. We observe that there is a +quasi-periodic behaviour with a period of roughly 20. In Figures (c) and (d), the quasi-periodic structure is +destroyed. The same parameters and initial condition are used as in Figure 3.8. +3.3 +Time Periodic Solutions +In the previous sections, it was essential that the background state of the system was stable, because +this allowed the dynamics to return to the rest state after an activation event. When we increase the +value of c1, the background state becomes unstable through a Hopf bifurcation, see Figure 3.1b. In +the phase plane, this transition is characterised by the fact that the background state is no longer +located on the lower branch of the u-nullcline, as in Figures 3.2b and 3.9b, instead, it lies on the +middle branch of the u-nullcline, see Figure 3.12b. Hence, after an excursion, the solution cannot +return to the unstable background state and is exited again, resulting in time-periodic motion. When +we start with a spatial homogeneous initial condition, the PDE simulation shows periodic oscillations +in time, see Figures 3.11 and 3.12. Both components still display slow-fast behaviour, however, this +time not in the spatial variable x but in the temporal variable t. In the case of nonhomogeneous +initial conditions, it takes several oscillations before they are all synced up spatially (not shown). The +observed behaviour has the characteristics of a relaxation oscillation as studied intensively for the +Van der Pol equation [32]. This is not a surprise as the Van der Pol equation formed the foundation +for the classic FitzHugh-Nagumo model and PDE (1.1) can be seen as a variation on this classic +model. +Time periodic solutions with noise +For small values of the noise σ, the observed period is +close to the deterministic version, but when the value of σ increases, the period also decreases +monotonically, as is expected. Note that after excitation, the inhibitor remains high preventing +activation events. When the noise is too high no patterns are observed. We can investigate the +relation between the reduction of the period and the intensity of the noise. In FigureA.2b, we plot +the estimated period versus the noise intensity. We indeed see that the period decreases monotonically +16 + +30 +20 +4 +10 +3 +X +0 +2 +-10 +1 +-20 +-30 +0 +20 +40 +60 +80 +100 +t30 +20 +4 +10 +X +0 +3 +-10 +-20 +2 +-30 +0 +20 +40 +60 +80 +100 +t30 +20 +5 +10 +4 +X +3 +-10 +-20 +-30 +0 +20 +40 +60 +80 +100 +t30 +0 +20 +5 +10 +4 +X +0 +3 +-10 +-20 +-30 +0 +20 +40 +60 +80 +100 +t(a) +(b) +Fig. 3.11: Simulation of the PDE (1.1), Figure (a) shows the activator u and Figure (b) the inhibitor v. By +measuring the distances between the maxima of the oscillations we find the estimate T = 8.14 for the period +of the oscillation. Note that this is significantly smaller than the quasi-periodic oscillations in Figure 3.10. +The parameters are set to Du = 0.1, a1 = 0.167, a2 = 16.67, a3 = 167, a4 = 1.44, a5 = 1.47, Dv = 1, +ε = 0.52, c1 = 0.4 and c2 = 3.9. +(a) +(b) +Fig. 3.12: Cross-section of Figure 3.11 at x = 0, together with the corresponding phase plane. It is clear that +the solution leaves the background state (marked by an asterisk), but does not return to it. +17 + +30 +4 +20 +10 +3 +X +0 +2 +-10 +1 +-20 +-30 +0 +5 +10 +15 +20 +25 +t30 +20 +4 +10 +X +0 +3 +-10 +-20 +2 +-30 +0 +5 +10 +15 +20 +25 +t(a) +(b) +Fig. 3.13: Simulation of the SPDE (2.10). Figure (a) shows the activator u and Figure (b) the inhibitor v. +When we average over the x-direction and measure the distance between the maxima, we find T ≈ 7.87. +Same parameters as in Figure 3.11 with σ = 0.01. +(a) Wild Type +(b) PTEN-null +Fig. 3.14: Two simulations of PDE (1.1) with parameters as in [1]; Du = 0.1, a1 = 0.167, a2 = 16.67, +a4 = 1.44, a5 = 1.47, Dv = 1, ε = 0.4, c1 = 0.1 and, for Figure (a), a3 = 167 and c2 = 2.1, while a3 = 300.6 +and c2 = 3 for Figure (b). The initial condition is equal to those in the previous figures. +with the noise. +3.4 +Wild-Type versus PTEN-null Cells. +Now that we have studied several different fundamental patterns, we can focus on understanding +the different cell shapes. In [1], two sets of parameters are compared, representing WT cells (i.e. +healthy cells) and PTEN-null cells where the tumour-suppressing gene PTEN has been switched +off[5]. First, we simulate the deterministic PDE (1.1) for both sets of parameters, see Figure 3.14. +We observe that in both parameter regimes, there are two counter-propagating travelling waves +but the specific profiles and speeds are different. Especially, note that the wave in Figure 3.14b is +significantly broader and higher than the wave in Figure 3.14a. +When noise is applied, the statistics of the dynamics shows a clear difference. In Figure 3.15, +we compare the SPDE simulations of (2.10) to the Gillespie simulations from [1]. Focusing on the +typical shape of the excitations, there is a clear qualitative correspondence between the two types +of simulation. Furthermore, in both types of simulation, the average pulse duration is longer in the +case of the PTEN-null cell simulations. Note that we show the SPDE simulations on a larger spatio- +temporal scale to get a better idea of the distribution in shapes and the zoom-boxes highlight the +detailed structure of a typical single activation event. In the case of PTEN-null cells, the background +state can be excited for much lower noise values (σ ≈ 0.007), while for WT cells, the noise needs to +be twice as large (σ ≈ 0.014) as a result of the increased values of c2 and a3. Hence, in PTEN-null +cells, an already existing pattern can more easily sustain itself, leading to the elongated shapes of +Figure 3.15d. +18 + +40 +6 +20 +4 +X +0 +2 +-20 +-40 +0 +0 +5 +10 +15 +20 +t30 +5 +20 +4 +10 +3 +X +0 +2 +-10 +-20 +-30 +0 +5 +10 +15 +20 +25 +t30 +20 +4 +10 +X +0 +3 +-10 +-20 +2 +-30 +0 +5 +10 +15 +20 +25 +t40 +6 +20 +4 +X +0 +2 +-20 +-40 +0 +0 +5 +10 +15 +20 +t(a) +(b) +(c) +(d) +Fig. 3.15: Comparison of the Gillespie model, Figures (a) and (c) from [1], versus the CLE approxima- +tion (2.10), Figures (b) and (d). The same parameters as in Figure 3.14, with σ = 0.06. The initial condi- +tion is (u∗, 2v∗). This can lead to an immediate excitation of the background state in Figure (d), while in +Figure (b), the excitation of the background is more spread out. The zoom-boxes highlight the details of a +single excitation. +4 +Discussion & Outlook +We set out to show how Stochastic Partial Differential Equations (CLE), or more specifically, Chem- +ical Langevin Equations, can be used to gain more insight into the dynamics of models for cell +motility. We have shown for an exemplary set of chemical reactions (see Tabel 1) that the CLE ap- +proach, combined with a basic analysis of the corresponding deterministic PDE, allows us to study +the different possible patterns with relative ease, both qualitative and quantitative, while remain- +ing close to the underlying chemical processes. To understand differences in cell behaviour, like the +difference between wild-type and cancerous cells as in [1], the study of the statistical properties of +the observed dynamics is essential. For instance, an essential characteristic differentiating wild-type +cells from cancerous cells is how long a pattern can survive after activation. The simulations in +the previous section show that the answer not only depends on the parameters of the system but +crucially on the interplay between the parameters and the noise. The CLE can be used to study +this interplay. A natural question to ask is if all the stochastic terms introduced in the CLE (2.9) +are really necessary. Could we, for example, ignore the noise term coming from the diffusion or +forget the derivation of the CLE altogether and just naively add an additive white noise term to the +equation for u? The histograms in Figure 3.7 indicate that the effects of the terms that come from +the diffusion are minimal (for the parameter values studied here) and therefore that these terms do +not contribute meaningfully to our understanding of the cell dynamics. Note that this would solve +the problem of the equation being ill-posed, see Remark 1, and would open up the possibilities for +more rigorous mathematical analysis based on the results in [19]. We also noted that adding just +additive white noise changes the statistics significantly, which indicates that completely abandoning +the CLE approach throws away too much detail. +In this paper, we studied a basic activator-inhibitor system with only a limited number of chemical +19 + +30 +20 +4 +10 +3 +X +0 +2 +-10 +-20 +-30 +0 +10 +20 +30 +40 +50 +t30 +20 +4 +10 +3 +X +0 +2 +-10 +-20 +-30 +0 +0 +10 +20 +30 +40 +50 +treactions. However, the derivation of the CLE (2.9) in §2 holds for any number of molecules and +for any number of chemical reactions. As such, one can see this paper as a proof of concept and the +methodology of this paper can be directly applied to more complex regulating systems, such as the +eight-component system designed in [2]. In subsequent work, we aim to work on these type of more +complex model to better understand the stochastic dynamics that causes the cell to move robustly +in one specific direction. +Furthermore, as shown in detail in Appendix B, the underlying deterministic RDE (1.1) is +amenable for rigorous mathematical analysis by using Geometric Singular Perturbation Theory [11, +20, 22, 23, e.g.]. We derived a first-order approximation for the jump location where, under certain +conditions, the standing wave has a sharp transition in its activator. This methodology could also +be used to, for instance, further analyse the travelling waves to derive approximations for the speed +of the waves. In other words, questions about the existence of localised solutions of (1.1) and bifur- +cations can thus be reduced to understanding relatively simple ODEs and the connections between +them. The details of these computations are left as future work. +References +[1] S. Bhattacharya, T. Banerjee, Y. Miao, H. Zhan, P. N. Devreotes, and P. A. Iglesias (2020), +Traveling and standing waves mediate pattern formation in cellular protrusions. Science Ad- +vances 6(32), eaay7682. +[2] D. Biswas, S. Bhattacharya and P. A. Iglesias (2022), Enhanced chemotaxis through spatially +regulated absolute concentration robustness. International Journal of Robust and Nonlinear +Control. +[3] P. C. Bressloff (2014), Stochastic Processes in Cell Biology, Vol. 41. Springer. +[4] P. Carter, B. de Rijk and B. 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Van der Pol (1926), On “relaxation-oscillations”. +The London, Edinburgh, and Dublin +Philosophical Magazine and Journal of Science 2(11), 978–992. +22 + +A +Numerical Methods +A.1 +(S)PDE Simulations +All the (S)PDE simulations in this paper were done using a semi-implicit Euler–Maruyama method +from [29, §10.5]. For the spatial directions, a standard 2nd-order central difference is used and for +the time stepping Euler-Maruyama. The deterministic linear part is evaluated at the next timestep, +making it semi-implicit. To be concrete, we study an SPDE of the form +du = [Lu + f(u)]dt + g(u)dWt, +where u is a vector, L is a linear differential operator, f, g are functions and dWt a white noise +vector. In comparison with the main text, the vector u equals (u, v) and L = D∂xx. +When we denote the numerical approximation of the linear part L with A, and the spatial +discretisation of u at time t with u(t), we find +u(t + dt) = u(t) + dt[Au(t + dt) + f(u(t))] + g(u(t))dWt. +(A.1) +The white noise step dWt is a vector where each random element is distributed as N(0, dt/h). Hence, +the approximation for the new value u(t + dt) becomes +(I − dtA) u(t + dt) = u(t) + dtf(u(t)) + g(u(t))dWt. +(A.2) +The equation for u(t + dt) is now a matrix equation and can be solved using standard solvers. It is +important to realise that the algorithms from [29] only work for Lipschitz noise terms. Hence, when +the term under the square root in (2.10) becomes close to zero, the algorithms become unstable. To +correct this, we take after every timestep the maximum of u(t + dt) and 0. +The specific models studied in the main text, even the PDE (1.1) can be very sensitive to the +size of the spatial discretisation h and temporal discretisation dt in certain parameter regimes. For +example, when c1 = 0.15 and the remaining parameters are equal to those in Figures 3.3 and 3.8, the +dynamics can differ significantly depending on the chosen size of the discretisation, see Figure A.1. +For the values used in the main text, c1 = 0.1 and c1 = 0.2, such a discrepancy was not observed +for reasonable discretisations. This is possibly related to the co-existence of travelling and standing +waves in this regime. +(a) +(b) +Fig. A.1: Simulation of the PDE (1.1). The spatial domain is [−60, 60] with 212 gridpoints, the time interval +is [0, 25]. In Figure (a), we used dt = 6.25 · 10−4 and in Figure (b) dt = 0.0025, i.e. 4 times larger. The +parameters were set at Du = 0.1, a1 = 0.167, a2 = 16.67, a3 = 167, a4 = 1.44, a5 = 1.47, Dv = 1, ε = 0.52, +c1 = 0.15 and c2 = 3.9. In both cases, the initial condition is equal to the one in the main text. +23 + +60 +40 +3 +20 +2 +X +0 +-20 +1 +-40 +-60 +0 +5 +10 +15 +20 +25 +t60 +40 +3 +20 +2 +X +0 +-20 +T +-40 +-60 +0 +5 +10 +15 +20 +25 +t(a) c1 = 0.2 +(b) c1 = 0.4 +Fig. A.2: This figure shows the period of the dynamics of SPDE (2.10) in two different regimes. In Figure (a), +with c1 = 0.2, we are in the regime of travelling waves with quasi-periodic movement as shown in Figure 3.10, +while in Figure (b), with c1 = 0.4, we are in the regime of oscillations in time as shown in Figure 3.13. In both +figures, the period is estimated by computing the average in the spatial direction and subsequently computing +the distances between the maxima in time. +A.2 +Pattern Recognition +For Figures 3.6 and 3.7, Matlab’s regionprops algorithm is used to identify the activation events +automatically. This proceeds in the following steps. First, we smooth the data using Matlab’s Gaus- +sian filter. Without smoothing, the algorithm detects multiple objects in a single event. Next, we +transform the data to a binary value by comparing it with a certain threshold: we say that u is +activated when it is five times its stationary value u∗. Then, the regionprops algorithm is applied +with the option BoundingBox. +One needs to take care of which initial condition to use. When we start with a spatial homoge- +neous initial condition (u∗, v∗), there is a lot of activation in the first instances of the simulation, see +Figure 3.5c, and it is not possible to define and detect individual activation events. Therefore, we +start not on the fixed point (u∗, v∗), but on (u∗, 4v∗) plus a small perturbation. The result is that +activation events only appear when v has decayed enough for excitations to happen. As the decay +is stochastic, and therefore not spatially homogeneous, the activation events start to appear more +spread out, making it possible to determine individual events. +B +Analysis +The deterministic PDE (1.1) has two components and ten parameters, making it difficult to directly +analyse mathematically, even for the simplest of localised structures simulated in the main text. +However, these simulations do reveal that the profiles of the two components of the PDE evolve on +a different spatial scale: the spatial changes in the slow v-component are more gradual than these of +the fast u-component, see, for instance, Figures 3.2a, 3.9a, and 3.12a. Furthermore, these simulations +also revealed that a large part of the spatial dynamics centres around the lower and upper branch of +the u-nullcline in the phase plane, with the u-profile making fast jumps in between. To amplify (and +exploit) this scale separation, we set Dv = 1 (as in [1]) and Du = ˜ε2, where ˜ε is a small parameter +that can be taken arbitrary small. Furthermore, we assume that our spatial domain is no longer +periodic but instead unbounded6. This transforms the PDE model (1.1) into +∂tu = ˜ε2∂xxu − a1u − a2uv + +a3u2 +a4 + u2 + a5 +∂tv = ∂xxv + ε(−c1v + c2u). +(B.1) +6It is relatively straightforward to generalise the results for the unbounded domain to the periodic domain for the +type of problems under consideration, see for example [9] +24 + +The small parameter ˜ε allows us to use Geometric Singular Perturbation Theory (GSPT) [11, 20, +22, 23, e.g.] to construct solutions that, to leading order in the small parameter, approximate the +localised structures of the main text. +In GSPT, the observation that the dynamics centres around the branches of u-nullcline is taken +to the extreme and we construct solutions whose slow dynamics in the singular limit, i.e. in the limit +of the small parameter ˜ε to zero, is confined to this nullcline, which we will refer to as the slow or +critical manifold. In contrast, during the fast jump in u, the slow component will not change in this +singular limit. These assumptions simplify the computations and allow us to compute parts of the +solution in the singular limit. The main theorems of GSPT [11, 20, 22, 23, e.g.], sometimes called +Fenichel Theorem 1-3, allow us to conclude that if the small parameter is small enough7, then there +indeed is a true solution of the PDE close to the one constructed in the singular limit. +Here, we only show the construction of the standing waves we found in §3.1. That is, we are +interested in the fixed points of the PDE dynamics +0 = ˜ε2∂xxu − a1u − a2uv + +a3u2 +a4 + u2 + a5 +0 = ∂xxv + ε(−c1v + c2u). +(B.2) +Upon defining ˜εu′ = p and v′ = q, where ′ denotes the derivative with respect to x, we can write +this equation as a system of four ODEs: +˜εu′ =p +˜εp′ =a1u + a2uv − +a3u2 +a4 + u2 − a5 +v′ =q +q′ =ε(c1v − c2u). +(B.3) +Taking the scale separation to the extreme, i.e. setting ˜ε = 0, significantly simplifies the equation: +0 =p +0 =a1u + a2uv − +a3u2 +a4 + u2 − a5 +v′ =q +q′ =ε(c1v − c2u). +(B.4) +We refer to this set of equations as the slow system. This system should be understood in the +following sense: on a large spatial scale, the dynamics of (v, q) is approximated by lines 3 and 4 +of the ODE above, and this approximation is valid in the region of the phase plane given by the +algebraic equations in lines 1 and 2. We refer to the solution of these algebraic equations as the slow +or critical manifold. When we try to explicitly compute the critical manifold as a function u(v), we +encounter a third-order polynomial, which can be solved exactly. However, this is not practical as +the graph u(v) cannot be represented by a single function, but it has three branches, the upper, +middle and lower branch. We will denote the upper branch with u+(v) and the lower branch with +u−(v). Hence, system (B.4) now becomes +v′ =q +q′ =ε(c1v − c2u±(v)). +(B.5) +We refer to this equation as the reduced slow system. +7Unfortunately, the theorems do not quantify what is meant by small enough. +25 + +(a) +(b) +Fig. B.1: In both figures, the red dots are a solution of (B.2), found by using Matlab’s bvp4c solver, with +an initial condition coming from a PDE simulation and the small parameter ˜ε2 was set to 10−4. Note that +this value corresponds to Du = 0.01, which is a factor ten smaller than in the main text. In Figure (a), we +compare the fast dynamics, i.e. the jump for variables u and p, with the predicted Hamiltonian (B.9), the +solid blue line. In Figure (b), the purple line is given by ¯v, the value of the jump as predicted by (B.10), +while the green and blue curves denote the nullclines. The parameters are a1 = 0.167, a2 = 16.67, a3 = 167, +a4 = 1.44, a5 = 1.47, ε = 0.52; c1 = 0.1 and c2 = 3.9. +When we are interested in the dynamics of u instead of v, we must zoom in to a smaller length +scale. Therefore, we define ˜εξ = x and use ˙ to denote the derivative with respect to ξ. System (B.3) +now becomes +˙u =p +˙p =a1u + a2uv − +a3u2 +a4 + u2 − a5 +˙v =˜εq +˙q =˜εε(c1v − c2u). +(B.6) +This system is called the fast system and is still equivalent to (B.3), but when we set ˜ε = 0, it is no +longer equivalent and reduces to +˙u =p +˙p =a1u + a2uv − +a3u2 +a4 + u2 − a5 +˙v =0 +˙q =0. +(B.7) +This shows that in the fast limit, the value of v is constant. When we denote the unknown value by +¯v, the system reduces to +˙u =p +˙p =a1u + a2u¯v − +a3u2 +a4 + u2 − a5. +(B.8) +This system is known as the reduced fast system. +How can we use both reduced systems to understand the dynamics of Figure B.1b? We observe +slow dynamics on the upper and lower branch of the critical manifold and a fast jump in between. +The reduced fast system describes the fast jump between the upper and lower branch of the critical +manifold. Therefore, a standing wave exists when this system has a heteroclinic orbit between the +upper and lower branch. The reduced fast system (B.8) is a Hamiltonian system with Hamiltonian +H(u, p) = 1 +2p2 − 1 +2 (a1 + a2¯v) u2 + a3(u − √a4 arctan(u/√a4)) + a5u. +(B.9) +26 + +Hence, a heteroclinic orbit exists when +H(u−(¯v), 0) = H(u+(¯v), 0). +(B.10) +We cannot solve this algebraic equation exactly, but it is a straightforward numerical problem. Note +that ¯v only depends on the parameters a1, ..., a5 and not on the parameters of the equation for v. +For this value of ¯v, the Hamiltonian overlaps with the fast dynamics, as is shown in Figure B.1a. +Furthermore, from Figure B.1b, it is clear that the value for ¯v is a good approximation for the +location of the jump for ˜ε = 10−2. +Now we have all the ingredients to construct the standing wave. We start at x = −∞ in the +background state of the reduced slow system on the lower branch. We follow the dynamics of the +reduced slow system (B.5) until we reach the value ¯v where we jump to the upper branch following +the reduced fast system (B.8). We will follow the slow (v, q)-dynamics on the upper branch until +we return to the value ¯v, but with the opposite sign for the derivative, i.e. we trace a curve from +(¯v, q(¯v)) to (¯v, −q(¯v)) in the reduced slow system. Then, we jump down again to the lower branch, +which we now trace back to the background state. We implicitly assume here that the maximum +value of v remains below the fold of the critical manifold (which is not the case for travelling waves, +see Figure 3.9). +This example shows how GSPT can be used to construct localised solutions of (1.1) and also +how to understand these solutions. Questions about the existence of localised solutions of (1.1) +and bifurcations can thus be reduced to understanding relatively simple ODEs and the connections +between them. +27 + diff --git a/XNFRT4oBgHgl3EQfNTdg/content/tmp_files/load_file.txt b/XNFRT4oBgHgl3EQfNTdg/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b063b03668eaf6a7a9fe00492bbb0e4de5758182 --- /dev/null +++ b/XNFRT4oBgHgl3EQfNTdg/content/tmp_files/load_file.txt @@ -0,0 +1,1119 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf,len=1118 +page_content='Waves in a Stochastic Cell Motility Model C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Hamster a,∗, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' van Heijster b, a Biometris - Wageningen University and Research Wageningen;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' The Netherlands Email: christian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='hamster@wur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='nl b Biometris - Wageningen University and Research Wageningen;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' The Netherlands Email: peter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='vanheijster@wur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='nl Abstract In Bhattacharya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' (Science Advances, 2020), a set of chemical reactions involved in the dynamics of actin waves in cells was studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Both at the microscopic level, where the individual chemical reactions are directly modelled using Gillespie-type algorithms, and on a macroscopic level where a deterministic reaction- diffusion equation arises as the large-scale limit of the underlying chemical reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' In this work, we derive, and subsequently study, the related mesoscopic stochastic reaction-diffusion system, or Chemical Langevin Equation, that arises from the same set of chemical reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' We explain how the stochastic patterns that arise from this equation can be used to understand the experimentally observed dynamics from Bhattacharya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' In particular, we argue that the mesoscopic stochastic model better captures the microscopic behaviour than the deterministic reaction-diffusion equation, while being more amenable for mathematical analysis and numerical simulations than the microscopic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Key words: Gillespie Algorithms, Cell Motility, Mesoscopic Patterns, SPDEs, Chemical Langevin Equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' 1 Introduction In order to move around, an amoeboid cell can change its shape by polymerising actin to curve the cell membrane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' The actin polymerisation is controlled by signalling molecules and experiments in Dictyostelium discoideum have shown that activation of these signalling molecules happens at localised patches that can move along the membrane like a wave [1, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' In wild-type (WT) cells, these waves move fast and die out, creating familiar-shaped pseudopods, while in cancerous cells these waves stick to a point, creating elongated protrusions [1], see Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' In absence of a signal, the formation of pseudopods happens at random places on the cell membrane, resulting in random motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' In contrast, when a cell senses a chemical signal, it can concentrate the random protrusions at the side of the cell where the signal comes from, leading to movement in the direction of the signal [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' As cells are small, the difference in signal strength between the front and the back of the cell (the gradient) is small as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Furthermore, the cell can only use discrete points at the membrane where the receptors are to estimate the direction of the signal [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Therefore, one of the ∗Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Preprint submitted to .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='. February 1, 2023 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='13509v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='AP] 31 Jan 2023 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='1: Stochastic simulations of the microscopic Gillespie-type model from [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' The figures on the left show stochastic simulations of the Ras activity for parameter values applicable to (A) wild-type cells and to (B) genetically modified cells, where the phosphatase PTEN has been switched off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' The figures on the right show typical cell shapes corresponding to the dynamics in the left figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' This shows that mutations in the gene that codes for PTEN lead to elongated protrusions typically associated with cancer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' The dotted yellow line is an indicator of the wave speed, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' the actin waves in (B) are slower and live longer than in (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Reproduced from [1] under creative commons license 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' main questions is “How can a cell use a small gradient in the signal to concentrate the actin activity in the front?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' This question has been studied intensively, but no complete description of all the microscopic chemical processes involved has been given yet, see [8] for a review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' In [1], the choice is made to describe the highly complex actin dynamics with a conceptual acti- vator u and inhibitor v that diffuse and react with each other as summarised in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' The species u and v are an abstraction of the dozens of components that regulate the actual cell movement, but the activator u can be thought of as Ras activity [1], which plays an important role in cell growth and differentiation [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' In particular, u is being activated by Reaction #3 and Reaction #4, while being inhibited by Reaction #1 and Reaction #2, with propensities as indicated in the table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' In addition, v is inhibited by Reaction #5, while Reaction #6 activates the inhibitor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' The information on the chemical reactions, in combination with the diffusion of both species, is generally used in one of two ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' First, there is a Gillespie-type algorithm [15, 16] which can be used to simulate the involved chemical reactions on a microscopic level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' For these simulations, (uk(tn), vk(tn)) (the solution at time tn at grid cell k) is treated as the number of molecules of type u and v at time tn in a grid cell with finite size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' For all these individual molecules the probabilities of diffusing to other grid cells or taking part in a chemical reaction are prescribed as by Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' To be precise, Reaction #1 implies that the time to the next reaction that degrades a u molecule in grid cell k is exponentially distributed with rate parameter (a1uk(tn))−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' See the panels on the left of Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='1 for examples of these simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' This Gillespie-type algorithm approach takes the stochastic nature of a single cell into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' However, it is computationally very expensive and difficult to analyse mathematically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Hence, it is hard to use this type of modelling approach to make valuable predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' 2 A Space (x) ime B Space (x) ime(t)No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Reaction Propensity u v 1 u → ∅ a1u −1 0 2 u → ∅ a2uv −1 0 3 ∅ → u a3u2/(a4 + u2) 1 0 4 ∅ → u a5 1 0 5 v → ∅ εc1v 0 −1 6 ∅ → v εc2u 0 1 Table 1: The chemical reactions that determine the actin wave dynamics from [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' A second way to use the reactions in Table 1 is to derive an average large-scale limit macro- scopic equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Hence, we assume that u and v are densities on a continuous domain, described by a reaction-rate equation with diffusion, also known as a Reaction-Diffusion Equation (RDE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' In particular, the RDE1 related to the chemical reactions in Table 1 is given by ∂tu = Du∂xxu − a1u − a2uv + a3u2 a4 + u2 + a5 , ∂tv = Dv∂xxv + ε(−c1v + c2u), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='1) which is a specific version of the general RDE we will encounter in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' This model is a variation on the classic FitzHugh-Nagumo model for neuron spiking [12, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Protrusions are formed at places with high activator u and u is inhibited by the terms −a1u and −a2uv, see Reaction #1 and Reaction #2 in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' This implies that an increase in u or v leads to a decrease in u, unless the increase is high enough such that activation from Reaction #3, modelled by a nonlinear Hill function a3u2/(a4+u2), takes over and negates the inhibiting effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Effectively, this means that a small increase in u can lead to a much larger increase in u, that is, the system is locally activated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Once u is large and the Hill function levels off at a fixed value a3, the amount of inhibitor v increases via the term εc2u (related to Reaction #6), leading to a fast decay in u by the −a2uv term (related to Reaction #2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' The inhibitor v then decays via Reaction #5 to the rest state and activation can happen again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' In addition, both species diffuse with diffusion coefficient Du, respectively Dv, where it is assumed that Du < Dv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' It is important to realise that, in both approaches, the modelled actin waves happen on the surface of the cell, and, as in [1], we only study a slice of this surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Therefore, the spatial domain must be thought of as an (approximate) circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' For deterministic RDEs like (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='1), a plethora of analytical tools are available (see, for instance, Appendix B) and numerical simulations are relatively straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' However, being a determin- istic equation, this RDE does not show the same stochastic dynamics as the Gillespie simulations and experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' A crucial difference between the macroscopic RDE model (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='1) and the Gillespie simulations revolves around the duration of the patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' In the RDE, an established pattern, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' a standing or travelling wave, will, if uninterrupted, remain there for a very long time, while these patterns are destroyed quickly both in stochastic simulations and experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Furthermore, when the rest state of the RDE (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='1) is stable, activation cannot come from the RDE itself, but it needs an external signal large enough to activate the nonlinear term a3u2/(a4 + u2) related to Reaction #3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' We generally refer to the activation of these patterns as activation events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' It is important to realise that the dynamics of the different chemical processes in the cell are inherently stochastic and at the size of a single cell chemical reactions are not well approximated by large-scale approximations, as Figures 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='1 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='2 show.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' In other words, treating the relevant enzymes and receptors like a continuous medium of infinitely many, infinitely small, particles is invalid, and the stochastic nature of reactions between individual molecules becomes important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' This so-called internal noise can serve as a signal to activate the dynamics if it is large enough at a 1Note that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='1) can also be obtained from a quasi-steady-state approximation from a more complex three- component model introduced in [2], commonly referred to as a Signal Transduction Excitation Network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' 3 (a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='2: Comparison of the deterministic model (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='1) and its stochastic counterpart (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' In Figure (a) we show a simulation of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='1), which is excited at t = 0, resulting in two counterpropagating travelling waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' In the stochastic simulation in Figure (b), the influence of the initial excitation quickly disappears and new pulses appear constantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' The same parameters are used as in the simulations shown in the second row of Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Observe the similarities in the shape of the pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' In Figure (a), the waves travel around the cell where they cancel each other, while in Figure (b) the waves cancel each other at a much shorter scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' See §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='4 for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' certain point in space and time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' As we noted before, the cell hence executes a random walk in the absence of a signal2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' This implies that an external signal does not necessarily activate the dynamics at a certain point on the membrane, but rather changes the random walk of the cell into a biased random walk in the direction of the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Using a more extended model than presented here, it is shown in [2] that coupling an external signal to the stochastic dynamics of the cell indeed can lead to movement in the direction of that signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Instead of studying the complex internal dynamics of the cell, it can be advantageous to perturb the deterministic RDE (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' For instance, in [1], an external source of noise is applied to the RDE (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='1), turning it into a Stochastic RDE (or Stochastic Partial Differential Equation (SPDE)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' While this approach can indeed activate the dynamics and make long-term deterministic waves collapse, it is inherently ad hoc and not a priori based on any of the involved biologically relevant processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' In between the macroscopic level of the RDE and the microscopic level of the chemical reactions, one can derive a mesoscopic SPDE, known as a Chemical Langevin Equation (CLE) [18], that also incorporates the internal noise of the cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' In §2, we will show that the SPDE associated with the chemical reactions as described in Table 1 plus diffusion is given by du = � Du∂xxu − (a1 + a2v)u + a3u2 a4 + u2 + a5 � dt + σ � (a1 + a2v)u + a3u2 a4 + u2 + a5 dW 1 t + σ∂x � 2Duu d ˜W 1 t , dv = (Dv∂xxv + ε(−c1v + c2u)) dt + σ � ε(−c1v + c2u) dW 2 t + σ∂x � 2Dvv d ˜W 2 t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='2) Here, (dW 1 t , dW 2 t ) and (d ˜W 1 t , d ˜W 2 t ) are two independent noise vectors with space-time white noise (each component is also independent of the other) and σ is a measure for the strength of the noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Indeed, in the no-noise limit σ → 0 the mesoscopic SPDE (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='2) reduces to the macroscopic RDE (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' In that sense, σ serves as a scale parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' The main advantage of the SPDE description is, on one hand, that the solutions still show the rich dynamics of the Gillespie models, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' the activation and destruction of waves, but are computationally significantly less expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' On the other hand, since the SPDE in the no-noise 2Describing the motion of free cells is a very subtle problem and random motion does not necessarily mean Brownian motion [27, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' 4 40 6 20 4 X 0 2 20 40 0 0 5 10 15 20 t30 20 4 10 3 X 0 2 10 20 30 0 10 20 30 40 50 tlimit reduces to the deterministic RDE model (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='1), we can use well-developed Partial Differential Equation (PDE) theory to gain insight into the dynamics of the RDE (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='1) and use this to study the closely related SPDE, see for instance [19, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' To give an idea of the differences between the deterministic and stochastic models we plot two simulations in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='2 that will be discussed later in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' It is clear that the simulation of the SPDE paints a much more dynamic picture than the deterministic one, which is more in line with the inherently noisy nature of the cell’s chemical processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Hence, SPDEs are an invaluable tool in unravelling the dynamics of a cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' This article is now organised as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' In §2 we explain how to derive the SPDE (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='2) from Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Subsequently, in §3 we study both the SPDE (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='2) and the RDE (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='1) numerically in different parameter regimes and qualitatively compare the observed dynamics to the Gillespie simulations from [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' In §4, we discuss the results and how they relate to the questions posed in this introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' 2 Derivation of the SPDE Our starting point to derive (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='2) is the set of chemical reactions as laid out in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' First, we introduce the column vector X(t) = (u(t), v(t)))T , where T indicates that we transpose the row vector, and the column vector R(X(t)) with the propensities of the six reactions: R(X(t)) = � a1u(t), a2u(t)v(t), a3u(t)2 a4 + u(t)2 , a5, εc1v(t), εc2u(t) �T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' The associated stoichiometric matrix S, which describes the change in X(t) for each reaction, is then given by S = � −1 −1 1 1 0 0 0 0 0 0 −1 1 � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='1) see the last two columns of Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' On top of these reactions, we assume that both variables also diffuse, so for a well-mixed solution in a large container we find the classic PDE ∂tX = D∂xxX + SR(X), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='2) where D is a diagonal diffusion matrix with coefficients Du and Dv on the diagonal [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' This PDE is identical to the RDE (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='1) and describes the dynamics of X(t), averaged over many individual reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' When the number of reacting molecules is large enough, and when we zoom out far enough such that all individual molecules become effectively a density, the macroscopic PDE gives a good approximation of the microscopic behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Statistically speaking, this means that the probability distribution of all possible states must be very sharply peaked around the average value described by the PDE, so the deviations from the mean can be ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='1 Motivating Example The assumption that we can ignore deviations from the mean is not always valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' For example, in population dynamics, we can write down birth-death models for several hundred individuals and with this number of individuals, random deviations from the mean are actually significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' To further exemplify, and to set the stage for the upcoming derivation, let us study such a simple discrete birth-death process: suppose a population is at time t in state X(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' In the next timestep dt, there are three possible outcomes: (i) the population grows by one individual with probability b(X(t))dt, (ii) the population decreases by one individual with probability d(X(t))dt, or (iii) nothing happens to the population with probability 1 − b(X(t))dt − d(X(t))dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Now, assume we have a continuous Stochastic Differential Equation (SDE) dx(t) = f(x(t))dt + g(x(t))dβt, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='3) 5 where βt is Brownian motion, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' we can think of dβt as a random step with average zero and variance dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' We now ask the question: “When is this continuous SDE a good approximation of the described discrete birth-death process?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Or, more precisely, “What should f(x) and g(x) be such that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='3) is a good approximation of the described discrete process?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Given a solution x of the SDE, we see that the average expected value at x(t + dt) is approximated, at lowest order in dt, by E[x(t + dt)] = x(t) + f(x(t))dt + O(dt2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' For the described birth-death process, we have that the expectation is E[X(t + dt)] = X(t) + [b(X(t)) − d(X(t))]dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Hence, the average expected jump size in population is identical for the SDE (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='3) and the birth- death process if we take f(x) := b(x) − d(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Next, we compute the deviation from the mean of the SDE (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='3) Var[x(t + dt)] = Var[g(x(t))dβt] + O(dt2) = g(x(t))2dt + O(dt2) , while this deviation for the birth-death process is Var[X(t + dt)] = b(x(t)) + d(x(t)) + O(dt2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Therefore, to make these deviations coincide at first order in dt, we must take g(x) := � b(x) + d(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Hence, the process x(t) described in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='3), which is continuous in population size and time, is a good approximation of the discrete process X(t) when dx(t) = (b(x(t)) − d(x(t)))dt + � b(x(t)) + d(x(t))dβt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='4) The stochastic process x(t) shares the average and variance with X(t) but differs in other points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Higher order moments of x(t) and X(t) will not be identical and x(t) can become negative, even when b and d are chosen such that this is not possible in the discrete model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' In order to link the SDE above to chemical reactions, we make the following observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' The birth of an individual can be thought of as the chemical reaction ∅ → X with propensity b(X) and stoichiometric value 1, while the death of an individual can be seen as the chemical reaction X → ∅ with propensity d(X) and stoichiometric value −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Next, we make an assumption which is called the leap condition [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' That is, we assume that, given a state X(t), enough reactions happen in the interval [t, t+dt] to describe the average jump size in [t, t+dt] by a Poisson process whose parameters depend on X(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' With this leap condition assumption, we implicitly also assume that X(t) is a good approximation of the solution in the whole time interval [t, t + dt].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' We now turn the discrete process X(t) into a continuous process x(t) by approximating the discrete Poisson process by a continuous Gaussian, see [24] for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' This approach results in an SDE similar to the SDE (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='4): dx(t) = (b(x(t)) − d(x(t)))dt + � b(x(t))dβ1 t − � d(x(t))dβ2 t , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='5) for two independent Brownian motions β1 t and β2 t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Although visually different from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='4), both SDEs have a noise term that is Gaussian with identical average and variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Therefore, both SDEs describe the same stochastic process and hence we can say that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='4) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='5) are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='2 Derivation of the CLE We have now gained some intuition for linking more general discrete chemical reactions to continuous S(P)DEs: if we have M different molecules in a vector X(t) with diffusion matrix D, N reactions given by a vector R(X(t)) and a stoichiometric matrix S, then the continuous SPDE for X(t) is given by dX(t) = (D∂xxX(t) + SR(X(t))) dt + 1 √ Ω S � diag(R(X(t)))dWt + 1 √ Ω ∂x � 2DX(t)d ˜Wt, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='6) 6 see [3, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' The equation is made of two parts, a local equation that describes the kinetics as in SDE (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='4) dX(t) = SR(X(t))dt + 1 √ Ω S � diag(R(X(t)))dWt (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='7) and a stochastic diffusion equation dX(t) = D∂xxX(t) + 1 √ Ω ∂x � 2DX(t)d ˜Wt, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='8) as derived in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Here, dWt and d ˜Wt are two independent vectors with space-time white noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' The vector dWt has N components coming from the N reactions, while d ˜Wt has the dimension M of X(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' SPDE (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='6) is known as the Chemical Langevin Equation (CLE) [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' The vector X(t) now describes the densities of the molecules involved, not the actual number of molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' How well the discrete number of molecules is approximated by a density is determined by the scale parameter Ω and is in that sense a measure for the noisiness of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' In the no-noise limit Ω → ∞, we recover the classic RDE (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' In contrast, for small Ω the dynamics of the discrete process is dominated by random events and the discrete process should be described in full detail by a chemical master equation [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' The CLE can be understood as the lowest order approximation of the chemical master equation for large Ω, see for more details [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' For an overview of all different paths leading from molecular kinetics to (S)PDEs, see [26, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' It is important to realise that SPDE (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='6) does not necessarily inherit all the statistical properties of the chemical master equation, only averages and variances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Another potential issue is that it does not necessarily ensures positivity of the solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Just as (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='4) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='5) are identical, we can rewrite (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='6) in the following way: dX(t) = (D∂xxX(t) + SR(X(t)))dt + 1 √ Ω � Sdiag(R(X(t)))ST dWt + 1 √ Ω ∂x � 2DX(t)d ˜Wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='9) This time, the noise vector dWt has just M components, reducing the number of random vectors that must be generated (when M < N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' The downside is that the computation of � Sdiag(R(X))ST is in general numerically more expensive then the computation of S � diag(R(X)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' However, in the present setting, there are no connections between the two variables in the stoichiometric matrix S (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='1) and the matrix Sdiag(R(X))ST is thus diagonal, making the computation of the square root trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Note that once we have the CLE (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='9), it can be applied to any set of chemical reactions and can therefore have widespread use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' For example, we can now return to Table 1 and apply the CLE to these reactions, which results in du = � Du∂xxu − (a1 + a2v)u + a3u2 a4 + u2 + a5 � dt + σ � (a1 + a2v)u + a3u2 a4 + u2 + a5dW 1 t + σ∂x � 2Duud ˜W 1 t , dv = (Dv∂xxv + ε(−c1v + c2u)) dt + σ � ε(−c1v + c2u)dW 2 t + σ∂x � 2Dvvd ˜W 2 t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='10) For notational convenience, we replaced 1/ √ Ω by a small parameter σ, resulting in the SPDE (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='2) from the introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' In the remainder of this work, we will study the SPDE above, mainly using numerical techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' It is important to realise that the SPDE above does not have a function-valued solution in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' The term ∂x � 2DX(t)d ˜Wt makes the equation ill-posed and solutions can only be understood in terms of distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Therefore, it is not a priori clear if the numerical solutions shown in the next section converge to a solution of the SPDE when the spatio-temporal discretisations dx and dt are sent to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' In §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='4, we will discuss the implications of omitting this term on the wave dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' 7 (a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='1: (a) The green line is the v-nullcline for c1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='18, while the red line is the nullcline for c1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' The blue line is the u-nullcline, independent of c1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' The u-axis is plotted logarithmically to better highlight the shape of the nullcline for small u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Note how the background state moved around the fold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' (b) Visual representation of the evolution of the two (complex) eigenvalues of the Jacobian matrix (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='2) for c1 varying from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='18 (dark blue) to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='35 (yellow), following the black arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' The other parameters are fixed at a1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='167, a2 = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='67, a3 = 167, a4 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='44, a5 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='47, ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='52 and c2 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' 3 Simulations In this section, we will numerically investigate the PDE (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='1) and SPDE (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' We investigate three of the main building blocks of the PDE dynamics: localised standing waves, localised travelling waves and time-periodic solutions, together with their counterparts in the SPDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' However, before we can investigate the dynamics, we must first establish some basic properties of the (S)PDE, like the existence, uniqueness and stability of the background state(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Since we are interested in localised waves and expect the activator to be in rest otherwise, we need for the existence of these localised waves that the spatially homogeneous background state is stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' In contrast, for the time-periodic solutions, we expect the background state to be unstable such that continuous excitations of the background state can happen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' The possible background states (u∗, v∗) of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='1) are given by the positive real solutions of the u-nullcline and v-nullcline 0 = −a1u − a2uv + a3u2 a4 + u2 + a5 , 0 = ε(−c1v + c2u) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='1) See Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='1a for a typical representation of the shape of the nullclines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Since the system parameters are all assumed to be positive, this is equivalent to finding the positive solutions u∗ of −a2c2 c1 u4 − a1u3 + � a3 + a5 − a2a4c2 c1 � u2 − a1a4u + a5a4 = 0 , with v∗ = c2u∗/c1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Due to the complexity of the general solution formula for quartic polynomials, it is not feasible to write down its solutions explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' However, by Descartes’ rule of signs [7] we know that there is only one positive real root if c1(a3 + a5) < a2a4c2 and one or three positive real roots otherwise3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' The stability of a background state (u∗, v∗) is then determined by the eigenvalues of the associated Jacobian matrix J(u∗, v∗) = � �−a1 − a2v∗ − 2a3a4u∗ (a4 + (u∗)2)2 −a2u∗ εc2 −εc1 � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='2) Since we do not have an explicit formula for (u∗, v∗), we must compute these eigenvalues numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' For example, when we allow one free parameter, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' c1, and fix the other values, then we can compute 3Note that the origin (0, 0) is only a background state if a5 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' 8 the background states and the associated eigenvalues of the Jacobian matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Taking the parameter values a1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='167, a2 = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='67, a3 = 167, a4 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='44, a5 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='47, ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='52 and c2 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='9 from [1] and letting c1 range from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='18 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='35, such that c1(a3 + a5) < a2a4c2, results in one admissible positive background state ranging from (u∗, v∗) ≈ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='077, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='669) to (u∗, v∗) ≈ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='142, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='586).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Initially, for the lower values of c1, the eigenvalues are real and negative, resulting in a stable background state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Increasing the value of c1 to approximately 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='25 results in complex eigenvalues, still with negative real parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' When we further increase the value of c1 to approximately 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='29, both eigenvalues cross the imaginary axis, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' the background state undergoes a Hopf bifurcation and we expect to see time- periodic solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' See Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='1b for a visual representation of the evolution of the eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' In Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='1a we show the nullclines for c1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='18 and c1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' The unique background state moved along the fold in the u-nullcline and as long as the background state is in between the two folds, the fixed point is unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' In the next sections, we will study localised standing and travelling waves for the same parameter set with c1 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='25 and for time-periodic solutions with c1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' The complex dynamics of pulse adding for c1-values in the intermediate regime between these two boundary values, where the eigenvalues of the Jacobian are stable but complex-valued, is outside the scope of this work, see for example [4] for more information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' So far, we only looked at background states, which are spatially homogeneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' However, we are interested in spatially nonhomogeneous patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' By definition, a localised wave is a fixed profile (Φu, Φv) that moves with a fixed speed c (possibly zero).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Therefore, when we change the spatial coordinate x to ξ = x − ct using the chain rule, the profile (Φu, Φv) is a stationary solution of the following shifted Ordinary Differential Equation (ODE): 0 = Du∂ξξΦu + c∂ξΦu − (a1 + a2Φv)Φu + a3Φ2 u a4 + Φ2u + a5, 0 = Dv∂ξξΦv + c∂ξΦv + ε(−c1Φv + c2Φu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='3) This ODE problem can be solved using numerical fixed-point algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' For these algorithms, a crude starting point is needed for the profile and the value of c, which can come from a PDE simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Note that this problem is translation invariant, meaning that we find a one-dimensional family of travelling waves, all shifted versions of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Hence, for the solver to converge, an extra condition to fix the location of the wave is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='1 Standing Waves In this section, we will study standing waves, which means we look for solutions of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='3) with c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' A solution to this ODE is shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' We observe that both components u and v indeed start at and return to their background state (u∗, v∗) ≈ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='0523, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='0394).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' We observe that the activator u changes rapidly in a small region in the spatial domain and we, therefore, call the activator u the fast variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' On the other hand, the inhibitor v is the slow variable as it changes more gradually over a larger spatial distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='2b shows the corresponding phase plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' The majority of the spatial dynamics happens near the lower branch of the u-nullcline before it has a fast excursion from the lower branch to the upper branch of this nullcline and, by the symmetry x �→ −x of the ODE (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='3), it then returns back to the lower branch in a similar fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' The fact that both components of the standing pulse evolve on a different spatial scale allows us to mathematically analyse this standing pulse, see Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' For instance, the value ¯v at which the activator u makes a sharp transition (approximately 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='8 in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='2b), can be approximated by the algebraic relation (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' The analysis also explains why the solution trajectory in the phase plane closely follows the lower branch of the u-nullcline for the most part of the trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' By assumption, the standing wave in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='2a is a stationary solution of the PDE (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' This can be confirmed by using the wave from the ODE as the initial condition for a PDE simulation (not shown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' However, we are not likely to find this single standing wave in a PDE simulation 9 (a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='2: Figure (a) shows a localised standing wave solution to ODE (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='3), found numerically with Matlab’s fsolve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' The green curve is the u-component and the red curve is the v-component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' In Figure (b), the u- nullcline (blue) and v-nullcline (green) of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='1) are shown together with the v-u phase plane of the standing wave from Figure (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' The phase plane is plotted on a semi-log scale to better highlight the dynamics for small u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' We observe that the standing wave starts from the background state (indicated by an asterisk) and initially follows the lower branch of the u-nullcline before jumping to the upper branch of the u-nullcline and follows the same track back to the background state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' The system parameters are taken from [1] and set to Du = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='1, Dv = 1, a1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='167, a2 = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='67, a3 = 167, a4 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='44, a5 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='47, ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='52, c1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='1, and c2 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' (a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='3: Simulation of the PDE (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='1), Figure (a) shows the activator u and Figure (b) the inhibitor v with an initial condition as described in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' The same parameters are used as in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Note that the v-component does not return to its rest state in the region between the two pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' 10 30 4 20 10 3 X 0 2 10 1 20 30 0 0 5 10 15 20 t30 4 20 10 3 X 0 2 10 1 20 30 0 0 5 10 15 20 t(a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='4: Same simulation as in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='3, but on different time scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Figure (a) shows the u-component, zoomed in to highlight the short-time dynamics, while Figure (b) shows the long-time dynamics of u high- lighting the pulse splitting phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Both simulations were done on a larger grid [−60, 60], so the waves would not affect each other on the other side of the domain on this large time scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' without a fine-tuned initial condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' As an example, we use for the simulation the initial condition u0 = u∗ + e−x2 and v0 = v∗ + 2/ cosh2(5x) as a crude approximation of the wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' The resulting simulation is shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' This initial condition splits in, what appears to be, two well- separated localised standing waves4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' However, the plot of the slow v-component makes clear that this is not the case, and that the two standing waves are connected through the slow component, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' the slow component is not in its rest state in between the two standing waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' For more details on the numerics of the (S)PDE simulations, see Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' The interaction between the two standing waves in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='3 through the slow v-component makes that the two standing waves repel each other on a very slow timescale as is made clear by taking long integration times, see Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='4b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' On an infinite domain, the two standing waves slowly drift apart forever, but on a periodic domain, we can expect them to stabilise once they are at an equal distance on both sides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' On the timescales of biological processes, this slow continuous splitting is probably not relevant and on short timescales, the term ‘standing waves’ for the solution at later times in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='3 is biologically justifiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Furthermore, note that for our understanding of the presented dynamics, it is essential to look at both components simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' In other words, for our understanding of Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='3a it is essential to also look at Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='3b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' We now take a closer look at the short-time dynamics presented in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='4a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' In [1], this splitting of the initial condition is described as two counter-propagating travelling waves, sometimes called trigger waves [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' By the formal mathematical definition, a travelling wave is a fixed profile moving with a fixed speed, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' a solution of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Therefore, mathematically speaking, these do not classify as travelling waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Instead, what we observe here would be classified as transient dynamics and pulse splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' However, it is clear that at t = 0, the activity of u is around x = 0, and after some time it moved to two different places, justifying the term ‘travelling’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' If we adopt the terms ‘standing’ and ‘travelling’, it is clear from Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='3a that around t = 3 a transition occurs from travelling to standing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Standing waves with noise For the same parameter values as in the previous paragraph, we now study the full SPDE (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' In Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='5, we plot realisations of the SPDE for different noise intensities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' For low noise levels, we see two quasi-stationary waves appear, like in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='3, before they are destroyed at different points in time by the noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Since the noise is low, no new activation events happen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' When we increase the noise intensity, the noise is able to activate the stable back- ground state, but the waves are also destroyed more quickly, resulting in a constant appearance and disappearance of waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Note the comparison between Figures 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='5c and the figures in [2], where a 4We also observe the evolution of the initial condition back to the stable background state (u∗, v∗), especially for initial conditions with smaller amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Simulations are not shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' 11 5 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='5 2 X0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='5 L 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='5 5 0 1 2 3 4 t10 3 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='5 2 X 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='5 1 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='5 10 0 200 400 600 t(a) σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='002 (b) σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='035 (c) σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='05 (d) σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='5: The u-component of the SPDE (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='10) for four different values of the noise σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' The other system parameters and initial conditions are the same as in the previous figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' In Figure (a), we only show the simulation of wave integrated up to T = 20 because the solution remains in the background state afterwards, the other three figures are shown up to T = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' similar model is studied using Gillespie algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' This activation of the background state is not possible in the deterministic PDE (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='1) without an external force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' In Figures 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='5b and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='5c, we see that in the first instances, many patterns are generated, causing the inhibitor to increase everywhere which blocks new activation events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' After this initial phase, new activation events appear, and sig- nificantly more for higher values of the noise as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' When we increase the noise even further, it becomes impossible to form patterns as every activation event is destroyed instantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Therefore, pattern formation happens at intermediate values of the noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' The idea that there is some ‘opti- mal’ value of the noise resulting in complex dynamics has been observed before in, for instance, the context of nerve impulses [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' In order to quantify this notion of optimality in the noise intensity we must first quantify the size and shape of the patterns in Figures 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='5b and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='5c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Using Matlab’s regionprops algorithm we can automatically detect the patches with a high value for the activator u (see Appendix A for details), giving us the possibility to compute the number of activation events and determine the width and duration of each event, see Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='6a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' In Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='6b we show the statistics for a range of σ values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' This figure shows that there is a clear cutoff for when activation events are likely to happen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' For values of σ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='035, the average number of events is lower than 1, and the number of activation events increases sharply after this value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' We observe that the width, the length and the maximum height of the events are all higher when the number of excitation events is low, but the variability in these values is also larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='7, we look at the statistics of the events for the specific value σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='046.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' The value of the maximum is sharply peaked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' This is something we expect, as the maximum is mainly determined by the deterministic dynamics after the excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' The width and length of the events are much more spread out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Especially for the width, we see a heavy tail towards zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' This is also expected because activation events come in two forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Most events result in two waves, but a small part of the events has the shape of just a single wave, which has a width of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='87 in the deterministic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' We checked whether or not these histograms are well approximated by a 12 30 5 20 4 10 3 X 0 2 10 1 20 30 0 0 5 10 15 20 t30 5 20 4 10 3 X 0 2 10 1 20 30 0 0 20 40 60 80 100 t30 5 20 4 10 3 X 0 10 20 30 0 20 40 60 80 100 t30 5 20 4 10 3 X 0 2 10 1 20 30 0 0 20 40 60 80 100 t(a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='6: In Figure (a) we show a simulation similar to those in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='5, but with σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='45 and a homogeneous initial condition with (u∗, 4v∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' The red boxes are the result of the pattern finding algorithm regionprops in Matlab;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' it identifies all the regions of excitations which we would also find by eye, see Appendix A for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' In Figure (b), we used this algorithm to find the length, width and maximum of these pulses (left axis), as well as the total number of activation events (right axis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' For each value of σ, the number of events is averaged over 100 simulations, and the length, width and maximum are averaged over all events in the 100 simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' We plot the average together with the standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Gaussian distribution, but this was rejected using a Kolmogorov-Smirnov test (p ∼ 10−14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Using the statistics on the width, length, and maximum, we can compare the solutions of SPDE (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='10) to SPDEs with the same deterministic part but different noise terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' First, we can set the ∂x √ 2DXdWt term coming from the diffusion to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' As noted in Remark 1, this term makes the mathematical analysis of the SPDE (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='10) significantly harder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Figures 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='7d-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='7f show that the statistics of the solutions do not change significantly when we delete this term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' This indicates that the noise coming from the reaction terms plays a more influential role in determining the shape of the patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' We are now also in the position to compare the CLE approach with the more ad hoc approach of adding additive white noise the to u-component to mimic the inherent noisiness of the system, see Figures 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='7g-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='7i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Indeed, the properties of the patterns are significantly different when we compare them to the full SPDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' In particular, with just white noise, the patterns are all short and narrow and do not reflect the complicated dynamics of the underlying chemical reactions and experiments (not shown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='2 Travelling Waves In order to find a travelling wave solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='1), understood as a solution of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='3) with c ̸= 0, we must ensure that the dynamics starting from the initial condition does not reach the standing phase or returns to the background state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' This can be achieved by increasing the value of c1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Increasing c1 results in a faster exponential decay of v back to the background state after an excitation, see Table 1, preventing the inhibitor from glueing the two waves together like in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Simulations for an increased value of c1, from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='1 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='25, are shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Note that the PDE (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='1) still only has one stable background state (u∗, v∗) ≈ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='0833, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='625).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' The initial condition splits into two counter-propagating travelling waves, but opposite to what happened with the standing wave before, they keep separating and move away from each other at a fixed speed until they collide and cancel each other out due to the periodicity of the domain, see Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' To find a single travelling wave, we again need to properly tune the initial condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' This can be done by selecting one of the two waves in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='8 and using it as the initial condition of the PDE simulation (not shown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' In Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='9 we show the travelling wave profile and its associated phase 5For values in between, say c1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='15, the numerics becomes very sensitive to the chosen discretisation, see Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' 13 60 40 C 3 20 X 0 2 20 1 40 60 C 0 0 20 40 60 80 100 t(a) (b) (c) (d) (e) (f) (g) (h) (i) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='7: Figures (a)-(c) show the histograms for the width, length and maximum of the pulses for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='046, for the same data as in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='6b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' For Figures (a) and (b), the bin width is fixed to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='25, and for Figure (c) to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Figures (d)-(f) show the same histograms, but in the simulations, the noise coming from the diffusion (the last term in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='10)) was set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' In Figures (g)-(i), we again show the same histograms, but with just white noise on the u-component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' In order to compare the noise levels, we did not choose the same σ value for the three cases but chose σ values such that the average number of activation events per simulation is approximately 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' For the Figures (d)-(f), this means σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='056, and for (g)-(i) σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' (a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='8: Simulation of the PDE (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='1), Figure (a) shows the activator u and Figure (b) the inhibitor v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' We observe the splitting of the initial condition in two counterpropagating travelling waves with a constant speed that exist until they cancel each other out due to the periodicity of the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' The slow inhibitor v decays back to its rest state in between the pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' The red dotted line has a speed of −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='10, which is close to the value of approximately −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='17 found by solving (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='3) using a fixed point method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' The parameters are Du = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='1, a1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='167, a2 = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='67, a3 = 167, a4 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='44, a5 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='47, Dv = 1, ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='52, c1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='2 and c2 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' 14 40 4 20 3 0 X 2 20 1 40 0 0 5 10 15 20 25 t40 4 20 3 0 X 2 20 1 40 0 0 5 10 15 20 25 t(a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='9: Profile of a single travelling wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Figure (a) shows both components u (green) and v (red) and Figure (b) the related phase plane, plotted on a semi-log scale to highlight the dynamics for small u, as well as the nullclines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' The asterisk indicates the fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' This solution is obtained as the endpoint of a PDE simulation (not shown), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' similar to Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='8, but with just one of the two waves as initial condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' As with the standing pulse, the dynamics around the u-nullcline is essential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' The solution trajectory starts from near the background state and follows the lower branch of the u-nullcline, jumps towards the upper branch of the nullcline and keeps following it until it falls off and returns to the lower branch to slowly evolve back towards the stable background state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' In contrast to the standing pulse, see Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='2, the travelling wave is no longer symmetric and it jumps back to the lower branch by falling off the edge of the upper branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' These travelling wave solutions could be analysed further using techniques similar to Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' It is important to realise that we do not expect to see travelling waves in practice as the travelling wave gets destroyed when it collides with another wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Therefore, in the stochastic simulations, it might not always be clear if we are looking at a travelling wave that collapses or at the transient dynamics towards a double pulse that subsequently gets destroyed by the noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Travelling waves with noise When we now return to SPDE (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='10), there are now four regimes for the same parameters as in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' For high values of the noise, we, as before, do not observe any patterns (not shown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' For low values of the noise, we just find the travelling wave (if the simulation is initiated by an appropriate initial condition) since the noise is not strong enough to destroy the wave, nor to activate another pattern, on the timescales of the simulation (not shown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' The interesting dynamics happens again at the intermediate levels of the noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' As Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='10a shows, the noise activates the dynamics, resulting in many counter-propagating travelling waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' A travelling wave is subsequently annihilated when it collides with a travelling wave coming from the other direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Hence, the collision dynamics of Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='8 is repeated many times on smaller spatial-temporal scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' We see in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='10 that after the annihilation of the travelling waves, the slow inhibitor v initially remains high preventing the activation of new counterpropagating travelling waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Only when after a certain time the inhibitor has sufficiently decayed, do we see the activation of new counterpropagating travelling waves by the noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' The creation and annihilation of travelling waves happen at a shorter time scale than the decay of the inhibitor, which makes the dynamics look synchronised, or even periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' In Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='2a we plot the approximate period versus the intensity of the noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' As expected, the period decreases with the intensity of the noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' It differs however significantly from the true time periodic motion we will discuss in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' When we increase the noise, the quasi-periodic pattern is broken up, as the counter-propagating travelling waves are destroyed before they collide and annihilate each other, so no synchronised patterns emerge, see Figures 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='10cand 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='10d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' These patterns become relevant when we discuss the comparison between the CLE and the Gillespie simulations in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='1, see §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='4 15 (a) (b) (c) (d) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='10: Simulation of the SPDE (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='10) for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='02, Figures (a) and (b), and σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='05, Figures (c) and (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' The red dashed line in (a) has a slope of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='05, close to the deterministic wave speed, but given the short time interval the wave exists, precise estimates are difficult to obtain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' We observe that there is a quasi-periodic behaviour with a period of roughly 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' In Figures (c) and (d), the quasi-periodic structure is destroyed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' The same parameters and initial condition are used as in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='3 Time Periodic Solutions In the previous sections, it was essential that the background state of the system was stable, because this allowed the dynamics to return to the rest state after an activation event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' When we increase the value of c1, the background state becomes unstable through a Hopf bifurcation, see Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' In the phase plane, this transition is characterised by the fact that the background state is no longer located on the lower branch of the u-nullcline, as in Figures 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='2b and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='9b, instead, it lies on the middle branch of the u-nullcline, see Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='12b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Hence, after an excursion, the solution cannot return to the unstable background state and is exited again, resulting in time-periodic motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' When we start with a spatial homogeneous initial condition, the PDE simulation shows periodic oscillations in time, see Figures 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='11 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Both components still display slow-fast behaviour, however, this time not in the spatial variable x but in the temporal variable t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' In the case of nonhomogeneous initial conditions, it takes several oscillations before they are all synced up spatially (not shown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' The observed behaviour has the characteristics of a relaxation oscillation as studied intensively for the Van der Pol equation [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' This is not a surprise as the Van der Pol equation formed the foundation for the classic FitzHugh-Nagumo model and PDE (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='1) can be seen as a variation on this classic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Time periodic solutions with noise For small values of the noise σ, the observed period is close to the deterministic version, but when the value of σ increases, the period also decreases monotonically, as is expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Note that after excitation, the inhibitor remains high preventing activation events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' When the noise is too high no patterns are observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' We can investigate the relation between the reduction of the period and the intensity of the noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' In FigureA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='2b, we plot the estimated period versus the noise intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' We indeed see that the period decreases monotonically 16 30 20 4 10 3 X 0 2 10 1 20 30 0 20 40 60 80 100 t30 20 4 10 X 0 3 10 20 2 30 0 20 40 60 80 100 t30 20 5 10 4 X 3 10 20 30 0 20 40 60 80 100 t30 0 20 5 10 4 X 0 3 10 20 30 0 20 40 60 80 100 t(a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='11: Simulation of the PDE (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='1), Figure (a) shows the activator u and Figure (b) the inhibitor v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' By measuring the distances between the maxima of the oscillations we find the estimate T = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='14 for the period of the oscillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Note that this is significantly smaller than the quasi-periodic oscillations in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' The parameters are set to Du = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='1, a1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='167, a2 = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='67, a3 = 167, a4 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='44, a5 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='47, Dv = 1, ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='52, c1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='4 and c2 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' (a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='12: Cross-section of Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='11 at x = 0, together with the corresponding phase plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' It is clear that the solution leaves the background state (marked by an asterisk), but does not return to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' 17 30 4 20 10 3 X 0 2 10 1 20 30 0 5 10 15 20 25 t30 20 4 10 X 0 3 10 20 2 30 0 5 10 15 20 25 t(a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='13: Simulation of the SPDE (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Figure (a) shows the activator u and Figure (b) the inhibitor v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' When we average over the x-direction and measure the distance between the maxima, we find T ≈ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Same parameters as in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='11 with σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' (a) Wild Type (b) PTEN-null Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='14: Two simulations of PDE (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='1) with parameters as in [1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Du = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='1, a1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='167, a2 = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='67, a4 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='44, a5 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='47, Dv = 1, ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='4, c1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='1 and, for Figure (a), a3 = 167 and c2 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='1, while a3 = 300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='6 and c2 = 3 for Figure (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' The initial condition is equal to those in the previous figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' with the noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='4 Wild-Type versus PTEN-null Cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Now that we have studied several different fundamental patterns, we can focus on understanding the different cell shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' In [1], two sets of parameters are compared, representing WT cells (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' healthy cells) and PTEN-null cells where the tumour-suppressing gene PTEN has been switched off[5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' First, we simulate the deterministic PDE (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='1) for both sets of parameters, see Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' We observe that in both parameter regimes, there are two counter-propagating travelling waves but the specific profiles and speeds are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Especially, note that the wave in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='14b is significantly broader and higher than the wave in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='14a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' When noise is applied, the statistics of the dynamics shows a clear difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' In Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='15, we compare the SPDE simulations of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='10) to the Gillespie simulations from [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Focusing on the typical shape of the excitations, there is a clear qualitative correspondence between the two types of simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Furthermore, in both types of simulation, the average pulse duration is longer in the case of the PTEN-null cell simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Note that we show the SPDE simulations on a larger spatio- temporal scale to get a better idea of the distribution in shapes and the zoom-boxes highlight the detailed structure of a typical single activation event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' In the case of PTEN-null cells, the background state can be excited for much lower noise values (σ ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='007), while for WT cells, the noise needs to be twice as large (σ ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='014) as a result of the increased values of c2 and a3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Hence, in PTEN-null cells, an already existing pattern can more easily sustain itself, leading to the elongated shapes of Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='15d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' 18 40 6 20 4 X 0 2 20 40 0 0 5 10 15 20 t30 5 20 4 10 3 X 0 2 10 20 30 0 5 10 15 20 25 t30 20 4 10 X 0 3 10 20 2 30 0 5 10 15 20 25 t40 6 20 4 X 0 2 20 40 0 0 5 10 15 20 t(a) (b) (c) (d) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='15: Comparison of the Gillespie model, Figures (a) and (c) from [1], versus the CLE approxima- tion (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='10), Figures (b) and (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' The same parameters as in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='14, with σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' The initial condi- tion is (u∗, 2v∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' This can lead to an immediate excitation of the background state in Figure (d), while in Figure (b), the excitation of the background is more spread out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' The zoom-boxes highlight the details of a single excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' 4 Discussion & Outlook We set out to show how Stochastic Partial Differential Equations (CLE), or more specifically, Chem- ical Langevin Equations, can be used to gain more insight into the dynamics of models for cell motility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' We have shown for an exemplary set of chemical reactions (see Tabel 1) that the CLE ap- proach, combined with a basic analysis of the corresponding deterministic PDE, allows us to study the different possible patterns with relative ease, both qualitative and quantitative, while remain- ing close to the underlying chemical processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' To understand differences in cell behaviour, like the difference between wild-type and cancerous cells as in [1], the study of the statistical properties of the observed dynamics is essential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' For instance, an essential characteristic differentiating wild-type cells from cancerous cells is how long a pattern can survive after activation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' The simulations in the previous section show that the answer not only depends on the parameters of the system but crucially on the interplay between the parameters and the noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' The CLE can be used to study this interplay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' A natural question to ask is if all the stochastic terms introduced in the CLE (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='9) are really necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Could we, for example, ignore the noise term coming from the diffusion or forget the derivation of the CLE altogether and just naively add an additive white noise term to the equation for u?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' The histograms in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='7 indicate that the effects of the terms that come from the diffusion are minimal (for the parameter values studied here) and therefore that these terms do not contribute meaningfully to our understanding of the cell dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Note that this would solve the problem of the equation being ill-posed, see Remark 1, and would open up the possibilities for more rigorous mathematical analysis based on the results in [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' We also noted that adding just additive white noise changes the statistics significantly, which indicates that completely abandoning the CLE approach throws away too much detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' In this paper, we studied a basic activator-inhibitor system with only a limited number of chemical 19 30 20 4 10 3 X 0 2 10 20 30 0 10 20 30 40 50 t30 20 4 10 3 X 0 2 10 20 30 0 0 10 20 30 40 50 treactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' However, the derivation of the CLE (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='9) in §2 holds for any number of molecules and for any number of chemical reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' As such, one can see this paper as a proof of concept and the methodology of this paper can be directly applied to more complex regulating systems, such as the eight-component system designed in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' In subsequent work, we aim to work on these type of more complex model to better understand the stochastic dynamics that causes the cell to move robustly in one specific direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Furthermore, as shown in detail in Appendix B, the underlying deterministic RDE (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='1) is amenable for rigorous mathematical analysis by using Geometric Singular Perturbation Theory [11, 20, 22, 23, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' We derived a first-order approximation for the jump location where, under certain conditions, the standing wave has a sharp transition in its activator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' This methodology could also be used to, for instance, further analyse the travelling waves to derive approximations for the speed of the waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' In other words, questions about the existence of localised solutions of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='1) and bifur- cations can thus be reduced to understanding relatively simple ODEs and the connections between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' The details of these computations are left as future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' References [1] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Bhattacharya, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Banerjee, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Miao, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Zhan, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Devreotes, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Iglesias (2020), Traveling and standing waves mediate pattern formation in cellular protrusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Science Ad- vances 6(32), eaay7682.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Hagedorn, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Mosler, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Larsen, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Cox, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Flyvbjerg (2008), Cell motility as random motion: A review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' The European Physical Journal Special Topics 157(1), 1–15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' [32] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Van der Pol (1926), On “relaxation-oscillations”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 2(11), 978–992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' 22 A Numerical Methods A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='1 (S)PDE Simulations All the (S)PDE simulations in this paper were done using a semi-implicit Euler–Maruyama method from [29, §10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' For the spatial directions, a standard 2nd-order central difference is used and for the time stepping Euler-Maruyama.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' The deterministic linear part is evaluated at the next timestep, making it semi-implicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' To be concrete, we study an SPDE of the form du = [Lu + f(u)]dt + g(u)dWt, where u is a vector, L is a linear differential operator, f, g are functions and dWt a white noise vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' In comparison with the main text, the vector u equals (u, v) and L = D∂xx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' When we denote the numerical approximation of the linear part L with A, and the spatial discretisation of u at time t with u(t), we find u(t + dt) = u(t) + dt[Au(t + dt) + f(u(t))] + g(u(t))dWt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='1) The white noise step dWt is a vector where each random element is distributed as N(0, dt/h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Hence, the approximation for the new value u(t + dt) becomes (I − dtA) u(t + dt) = u(t) + dtf(u(t)) + g(u(t))dWt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='2) The equation for u(t + dt) is now a matrix equation and can be solved using standard solvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' It is important to realise that the algorithms from [29] only work for Lipschitz noise terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Hence, when the term under the square root in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='10) becomes close to zero, the algorithms become unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' To correct this, we take after every timestep the maximum of u(t + dt) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' The specific models studied in the main text, even the PDE (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='1) can be very sensitive to the size of the spatial discretisation h and temporal discretisation dt in certain parameter regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' For example, when c1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='15 and the remaining parameters are equal to those in Figures 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='3 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='8, the dynamics can differ significantly depending on the chosen size of the discretisation, see Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' For the values used in the main text, c1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='1 and c1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='2, such a discrepancy was not observed for reasonable discretisations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' This is possibly related to the co-existence of travelling and standing waves in this regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' (a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='1: Simulation of the PDE (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' The spatial domain is [−60, 60] with 212 gridpoints, the time interval is [0, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' In Figure (a), we used dt = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='25 · 10−4 and in Figure (b) dt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='0025, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' 4 times larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' The parameters were set at Du = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='1, a1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='167, a2 = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='67, a3 = 167, a4 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='44, a5 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='47, Dv = 1, ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='52, c1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='15 and c2 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' In both cases, the initial condition is equal to the one in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' 23 60 40 3 20 2 X 0 20 1 40 60 0 5 10 15 20 25 t60 40 3 20 2 X 0 20 T 40 60 0 5 10 15 20 25 t(a) c1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='2 (b) c1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='4 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='2: This figure shows the period of the dynamics of SPDE (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='10) in two different regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' In Figure (a), with c1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='2, we are in the regime of travelling waves with quasi-periodic movement as shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='10, while in Figure (b), with c1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='4, we are in the regime of oscillations in time as shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' In both figures, the period is estimated by computing the average in the spatial direction and subsequently computing the distances between the maxima in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='2 Pattern Recognition For Figures 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='6 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='7, Matlab’s regionprops algorithm is used to identify the activation events automatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' This proceeds in the following steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' First, we smooth the data using Matlab’s Gaus- sian filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Without smoothing, the algorithm detects multiple objects in a single event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Next, we transform the data to a binary value by comparing it with a certain threshold: we say that u is activated when it is five times its stationary value u∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Then, the regionprops algorithm is applied with the option BoundingBox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' One needs to take care of which initial condition to use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' When we start with a spatial homoge- neous initial condition (u∗, v∗), there is a lot of activation in the first instances of the simulation, see Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='5c, and it is not possible to define and detect individual activation events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Therefore, we start not on the fixed point (u∗, v∗), but on (u∗, 4v∗) plus a small perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' The result is that activation events only appear when v has decayed enough for excitations to happen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' As the decay is stochastic, and therefore not spatially homogeneous, the activation events start to appear more spread out, making it possible to determine individual events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' B Analysis The deterministic PDE (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='1) has two components and ten parameters, making it difficult to directly analyse mathematically, even for the simplest of localised structures simulated in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' However, these simulations do reveal that the profiles of the two components of the PDE evolve on a different spatial scale: the spatial changes in the slow v-component are more gradual than these of the fast u-component, see, for instance, Figures 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='2a, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='9a, and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='12a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Furthermore, these simulations also revealed that a large part of the spatial dynamics centres around the lower and upper branch of the u-nullcline in the phase plane, with the u-profile making fast jumps in between.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' To amplify (and exploit) this scale separation, we set Dv = 1 (as in [1]) and Du = ˜ε2, where ˜ε is a small parameter that can be taken arbitrary small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Furthermore, we assume that our spatial domain is no longer periodic but instead unbounded6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' This transforms the PDE model (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='1) into ∂tu = ˜ε2∂xxu − a1u − a2uv + a3u2 a4 + u2 + a5 ∂tv = ∂xxv + ε(−c1v + c2u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='1) 6It is relatively straightforward to generalise the results for the unbounded domain to the periodic domain for the type of problems under consideration, see for example [9] 24 The small parameter ˜ε allows us to use Geometric Singular Perturbation Theory (GSPT) [11, 20, 22, 23, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='] to construct solutions that, to leading order in the small parameter, approximate the localised structures of the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' In GSPT, the observation that the dynamics centres around the branches of u-nullcline is taken to the extreme and we construct solutions whose slow dynamics in the singular limit, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' in the limit of the small parameter ˜ε to zero, is confined to this nullcline, which we will refer to as the slow or critical manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' In contrast, during the fast jump in u, the slow component will not change in this singular limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' These assumptions simplify the computations and allow us to compute parts of the solution in the singular limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' The main theorems of GSPT [11, 20, 22, 23, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' ], sometimes called Fenichel Theorem 1-3, allow us to conclude that if the small parameter is small enough7, then there indeed is a true solution of the PDE close to the one constructed in the singular limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Here, we only show the construction of the standing waves we found in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' That is, we are interested in the fixed points of the PDE dynamics 0 = ˜ε2∂xxu − a1u − a2uv + a3u2 a4 + u2 + a5 0 = ∂xxv + ε(−c1v + c2u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='2) Upon defining ˜εu′ = p and v′ = q, where ′ denotes the derivative with respect to x, we can write this equation as a system of four ODEs: ˜εu′ =p ˜εp′ =a1u + a2uv − a3u2 a4 + u2 − a5 v′ =q q′ =ε(c1v − c2u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='3) Taking the scale separation to the extreme, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' setting ˜ε = 0, significantly simplifies the equation: 0 =p 0 =a1u + a2uv − a3u2 a4 + u2 − a5 v′ =q q′ =ε(c1v − c2u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='4) We refer to this set of equations as the slow system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' This system should be understood in the following sense: on a large spatial scale, the dynamics of (v, q) is approximated by lines 3 and 4 of the ODE above, and this approximation is valid in the region of the phase plane given by the algebraic equations in lines 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' We refer to the solution of these algebraic equations as the slow or critical manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' When we try to explicitly compute the critical manifold as a function u(v), we encounter a third-order polynomial, which can be solved exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' However, this is not practical as the graph u(v) cannot be represented by a single function, but it has three branches, the upper, middle and lower branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' We will denote the upper branch with u+(v) and the lower branch with u−(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Hence, system (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='4) now becomes v′ =q q′ =ε(c1v − c2u±(v)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='5) We refer to this equation as the reduced slow system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' 7Unfortunately, the theorems do not quantify what is meant by small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' 25 (a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='1: In both figures, the red dots are a solution of (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='2), found by using Matlab’s bvp4c solver, with an initial condition coming from a PDE simulation and the small parameter ˜ε2 was set to 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Note that this value corresponds to Du = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='01, which is a factor ten smaller than in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' In Figure (a), we compare the fast dynamics, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' the jump for variables u and p, with the predicted Hamiltonian (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='9), the solid blue line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' In Figure (b), the purple line is given by ¯v, the value of the jump as predicted by (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='10), while the green and blue curves denote the nullclines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' The parameters are a1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='167, a2 = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='67, a3 = 167, a4 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='44, a5 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='47, ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='52;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' c1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='1 and c2 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' When we are interested in the dynamics of u instead of v, we must zoom in to a smaller length scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Therefore, we define ˜εξ = x and use ˙ to denote the derivative with respect to ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' System (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='3) now becomes ˙u =p ˙p =a1u + a2uv − a3u2 a4 + u2 − a5 ˙v =˜εq ˙q =˜εε(c1v − c2u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='6) This system is called the fast system and is still equivalent to (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='3), but when we set ˜ε = 0, it is no longer equivalent and reduces to ˙u =p ˙p =a1u + a2uv − a3u2 a4 + u2 − a5 ˙v =0 ˙q =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='7) This shows that in the fast limit, the value of v is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' When we denote the unknown value by ¯v, the system reduces to ˙u =p ˙p =a1u + a2u¯v − a3u2 a4 + u2 − a5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='8) This system is known as the reduced fast system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' How can we use both reduced systems to understand the dynamics of Figure B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='1b?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' We observe slow dynamics on the upper and lower branch of the critical manifold and a fast jump in between.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' The reduced fast system describes the fast jump between the upper and lower branch of the critical manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Therefore, a standing wave exists when this system has a heteroclinic orbit between the upper and lower branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' The reduced fast system (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='8) is a Hamiltonian system with Hamiltonian H(u, p) = 1 2p2 − 1 2 (a1 + a2¯v) u2 + a3(u − √a4 arctan(u/√a4)) + a5u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='9) 26 Hence, a heteroclinic orbit exists when H(u−(¯v), 0) = H(u+(¯v), 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='10) We cannot solve this algebraic equation exactly, but it is a straightforward numerical problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Note that ¯v only depends on the parameters a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=', a5 and not on the parameters of the equation for v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' For this value of ¯v, the Hamiltonian overlaps with the fast dynamics, as is shown in Figure B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Furthermore, from Figure B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='1b, it is clear that the value for ¯v is a good approximation for the location of the jump for ˜ε = 10−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Now we have all the ingredients to construct the standing wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' We start at x = −∞ in the background state of the reduced slow system on the lower branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' We follow the dynamics of the reduced slow system (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='5) until we reach the value ¯v where we jump to the upper branch following the reduced fast system (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' We will follow the slow (v, q)-dynamics on the upper branch until we return to the value ¯v, but with the opposite sign for the derivative, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' we trace a curve from (¯v, q(¯v)) to (¯v, −q(¯v)) in the reduced slow system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Then, we jump down again to the lower branch, which we now trace back to the background state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' We implicitly assume here that the maximum value of v remains below the fold of the critical manifold (which is not the case for travelling waves, see Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' This example shows how GSPT can be used to construct localised solutions of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='1) and also how to understand these solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' Questions about the existence of localised solutions of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content='1) and bifurcations can thus be reduced to understanding relatively simple ODEs and the connections between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} +page_content=' 27' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFRT4oBgHgl3EQfNTdg/content/2301.13509v1.pdf'} diff --git a/XdFQT4oBgHgl3EQfdDZR/content/2301.13329v1.pdf b/XdFQT4oBgHgl3EQfdDZR/content/2301.13329v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..2d2b946151ddf4d017523dcfa41fde9d331b301e --- /dev/null +++ b/XdFQT4oBgHgl3EQfdDZR/content/2301.13329v1.pdf @@ -0,0 +1,3 @@ +version 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mode 100644 index 0000000000000000000000000000000000000000..4b3eee869e7913bf587333e90389eb46033466f1 --- /dev/null +++ b/ZNE1T4oBgHgl3EQfKAM7/content/tmp_files/2301.02958v1.pdf.txt @@ -0,0 +1,724 @@ +A fresh look at the generalized parton distributions of light pseudoscalar mesons +Zanbin Xing,1 Minghui Ding,2 Kh´epani Raya,3 and Lei Chang1 +1School of Physics, Nankai University, Tianjin 300071, China +2Helmholtz-Zentrum Dresden-Rossendorf, Bautzner Landstraße 400, 01328 Dresden, Germany +3Department of Integrated Sciences and Center for Advanced Studies in Physics, +Mathematics and Computation, University of Huelva, E-21071 Huelva, Spain. +(Dated: January 10, 2023) +We present a symmetry-preserving scheme to derive the pion and kaon generalized parton distri- +butions (GPDs) in Euclidean space. The key to maintaining crucial symmetries under this approach +is the treatment of the scattering amplitude, such that it contains both the traditional leading-order +contributions and the scalar/vector pole contribution automatically, the latter being necessary to +ensure the soft-pion theorem. The GPD is extracted analytically via the uniqueness and definition +of the Mellin moments and we find that it naturally matches the double distribution; consequently, +the polynomiality condition and sum rules are satisfied. The present scheme thus paves the way for +the extraction of the GPD in Euclidean space using the Dyson-Schwinger equation framework or +similar continuum approaches. +Introduction— The question of how partons inside +hadrons are distributed in momentum and position space +has surrounded physicists for decades, and successive at- +tempts have been made to answer this question through +both experimental and theoretical methods [1, 2]. +A +quantity that encodes answers to this fundamental ques- +tion is the generalized parton distribution (GPD) [3–5], +which is non-perturbative and contains information on +both the longitudinal-momentum and the transverse spa- +tial distributions. In addition, the GPD is deeply con- +nected to hadron properties [6], for example, lower order +moments of the GPD can be linked to hadron form fac- +tors and the energy-momentum tensor, and so charge and +mass distributions, as well as pressure and shear forces +inside hadrons [7]. +Being a non-perturbative quantity, an investigation of +the GPD requires a sensible non-perturbative approach. +A traditional way to studying the GPD is to derive +all physical quantities required in the light-front coor- +dinate system [8–10]. +However, the non-perturbative +quantum chromodynamics (QCD) is commonly formu- +lated in Euclidean space. Another approach, which in +fact projects Euclidian quantities onto the light-front, is +to use the overlap representation of the light-front wave +function [11], however this limits the domain in which +the GPD can be computed and sophisticated methods +for extrapolation must be employed [12]. Thus it is very +challenging to study the GPD directly in the light-front +coordinate system. +Alternatively, other methods that make it possible +to compute the GPD directly in Euclidean space have +been also investigated, such as implementations in lat- +tice QCD [13, 14] and in continuum field theory meth- +ods [8, 15]. The discussion here concentrates on the im- +plementation in the continuum field theory approach, i.e., +using Dyson-Schwinger equations (DSEs) [16, 17]. The +main idea is to calculate the Mellin moments and then +extract the GPD via the uniqueness property. In early +explorations following this path, for instance Ref. [18], +only the contribution of a triangle diagram was consid- +ered, and when one did so, it was found that the sym- +metry restrictions on the GPD required by QCD, such +as the polynomiality condition, sum rule and soft-pion +theorems, were not all fulfilled. It was then realized that +it was not sufficient to consider only the contribution of +the triangle diagram; there had to be contributions from +other diagrams to ensure the relevant symmetries. For +the parton distribution function (PDF), obtained as the +forward limit of the GPD, this was first fixed in Ref. [19], +but the challenge for off-forward kinamatics remained. +These ideas have been recently revisited, and a novel +perspective is provided in Ref. [20]. There in it has been +shown that one can directly solve for the dressed meson- +meson scattering amplitude, and then use it to calculate +meson gravitational form factors. Capitalizing on these +recent findings, we observe that it is possible to perform a +symmetry-preserving calculation of the GPD using DSEs +in Euclidean space. +In general, the discussion here is universal for all +mesons. However, we have special interests in light pseu- +doscalars, particularly pion and kaon. Contemporary re- +search has shown that there are various connections be- +tween the properties of pion and kaon and the emergent +hadronic masses (EHM) [21]; connections that have been +firmly established empirically. +Therefore, in this arti- +cle, we will adopt the symmetry-preserving scheme pro- +posed in [20], i.e., consider the full meson-meson scatter- +ing amplitude and use its results, to calculate the Mellin +moments of the pion and kaon GPDs; subsequently, the +GPDs are then identified from the definition of its Mellin +moments. +Scattering amplitude: truncation and model— We con- +sider as an example the up-quark leading-twist vector +GPD in π+, i.e., Hπ +u (x, ξ, Q2), (whose isospin decompo- +sition can be obtained accordingly), which can be ex- +arXiv:2301.02958v1 [hep-ph] 8 Jan 2023 + +2 +Mπ +u(q−, p, k) = +p +q− + p +q− + k +k +ladder +−−−−−−−−−→ +approximation +1 +1 ++ +1 +1 ++ +1 +1 ++ · · · +contact +−−−−−−−→ +interaction +1 +1 ++ +1 +1 +FIG. 1. Graphical representation of the scattering amplitude Mπ +u(q−, p, k). The first row denotes the scattering amplitude +under the ladder truncation. The second row denotes the scattering amplitude calculated in a symmetry preserving manner +under contact interaction, and the contributions of the two diagrams are Mπ +u,1(q−, p, k) and Mπ +u,2(q−, p, k) respectively, i.e., +Mπ +u = Mπ +u,1 + Mπ +u,2. Solid, double-solid, dotted lines represent quark, pion, and the effective scalar/vector meson propagator +∆s,v, respectively. Filled and empty circles represent pion BSA and bare quark-scalar/vector vertex, respectively. +pressed in the form +2Hπ +u (x, ξ, Q2) = Nctr +� +q +iγnMπ +u(q−, p, k) , +(1) +where Mπ +u(q−, p, k) is the off-forward scattering ampli- +tude; in the Euclidean metric, p = P − Q/2 and k = +P + Q/2 are the momentum of the incoming and outgo- +ing pion, respectively, consequently, Q is the momentum +transfer and P is the average momentum; q− = q − P; +Nc = 3 is the color degree of freedom and tr indi- +cates a trace over spinor indices; γn = δx +n(q)γ · n is +a generalization of the bare quark-photon vertex γµ, +δx +n(q) = δ(n · q − xn · P), and n is a light-like vector, i.e., +n2 = 0; the ‘skewness’, defining the longitudinal momen- +tum transfer, is defined as ξ = − n·Q +2n·P and we focus on the +0 < ξ < 1 domain. The range x ∈ [ξ, 1] is the Dokshitzer- +Gribov-Lipatov-Altarelli-Parisi (DGLAP) region, while +x ∈ [−ξ, ξ] is the Efremov-Radyushkin-Brodsky-Lepage +(ERBL) region. +To obtain the scattering amplitude Mπ +u(q−, p, k), a +non-perturbativeapproach is required, such as lattice +QCD simulation [13, 22], and the DSEs framework based +on continuum field theory [23]. +In the framework of +DSEs, the leading-order truncation is the ladder approx- +imation, which is sufficient to guarantee charge conserva- +tion and the emergence of pions as Goldstone bosons of +dynamical chiral symmetry breaking [24]. Mπ +u(q−, p, k) +under the ladder approximation is graphically repre- +sented in the first row of Fig. 1, which can be interpreted +as containing an infinite number of gluon exchange con- +tributions. It can be immediately read from Fig. 1 that +there are two options for calculating the GPD, either +dressing γn [18, 25] or dressing the scattering amplitude +Mπ +u(q−, p, k) [20] directly. +Furthermore, the gluon interaction form in Fig. 1 is +required. Although one can calculate the scattering am- +plitude directly using realistic gluon interactions, this is +numerically cumbersome. In this article, as the first work +to calculate the GPD consistently in the framework of +DSEs, we tend to make an attempt first using the sim- +ple model [26, 27] to demonstrate the process, and the +techniques developed here can be extended to realistic +interaction situations without difficulty. +In the simple +model, i.e., contact interaction, the gluon interactions ex- +changed between quarks are approximated as zero-range +interactions, Gµν(k−q) = δµν/m2 +G, with mG a gluon mass +scale. Due to the momentum-independent nature of the +gluon interaction, the quark propagator, and the Bethe- +Salpeter amplitude (BSA) of the pseudoscalar meson, are +simple, +Sf(k) = (−iγ · k + Mf)/D(k) , +(2a) +ΓPS(P) = iγ5EPS(P) + γ5γ · P +Mfg +FPS(P) , +(2b) +where D(k) = k2 + M 2 +f , Mfg = +2Mf Mg +Mf +Mg , f and g de- +note quark flavors, Mf,g is the constituent quark mass, +EPS and FPS are two scalar functions that depend only +on the total momentum of the pseudoscalar meson. In +addition, the regularization procedure has to be imple- +mented when using the contact interaction, we adopt a +symmetry preserving regularization scheme [28] and use +the same parameters therein. +The scattering amplitude Mπ +u(q−, p, k) can be system- +atically calculated, the result of which is illustrated in +the second row of Fig. 1, where the effective scalar/vector +meson propagator (the dotted line) can be expressed as +∆s,v(Q2) = +1 +3m2 +G/4 + f s,v(Q2) , +(3) +with f s,v the one-loop scalar/vector vacuum polariza- +tion. As with the quark propagator and the pseudoscalar +meson BSA, the effective scalar/vector meson propaga- +tor can be solved numerically by the consistent rainbow- +ladder truncation of the corresponding equation. Up to +this point, the scattering amplitude Mπ +u(q−, p, k) under +the contact interaction is obtained, and calculating the +GPD seems straightforward. +Mellin moments and double distributions— In Eu- +clidean space, one convention is to start with the Mellin +moments of the GPD [29], which can be written as +2⟨xm⟩π +u(ξ, Q2) = Nctr +� +q +(n · q)m +(n · P)m+1 iγ · nMu(q−, p, k) , +(4) + +3 +from the definition ⟨xm⟩π +u(ξ, Q2) = +� +xmHπ +u (x, ξ, Q2)dx. +For convenience, we label the two components of ⟨xm⟩π +u +corresponding to the two diagrams in the second row of +Fig. 1 as ⟨xm⟩π +u,1 and ⟨xm⟩π +u,2, and similarly the two com- +ponents of Hπ +u are Hπ +u,1 and Hπ +u,2. If Mellin moments are +calculated, they can then be used to reconstruct the GPD +numerically [30] or analytically [31, 32]. +We first observe that the denominators of ⟨xm⟩π +u,1 and +⟨xm⟩π +u,2 can be parameterized respectively as +1 +D−Q/2DQ/2DP += +� +Ωu +du1du2 +[(q + αQ − βP)2 + ω3]3 ,(5a) +1 +D−Q/2DQ/2 += +� +Ωu +du1du2δ(β) +[(q + αQ)2 + ω2]2 , +(5b) +where DX = D(q − X), α = u1 − u2, β = 1 − u1 − u2 +and Ωu = {(u1, u2)|0 < u1 < 1, 0 < u2 < 1 − u1} +is the domain of integration in the Feynman parame- +ter space. +ω3 = ω(β, α), ω2 = ω(0, α) and ω(β, α) = +M 2 − ¯ββm2 +π + 1 +4Q2 �¯β2 − α2� +. Here we use the isospin +symmetry Mu/d = M, and ¯β = 1 − β. +We then observe that the numerators of ⟨xm⟩π +u,1 and +⟨xm⟩π +u,2, when shifting the loop moment q → q−αQ+βP, +can both be expressed generally as +1 +n · P +� n · q +n · P + β + ξα +�m � +a,b,c +(n·q)a(P·q)b(Q·q)cfabc(q2) , +(6) +where the coefficient fabc(q2) is an even function of the +momentum q, and the powers are restricted by the trace +so that a ∈ {0, 1} and b + c ∈ {0, 1, 2}. +Considering +n2 = 0, the numerators can be simplified by the fact +that the integral +�m +j +� +(αξ +β)m−j +� +q +(n·q)j+a(P ·q)b(Q·q)cfabc(q2) , (7) +survives if and only if +0 ≤ j + a ≤ b + c , +j + a + b + c ∈ even number . (8) +By performing the regularization procedure [28] and +changing the integration variable (u1, u2) to (β, α), the +integration domain changed accordingly from Ωu to Ω = +{(α, β)|0 < β < 1, −¯β < α < ¯β}, we obtain the Mellin +moments ⟨xm⟩π +u,1 and ⟨xm⟩π +u,2 respectively as +⟨xm⟩π +u,1 = +� +Ω +dβdα [(β + ξα)m(ha0 + ξha1) ++m(β + ξα)m−1(hb0 + ξhb1 + ξ2hb2) +� +,(9a) +⟨xm⟩π +u,2 = +� +Ω +dβdα (β + ξα)m(hc0 + ξhc1)δ(β) , +(9b) +where the kernel hi, i ∈ {a0, a1, b0, b1, b2, c0, c1} depends +on (β, α, Q2). We would like to emphasize that the co- +efficient function of (β + ξα)m can always be expressed +in the linear form of ξ, while the coefficient function of +m(β + ξα)m−1 can always be expressed in the quadratic +form of ξ, following ha0,b0,b2,c0/a1,b1,c1 is the even/odd +function of α. Furthermore, we would prefer to point out +that hc0/c1 contains the vector/scalar pole contribution +respectively. +The second line of Eq. (9a) needs special attention. +Noting the fact (we denote (β + ξα) as (·)) that m(·)m−1 +can be written as ∂(·)m +∂β +or ∂(·)m +ξ∂α , one can turn m(·)m−1 +into the form associated with (·)m by using integration +by parts. Since the kernel contains two variables, α and +β, the arbitrariness of the differentiation is unavoidable. +We choose a particular format for illustration, namely +m(·)m−1(hb0 + ξhb1 + ξ2hb2) += ∂ [(·)m(hb0 + ηξhb1)] +∂β +− (·)m ∂(hb0 + ηξhb1) +∂β ++ ∂ [(·)m(¯ηhb1 + ξhb2)] +∂α +− (·)m ∂(¯ηhb1 + ξhb2) +∂α +, (10) +with the arbitrary parameter η, and η + ¯η = 1. Substi- +tuting Eq. (10) into the second line of Eq. (9a), we finally +obtain the Mellin moments +⟨xm⟩π +u = +� +Ω +dβdα(β+ξα)m � +F(β, α, Q2) + ξG(β, α, Q2) +� +, +(11) +where +F(β, α, Q2) = ha0 − ∂hb0 +∂β − ¯η ∂hb1 +∂α + hc0δ(β) ++hb0 +� +δ(¯β − |α|) − δ(β) +� ++¯ηhb1 +� +δ(¯β − α) − δ(¯β + α) +� +, (12a) +G(β, α, Q2) = ha1 − η ∂hb1 +∂β − ∂hb2 +∂α + hc1δ(β) ++ηhb1 +� +δ(¯β − |α|) − δ(β) +� ++hb2 +� +δ(¯β − α) − δ(¯β + α) +� +. +(12b) +This is nothing but the Mellin moments of the double dis- +tribution [33], and the appearance of η exhibits the am- +biguity of the double distribution. Comparing Eq. (11) +with Eq. (4), the GPD is obtained analytically from the +double distribution integrated along a line of equation +x − β − ξα = 0, i.e., +Hπ +u (x, ξ, Q2) = +� +Ω +dβdα δ(x − β − ξα) +× +� +F(β, α, Q2) + ξG(β, α, Q2) +� +. (13) +As a complement to the scheme of the α representation of +propagators [9, 34], we have provided a double distribu- +tion calculation scheme based on the Feynman technique. +Discussions— In this section, we summarize some +properties of the GPD that we have obtained. +Polynomiality condition: +Based on the symmetry +properties of hi, one would obtain that the double +distribution functions F(β, α, Q2) = F(β, −α, Q2) and + +4 +�1.0 +�0.5 +0.0 +0.5 +1.0 +x +0.0 +0.5 +1.0 +Ξ +0.0 +0.5 +1.0 +HΠu�x,Ξ,0� +�1.0 +�0.5 +0.0 +0.5 +1.0 +x +0.0 +0.5 +1.0 +Ξ +0.0 +0.5 +1.0 +HKu�x,Ξ,0� +FIG. 2. +Up quark GDPs in pion (upper panel) and kaon +(lower panel) with zero momentum transfer Q2 = 0. +G(β, α, Q2) = −G(β, −α, Q2) on the domain Ω. +Con- +sequently, the Mellin moments fulfill the polynomiality +condition and are even functions of ξ, with the power of +ξ in the polynomial at most m + 1. +Sum rules: +Particularly, the zeroth Mellin moment of +the GPD corresponds to the electromagnetic form factor +⟨x0⟩PS +f (ξ, Q2) = F em,PS +f +(Q2) , +(14) +and the first Mellin moment corresponds to the gravita- +tional form factors +⟨x1⟩PS +f (ξ, Q2) = APS +f (Q2) + ξ2DPS +f (Q2) . +(15) +Following the regularization procedure in Ref. [28], we +have analytically checked the equivalence between the +form factors calculated in Ref. [20] and the results ex- +tracted here using Mellin moments. Thus, the sum rules +hold. +Soft-pion theorem: +In the chiral limit mπ = 0 and at +Q2 = 0, the two components of the GPD corresponding +to the two diagrams in the second row of Fig. 1 can be +written analytically as +Hc.l. +u,1(x, ξ, 0) = (ξ + x)EPS − 2ξFPS +2ξ(EPS − 2FPS) +θ(ξ − x, x + ξ) ++θ(1 − x, x − ξ) , +(16a) +Hc.l. +u,2(x, ξ, 0) = +−xEPS +2ξ(EPS − 2FPS)θ(ξ − x, x + ξ) , (16b) +and added up to +Hc.l. +u (x, ξ, 0) = 1 +2θ(ξ − x, x + ξ) + θ(1 − x, x − ξ) . (17) +From Eq. (17) and the fact of constant behavior of parton +distribution amplitude, we note that the soft-pion limit +at ξ = 1 is well satisfied. Additionally, it is worth empha- +sizing that the scalar pole contribution is important for +achieving such limit, which cannot be true if only Hc.l. +u,1 +is considered [35]. +The up-quark GPD in pion and kaon beyond the +chiral limit have been depicted in Fig. 2. +We see +that Hπ +u (x, 0, 0) is symmetric around x = 1/2, while +HK +u (x, 0, 0) is skewed, peaking at x < 1/2. The GPD +in the −ξ < x < ξ domain, i.e., the ERBL region, is a +direct result of this approach, and is positive. Fig. 2 also +shows that both GPDs vanish at x < −ξ. Additionally, +it is noting that GPDs are discontinuous at x = ±ξ and +non-vanishing at x = 1, typical results of the contact in- +teraction used here. In the case of realistic interactions, +the continuity of GPD is to be expected. +Summary— In this work, we show the procedure and +numerical results for the calculation of light pseudoscalar +meson GPDs in Euclidean space within a consistent +scheme. The symmetry-preserving treatment of the scat- +tering amplitude is crucial in this computational process, +and this treatment allows the scattering amplitude to in- +clude not only the contribution of the conventional trian- +gle diagram but also that of an additional diagram con- +taining the scalar/vector meson poles. In doing so, it is +seen that the latter is a necessary part that cannot be ne- +glected, otherwise soft-pion theorem is violated. In order +to extract the GPD directly in the Euclidean space, we +evaluate the corresponding Mellin moments and, by using +Feynman parameters and other algebraic approaches, we +systematically systematically identify the GPD and ob- +tain the corresponding double distribution. Specifically, +there are several novel findings in this computational pro- +cess: (i) The symmetry-preserving regularization scheme +allows us to include all terms after tracing without the +need to reduce the common factors in the numerator and +denominator in the GPD Mellin moment formula, as was +commonly performed in Ref. [9, 15]. (ii) In this process, +a form of m(·)m−1 arises, and different handles of this +term can reflect the ambiguity of the double distribution. +(iii) Since the GPD we obtained matches the double dis- +tribution, the symmetry restrictions required by QCD, +such as the polynomiality condition and sum rules are + +5 +satisfied. 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C 81, 6 (2021), arXiv:2009.11384 [hep-ph]. + diff --git a/ZNE1T4oBgHgl3EQfKAM7/content/tmp_files/load_file.txt b/ZNE1T4oBgHgl3EQfKAM7/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..4129181de1f9207646da14672ed2947fc6bce008 --- /dev/null +++ b/ZNE1T4oBgHgl3EQfKAM7/content/tmp_files/load_file.txt @@ -0,0 +1,486 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf,len=485 +page_content='A fresh look at the generalized parton distributions of light pseudoscalar mesons Zanbin Xing,1 Minghui Ding,2 Kh´epani Raya,3 and Lei Chang1 1School of Physics, Nankai University, Tianjin 300071, China 2Helmholtz-Zentrum Dresden-Rossendorf, Bautzner Landstraße 400, 01328 Dresden, Germany 3Department of Integrated Sciences and Center for Advanced Studies in Physics, Mathematics and Computation, University of Huelva, E-21071 Huelva, Spain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' (Dated: January 10, 2023) We present a symmetry-preserving scheme to derive the pion and kaon generalized parton distri- butions (GPDs) in Euclidean space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' The key to maintaining crucial symmetries under this approach is the treatment of the scattering amplitude, such that it contains both the traditional leading-order contributions and the scalar/vector pole contribution automatically, the latter being necessary to ensure the soft-pion theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' The GPD is extracted analytically via the uniqueness and definition of the Mellin moments and we find that it naturally matches the double distribution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' consequently, the polynomiality condition and sum rules are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' The present scheme thus paves the way for the extraction of the GPD in Euclidean space using the Dyson-Schwinger equation framework or similar continuum approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' Introduction— The question of how partons inside hadrons are distributed in momentum and position space has surrounded physicists for decades, and successive at- tempts have been made to answer this question through both experimental and theoretical methods [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' A quantity that encodes answers to this fundamental ques- tion is the generalized parton distribution (GPD) [3–5], which is non-perturbative and contains information on both the longitudinal-momentum and the transverse spa- tial distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' In addition, the GPD is deeply con- nected to hadron properties [6], for example, lower order moments of the GPD can be linked to hadron form fac- tors and the energy-momentum tensor, and so charge and mass distributions, as well as pressure and shear forces inside hadrons [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' Being a non-perturbative quantity, an investigation of the GPD requires a sensible non-perturbative approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' A traditional way to studying the GPD is to derive all physical quantities required in the light-front coor- dinate system [8–10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' However, the non-perturbative quantum chromodynamics (QCD) is commonly formu- lated in Euclidean space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' Another approach, which in fact projects Euclidian quantities onto the light-front, is to use the overlap representation of the light-front wave function [11], however this limits the domain in which the GPD can be computed and sophisticated methods for extrapolation must be employed [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' Thus it is very challenging to study the GPD directly in the light-front coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' Alternatively, other methods that make it possible to compute the GPD directly in Euclidean space have been also investigated, such as implementations in lat- tice QCD [13, 14] and in continuum field theory meth- ods [8, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' The discussion here concentrates on the im- plementation in the continuum field theory approach, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=', using Dyson-Schwinger equations (DSEs) [16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' The main idea is to calculate the Mellin moments and then extract the GPD via the uniqueness property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' In early explorations following this path, for instance Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' [18], only the contribution of a triangle diagram was consid- ered, and when one did so, it was found that the sym- metry restrictions on the GPD required by QCD, such as the polynomiality condition, sum rule and soft-pion theorems, were not all fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' It was then realized that it was not sufficient to consider only the contribution of the triangle diagram;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' there had to be contributions from other diagrams to ensure the relevant symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' For the parton distribution function (PDF), obtained as the forward limit of the GPD, this was first fixed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' [19], but the challenge for off-forward kinamatics remained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' These ideas have been recently revisited, and a novel perspective is provided in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' There in it has been shown that one can directly solve for the dressed meson- meson scattering amplitude, and then use it to calculate meson gravitational form factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' Capitalizing on these recent findings, we observe that it is possible to perform a symmetry-preserving calculation of the GPD using DSEs in Euclidean space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' In general, the discussion here is universal for all mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' However, we have special interests in light pseu- doscalars, particularly pion and kaon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' Contemporary re- search has shown that there are various connections be- tween the properties of pion and kaon and the emergent hadronic masses (EHM) [21];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' connections that have been firmly established empirically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' Therefore, in this arti- cle, we will adopt the symmetry-preserving scheme pro- posed in [20], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=', consider the full meson-meson scatter- ing amplitude and use its results, to calculate the Mellin moments of the pion and kaon GPDs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' subsequently, the GPDs are then identified from the definition of its Mellin moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' Scattering amplitude: truncation and model— We con- sider as an example the up-quark leading-twist vector GPD in π+, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=', Hπ u (x, ξ, Q2), (whose isospin decompo- sition can be obtained accordingly), which can be ex- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content='02958v1 [hep-ph] 8 Jan 2023 2 Mπ u(q−, p, k) = p q− + p q− + k k ladder −−−−−−−−−→ approximation 1 1 + 1 1 + 1 1 + · · · contact −−−−−−−→ interaction 1 1 + 1 1 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' Graphical representation of the scattering amplitude Mπ u(q−, p, k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' The first row denotes the scattering amplitude under the ladder truncation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' The second row denotes the scattering amplitude calculated in a symmetry preserving manner under contact interaction, and the contributions of the two diagrams are Mπ u,1(q−, p, k) and Mπ u,2(q−, p, k) respectively, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=', Mπ u = Mπ u,1 + Mπ u,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' Solid, double-solid, dotted lines represent quark, pion, and the effective scalar/vector meson propagator ∆s,v, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' Filled and empty circles represent pion BSA and bare quark-scalar/vector vertex, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' pressed in the form 2Hπ u (x, ξ, Q2) = Nctr � q iγnMπ u(q−, p, k) , (1) where Mπ u(q−, p, k) is the off-forward scattering ampli- tude;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' in the Euclidean metric, p = P − Q/2 and k = P + Q/2 are the momentum of the incoming and outgo- ing pion, respectively, consequently, Q is the momentum transfer and P is the average momentum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' q− = q − P;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' Nc = 3 is the color degree of freedom and tr indi- cates a trace over spinor indices;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' γn = δx n(q)γ · n is a generalization of the bare quark-photon vertex γµ, δx n(q) = δ(n · q − xn · P), and n is a light-like vector, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=', n2 = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' the ‘skewness’, defining the longitudinal momen- tum transfer, is defined as ξ = − n·Q 2n·P and we focus on the 0 < ξ < 1 domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' The range x ∈ [ξ, 1] is the Dokshitzer- Gribov-Lipatov-Altarelli-Parisi (DGLAP) region, while x ∈ [−ξ, ξ] is the Efremov-Radyushkin-Brodsky-Lepage (ERBL) region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' To obtain the scattering amplitude Mπ u(q−, p, k), a non-perturbativeapproach is required, such as lattice QCD simulation [13, 22], and the DSEs framework based on continuum field theory [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' In the framework of DSEs, the leading-order truncation is the ladder approx- imation, which is sufficient to guarantee charge conserva- tion and the emergence of pions as Goldstone bosons of dynamical chiral symmetry breaking [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' Mπ u(q−, p, k) under the ladder approximation is graphically repre- sented in the first row of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' 1, which can be interpreted as containing an infinite number of gluon exchange con- tributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' It can be immediately read from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' 1 that there are two options for calculating the GPD, either dressing γn [18, 25] or dressing the scattering amplitude Mπ u(q−, p, k) [20] directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' Furthermore, the gluon interaction form in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' 1 is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' Although one can calculate the scattering am- plitude directly using realistic gluon interactions, this is numerically cumbersome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' In this article, as the first work to calculate the GPD consistently in the framework of DSEs, we tend to make an attempt first using the sim- ple model [26, 27] to demonstrate the process, and the techniques developed here can be extended to realistic interaction situations without difficulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' In the simple model, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=', contact interaction, the gluon interactions ex- changed between quarks are approximated as zero-range interactions, Gµν(k−q) = δµν/m2 G, with mG a gluon mass scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' Due to the momentum-independent nature of the gluon interaction, the quark propagator, and the Bethe- Salpeter amplitude (BSA) of the pseudoscalar meson, are simple, Sf(k) = (−iγ · k + Mf)/D(k) , (2a) ΓPS(P) = iγ5EPS(P) + γ5γ · P Mfg FPS(P) , (2b) where D(k) = k2 + M 2 f , Mfg = 2Mf Mg Mf +Mg , f and g de- note quark flavors, Mf,g is the constituent quark mass, EPS and FPS are two scalar functions that depend only on the total momentum of the pseudoscalar meson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' In addition, the regularization procedure has to be imple- mented when using the contact interaction, we adopt a symmetry preserving regularization scheme [28] and use the same parameters therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' The scattering amplitude Mπ u(q−, p, k) can be system- atically calculated, the result of which is illustrated in the second row of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' 1, where the effective scalar/vector meson propagator (the dotted line) can be expressed as ∆s,v(Q2) = 1 3m2 G/4 + f s,v(Q2) , (3) with f s,v the one-loop scalar/vector vacuum polariza- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' As with the quark propagator and the pseudoscalar meson BSA, the effective scalar/vector meson propaga- tor can be solved numerically by the consistent rainbow- ladder truncation of the corresponding equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' Up to this point, the scattering amplitude Mπ u(q−, p, k) under the contact interaction is obtained, and calculating the GPD seems straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' Mellin moments and double distributions— In Eu- clidean space, one convention is to start with the Mellin moments of the GPD [29], which can be written as 2⟨xm⟩π u(ξ, Q2) = Nctr � q (n · q)m (n · P)m+1 iγ · nMu(q−, p, k) , (4) 3 from the definition ⟨xm⟩π u(ξ, Q2) = � xmHπ u (x, ξ, Q2)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' For convenience, we label the two components of ⟨xm⟩π u corresponding to the two diagrams in the second row of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' 1 as ⟨xm⟩π u,1 and ⟨xm⟩π u,2, and similarly the two com- ponents of Hπ u are Hπ u,1 and Hπ u,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' If Mellin moments are calculated, they can then be used to reconstruct the GPD numerically [30] or analytically [31, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' We first observe that the denominators of ⟨xm⟩π u,1 and ⟨xm⟩π u,2 can be parameterized respectively as 1 D−Q/2DQ/2DP = � Ωu du1du2 [(q + αQ − βP)2 + ω3]3 ,(5a) 1 D−Q/2DQ/2 = � Ωu du1du2δ(β) [(q + αQ)2 + ω2]2 , (5b) where DX = D(q − X), α = u1 − u2, β = 1 − u1 − u2 and Ωu = {(u1, u2)|0 < u1 < 1, 0 < u2 < 1 − u1} is the domain of integration in the Feynman parame- ter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' ω3 = ω(β, α), ω2 = ω(0, α) and ω(β, α) = M 2 − ¯ββm2 π + 1 4Q2 �¯β2 − α2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' Here we use the isospin symmetry Mu/d = M, and ¯β = 1 − β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' We then observe that the numerators of ⟨xm⟩π u,1 and ⟨xm⟩π u,2, when shifting the loop moment q → q−αQ+βP, can both be expressed generally as 1 n · P � n · q n · P + β + ξα �m � a,b,c (n·q)a(P·q)b(Q·q)cfabc(q2) , (6) where the coefficient fabc(q2) is an even function of the momentum q, and the powers are restricted by the trace so that a ∈ {0, 1} and b + c ∈ {0, 1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' Considering n2 = 0, the numerators can be simplified by the fact that the integral �m j � (αξ +β)m−j � q (n·q)j+a(P ·q)b(Q·q)cfabc(q2) , (7) survives if and only if 0 ≤ j + a ≤ b + c , j + a + b + c ∈ even number .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' (8) By performing the regularization procedure [28] and changing the integration variable (u1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' u2) to (β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' α),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' the integration domain changed accordingly from Ωu to Ω = {(α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' β)|0 < β < 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' −¯β < α < ¯β},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' we obtain the Mellin moments ⟨xm⟩π u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content='1 and ⟨xm⟩π u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content='2 respectively as ⟨xm⟩π u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content='1 = � Ω dβdα [(β + ξα)m(ha0 + ξha1) +m(β + ξα)m−1(hb0 + ξhb1 + ξ2hb2) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content='(9a) ⟨xm⟩π u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content='2 = � Ω dβdα (β + ξα)m(hc0 + ξhc1)δ(β) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' (9b) where the kernel hi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' i ∈ {a0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' a1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' b0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' b1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' b2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' c0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' c1} depends on (β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' Q2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' We would like to emphasize that the co- efficient function of (β + ξα)m can always be expressed in the linear form of ξ, while the coefficient function of m(β + ξα)m−1 can always be expressed in the quadratic form of ξ, following ha0,b0,b2,c0/a1,b1,c1 is the even/odd function of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' Furthermore, we would prefer to point out that hc0/c1 contains the vector/scalar pole contribution respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' The second line of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' (9a) needs special attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' Noting the fact (we denote (β + ξα) as (·)) that m(·)m−1 can be written as ∂(·)m ∂β or ∂(·)m ξ∂α , one can turn m(·)m−1 into the form associated with (·)m by using integration by parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' Since the kernel contains two variables, α and β, the arbitrariness of the differentiation is unavoidable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' We choose a particular format for illustration, namely m(·)m−1(hb0 + ξhb1 + ξ2hb2) = ∂ [(·)m(hb0 + ηξhb1)] ∂β − (·)m ∂(hb0 + ηξhb1) ∂β + ∂ [(·)m(¯ηhb1 + ξhb2)] ∂α − (·)m ∂(¯ηhb1 + ξhb2) ∂α , (10) with the arbitrary parameter η, and η + ¯η = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' Substi- tuting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' (10) into the second line of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' (9a), we finally obtain the Mellin moments ⟨xm⟩π u = � Ω dβdα(β+ξα)m � F(β, α, Q2) + ξG(β, α, Q2) � , (11) where F(β, α, Q2) = ha0 − ∂hb0 ∂β − ¯η ∂hb1 ∂α + hc0δ(β) +hb0 � δ(¯β − |α|) − δ(β) � +¯ηhb1 � δ(¯β − α) − δ(¯β + α) � , (12a) G(β, α, Q2) = ha1 − η ∂hb1 ∂β − ∂hb2 ∂α + hc1δ(β) +ηhb1 � δ(¯β − |α|) − δ(β) � +hb2 � δ(¯β − α) − δ(¯β + α) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' (12b) This is nothing but the Mellin moments of the double dis- tribution [33], and the appearance of η exhibits the am- biguity of the double distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' Comparing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' (11) with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' (4), the GPD is obtained analytically from the double distribution integrated along a line of equation x − β − ξα = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=', Hπ u (x, ξ, Q2) = � Ω dβdα δ(x − β − ξα) × � F(β, α, Q2) + ξG(β, α, Q2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' (13) As a complement to the scheme of the α representation of propagators [9, 34], we have provided a double distribu- tion calculation scheme based on the Feynman technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' Discussions— In this section, we summarize some properties of the GPD that we have obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' Polynomiality condition: Based on the symmetry properties of hi, one would obtain that the double distribution functions F(β, α, Q2) = F(β, −α, Q2) and 4 �1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content='0 �0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content='0 x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content='0 Ξ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content='0 HΠu�x,Ξ,0� �1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content='0 �0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content='0 x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content='0 Ξ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content='0 HKu�x,Ξ,0� FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' Up quark GDPs in pion (upper panel) and kaon (lower panel) with zero momentum transfer Q2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' G(β, α, Q2) = −G(β, −α, Q2) on the domain Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' Con- sequently, the Mellin moments fulfill the polynomiality condition and are even functions of ξ, with the power of ξ in the polynomial at most m + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' Sum rules: Particularly, the zeroth Mellin moment of the GPD corresponds to the electromagnetic form factor ⟨x0⟩PS f (ξ, Q2) = F em,PS f (Q2) , (14) and the first Mellin moment corresponds to the gravita- tional form factors ⟨x1⟩PS f (ξ, Q2) = APS f (Q2) + ξ2DPS f (Q2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' (15) Following the regularization procedure in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' [28], we have analytically checked the equivalence between the form factors calculated in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' [20] and the results ex- tracted here using Mellin moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' Thus, the sum rules hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' Soft-pion theorem: In the chiral limit mπ = 0 and at Q2 = 0, the two components of the GPD corresponding to the two diagrams in the second row of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' 1 can be written analytically as Hc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' u,1(x, ξ, 0) = (ξ + x)EPS − 2ξFPS 2ξ(EPS − 2FPS) θ(ξ − x, x + ξ) +θ(1 − x, x − ξ) , (16a) Hc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' u,2(x, ξ, 0) = −xEPS 2ξ(EPS − 2FPS)θ(ξ − x, x + ξ) , (16b) and added up to Hc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' u (x, ξ, 0) = 1 2θ(ξ − x, x + ξ) + θ(1 − x, x − ξ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' (17) From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' (17) and the fact of constant behavior of parton distribution amplitude, we note that the soft-pion limit at ξ = 1 is well satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' Additionally, it is worth empha- sizing that the scalar pole contribution is important for achieving such limit, which cannot be true if only Hc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' u,1 is considered [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' The up-quark GPD in pion and kaon beyond the chiral limit have been depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' We see that Hπ u (x, 0, 0) is symmetric around x = 1/2, while HK u (x, 0, 0) is skewed, peaking at x < 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' The GPD in the −ξ < x < ξ domain, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=', the ERBL region, is a direct result of this approach, and is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' 2 also shows that both GPDs vanish at x < −ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' Additionally, it is noting that GPDs are discontinuous at x = ±ξ and non-vanishing at x = 1, typical results of the contact in- teraction used here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' In the case of realistic interactions, the continuity of GPD is to be expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' Summary— In this work, we show the procedure and numerical results for the calculation of light pseudoscalar meson GPDs in Euclidean space within a consistent scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' The symmetry-preserving treatment of the scat- tering amplitude is crucial in this computational process, and this treatment allows the scattering amplitude to in- clude not only the contribution of the conventional trian- gle diagram but also that of an additional diagram con- taining the scalar/vector meson poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' In doing so, it is seen that the latter is a necessary part that cannot be ne- glected, otherwise soft-pion theorem is violated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' In order to extract the GPD directly in the Euclidean space, we evaluate the corresponding Mellin moments and, by using Feynman parameters and other algebraic approaches, we systematically systematically identify the GPD and ob- tain the corresponding double distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' Specifically, there are several novel findings in this computational pro- cess: (i) The symmetry-preserving regularization scheme allows us to include all terms after tracing without the need to reduce the common factors in the numerator and denominator in the GPD Mellin moment formula, as was commonly performed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' [9, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' (ii) In this process, a form of m(·)m−1 arises, and different handles of this term can reflect the ambiguity of the double distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' (iii) Since the GPD we obtained matches the double dis- tribution, the symmetry restrictions required by QCD, such as the polynomiality condition and sum rules are 5 satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' Finally, it is worth noting that our approach is universal for all sorts of mesons, and although we use the contact interaction model to illustrate all the computa- tional processes of light pseudoscalar mesons, the present approach opens a window for studying meson GPDs with sophisticated interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} +page_content=' 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+page_content='11384 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfKAM7/content/2301.02958v1.pdf'} diff --git a/ZNE2T4oBgHgl3EQfEgZ2/vector_store/index.pkl b/ZNE2T4oBgHgl3EQfEgZ2/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..2b174d8f802ec36fed2ec647d97ac7aba7794a8a --- /dev/null +++ b/ZNE2T4oBgHgl3EQfEgZ2/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:e7e34037906cc77446cc6532528f49afc2c6664d99c93dc5c6e52e7aa9bfe907 +size 133288 diff --git a/aNAyT4oBgHgl3EQfifjx/content/tmp_files/2301.00399v1.pdf.txt b/aNAyT4oBgHgl3EQfifjx/content/tmp_files/2301.00399v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..3dc1448749b501f4be2f41ff660513f99f8f8914 --- /dev/null +++ b/aNAyT4oBgHgl3EQfifjx/content/tmp_files/2301.00399v1.pdf.txt @@ -0,0 +1,678 @@ +Semantic Operator Prediction and +Applications +Farshad Noravesh* +January 3, 2023 +Abstract +In the present paper, semantic parsing challenges are briefly introduced +and QDMR formalism in semantic parsing is implemented using sequence +to sequence model with attention but uses only part of speech(POS) as a rep- +resentation of words of a sentence to make the training as simple and as fast +as possible and also avoiding curse of dimensionality as well as overfitting. +It is shown how semantic operator prediction could be augmented with other +models like the CopyNet model or the recursive neural net model. +1 +Introduction +Semantic parsing and question answering have become coupled in recent years +due to many reasons such as a technical reason, namely distant supervision, since +creating a dataset for question answering pairs are much simpler than treebanks. +Another type of weak supervision is to consider logical form of highest node in +the tree as the only source of supervision as is done in (Herzig & Berant 2021). +State of the art models to knowledge based question answering(KBQA) is +observed to be based on semantic parsing to produce logical forms that can be +easily executed on these knowledge graphs as is mentioned in (Gu et al. 2022) , +(Gu et al. 2021),(Berant et al. 2013) or separating semantic parsing task from the +knowledge base interaction which is proposed in (Ravishankar et al. 2021). +*Email: noraveshfarshad@gmail.com +1 +arXiv:2301.00399v1 [cs.CL] 1 Jan 2023 + +Traditionally, entity linking(finding entities mentioned in the given question) +has been considered as subproblem of semantic parsing and it is assumed that it +is done beforehand, while (Krishnamurthy et al. 2017) has combined entity link- +ing with semantic parsing. There are many formalisms in semantic parsing like +abstract meaning representation(AMR), discourse representation structure(DRS), +structured query language (e.g., SQL), and lambda calculus. (Kapanipathi et al. +2020) uses AMR for KBQA which integrates multiple, reusable modules like se- +mantic parser, entity and relationship linkers, and neuro-symbolic reasoner. In- +stead of KBQA, the source of data could be based on tables. In this category, the +answer format could be short term entity. It could also be a free form text like the +FeTaQA dataset in (Nan et al. 2022) . +Semantic parsing is the building block of many challenging problems in ar- +tificial intelligence such as dialogue systems, question answering , and enhances +technologies based on conversational AI, reading comprehension, and story gener- +ation. Traditional approaches to question answering such as (Zhou et al. 2018),(Shi +et al. 2021),(Zhang et al. 2022),(Ren et al. 2021) do not leverage semantic parsing +and therefore their approaches are less explainable and interpretable and is hard to +generalize. This is even harder for open domain question answering such as (Sun +et al. 2018),(Sun et al. 2019). There are three approaches to parsing in general, +namely top-to-bottom, bottom-up, hybrid. Although the bottom up constituency +parsing introduced in (Yang & Tu 2022) is efficient but it is not scalable since +getting these complex sequence annotations is expensive from crowdsourcing per- +spective. This suggests two paradigms to handle this problem. The first idea is +to use distant supervision such that the error from question answering problem is +backpropagated down to semantic parsing. The second idea which is proposed by +(Wolfson et al. 2020) creates a middle layer to make the crowdsourcing cheaper +and more scalable. Section 2 demonstrates why some approaches are not scalable +and are expensive to be implemented in practice. Section 3 shows how QDMR +formalism is helpful for any scalable algorithm for semantic parsing. In section 4 +a model is suggested. One of the main contributions of the present paper is em- +phasizing on lexicon-style alignments and disentangled information processing. +In recent years, there has been interest on leveraging semantic tagging for seman- +tic parsing as is done in (Zheng & Lapata 2020) by first seeing semantic tags as +latent variables and then using these semantic tags sequence to learn the logical +form like either SQL type or lambda calculus. The training can be done either +separately or jointly in an End-to-End way. +2 + +2 +Expensive Semantic Parsing +Creating treebank is the first major challenge of current semantic parsing methods. +This issue becomes even more dramatic than creating treebanks in syntactic pars- +ing, since apart from ambiguities, the crowdsourcing agents are more expensive +and each sentence takes more time to be annotated and annotators should be famil- +iar with complex formalisms like combinatory categorial grammar(CCG),lambda +calculus, type raising and composition in combinatory categorial grammar (CCG). +Using pointer network as is done in (Yang & Tu 2022) needs expensive crowd- +Figure 1: expensive dataset +sourcing. +Although it was used for syntactic parsing, one can use the same +methodology and apply it to semantic parsing by creating a dataset like figure 1 +which is very expensive in practice. Such an imaginary expensive model is shown +in figure 2. +A typical sentence in QDMR dataset (Wolfson et al. 2020) can be parsed us- +ing semantic operators as is shown in figure 3. For example, when the decoder +is at cursor 6, it points to boundary 12(of making database systems usable) and +semantic operator label for it is "filter" . +Recently, semantic parsing modeling is being done in stages like (Dong & La- +pata 2018) which handles the input utterance in some steps ranging from coarse +level to fine details. Thus, they first generate a rough sketch of its meaning, where +variable names and arguments is glossed over. Then, missing details in sketch +itself is filled in appropriately by the details inside the input utterance. Another +example of staging is (Wolfson et al. 2020) which tackles the problem from a +different perspective by creating a middle layer that is much easier to annotate +for crowdsourcing and does not need any expertise in complex logical forms, +lambda calculus and CCG. The next section shows how this staging mechanism +in BREAK dataset could accelerate annotation process and create a big dataset +relatively cheaply. +3 + +question_id,question_text,decoder_input_span,decoder_input_label,decoder_output_pointed,decoder_output label +ACADEMIc_train_0,return me the homepage of PVLDB .,start_from start_to 0 2 2 4,start span select,2 4 7,span +select filter +AcADEMIc_train_1,return me the homepage of H.V.Jagadish .,start_from start_to 0 2 2 4,start span select,2 4 +7,span select filter +AcADEMIc_train_1o,return me the number of references of making database systems usable .,start_from start_to +0 2 2 5 5 6 6 12 12 2,start span skip select filter skip,2 5 6 12 2 5,span skip select filter skip aggregate +Retrieval .,start_from start_to 0 2 2 4 4 6 6 9 9 14 14 2,start span skip select filter filter skip,2 4 6 9 +14 2 4,span skip select filter filter skip aggregate +.,start_from start_to 0 2 2 8 8 9 9 14 14 4 4 6 6 2,start span skip select filter skip project skip,2 8 9 14 +4 6 2 4,span skip select filter skip project skip aggregateFigure 2: expensive model +3 +Question Decomposition +Many of methods for question answering like (Yavuz et al. 2022) are not using +semantic parsing. One reason is because current semantic parsing formalisms +are expensive from dataset development perspective and also hard to implement. +Question decompositon research is rapidly growing as is mentioned in (Min et al. +2019),(Perez et al. 2020). A good approach to distant supervision in semantic +parsing is to use backpropagation of errors that are generated from the gold solu- +tion in Figure 4. Thus the semantic logical rules in each subproblem in this ques- +tion decomposition are considered as latent variables and are not directly involved +in supervision. Question decomposition is so inspiring that (Wolfson et al. 2020) +introduced BREAK dataset and defined QDMR(Question Decomposition Mean- +ing Representation) and contains over 83K pairs of questions and their QDMRs +which can also be used for open domain question answering. BREAK dataset has +thirteen operators and five of them is shown in Figure 5. By leveraging CopyNet in +(Gu et al. 2016) for BREAK dataset, this semantic parsing problem can be seen as +a machine translation problem. Although problem seems to be solved with more +4 + +embedding +encoder +decoder +label embedding +span embedding +pointed,label +cursor +pointed +5, aggregate +2, skip +12, filter +6, select +5, skip +2, span +0 +2 +6 +embedding +[0,2] +[6,12] +[12,2] + +2 +5 +5. +number +making +database + +return +me +the +4. +references +systems +usable + +start +span +skip +select +filter +skipFigure 3: expensive parsing +than 70 percent accuracy, but this approach to modeling lacks interpretability and +therefore compositionality is necessary for better generalization. +4 +Model +The motivation of using POS tags instead of the word tokens is reducing the com- +plexity of the model. There is an even better representation that POS which is +called "universal semantic tags" in (Abzianidze & Bos 2017) as it includes se- +mantic virtues of POS-tags and Named Entity (NE) classes but is not used in the +present paper. +Using word2vec for words of the sentence would assign a high dimensional +vector to each word while assigning a small size vector to each POS tag expresses +that tag sufficiently and there is no need to represent words by vectors which +reduces the curse of dimensionality. As will be shown in the experiments in the +next section, It suffices that two or three dimensional vector for each tag capture an +expressive representation. POS tags embeddings are learnt jointly with semantic +operators. Thus, this would lead to a good tradeoff for model complexity to have +less prediction error and also avoiding overfitting. +5 + +aggregate +filter +the number +[2,5] +select +of making database systems usable +[6,12] +return me +references +[0,2] +[5,6]Figure 4: question decomposition +Figure 5: BREAK +4.1 +Model Overview +The proposed model is just like a standard encoder decoder network with attention +like (Bahdanau et al. 2014). xt is the input POS tag sequence of a sentence, and yt +is the resulting semantic operator sequence. Gated recurrent units(GRU) is used +for both encoder and decoder since they are relatively faster than LSTM and they +have less parameters. The hidden vectors of the encoder are: +h(t) = f(xt, ht−1) +(4.1) +6 + +Question: What feature lends its name to the county sharing a bridge with the county +containing Harrison Township? +Answer: Ohio River +Subquestion 1: What is the county containing Harrison Township? Y +Subquestion 2: What is the county sharing a bridge with Y? X +Subquestion 3: What feature lends its name to X? Answer to the main question +Question: What is the capital of the county, that borders another county, that is +next to a third county, where Scott Township is located? +Answer: Stroudsburg +Subquestion 1: what is the county where Scott Township is located? Z +Subquestion 2: what is the county that is next to Z? Y +Subquestion 3: what is the county that borders Y? X +Subquestion 4: what is the capital of X? Answer to the main questionOperator +Template / Signature +Question +Decomposition +Return [entities] +How many touchdowns were scored +1. Return touchdowns +Select +s个M +overall? +2. Return the number of #1 +Return [ref] [condition] +I would like a flight from Toronto to +1. Return flights +Filter +So,w→ So +San Diego please. +2. Return #1 from Toronto +3. Return #2 to San Dieg0 +Return [relation] of [ref] +Who is the head coach of the Los +1. Return the Los Angeles Lakers +Project +W,Se → So +Angeles Lakers? +2. Return the head coach of #1 +Return [aggregate] of [ref] +How many states border Colorado? +1. Return Colorado +Aggregate +u← s^66em +2. Return border states of #1 +3. Return the number of #2 +Return [aggregate] [ref1] for each +How many female students are there +1. Return clubs +Group +[ref2] +in each club? +2. Return female students of #1 +us← 's*0s6bem +3. Return the number of #2 for each #1The probability of each semantic operator sequence is : +p(y) = +T� +t=1 +p(yt|y1, . . . , yt−1, x) +p(yt|y1, . . . , yt−1, x) = g(yt−1, st, ct) +(4.2) +where st is the hidden state of the decoder at time t and ct is the context at time +t. The weights of Bahdanau attention (Bahdanau et al. 2014) is used to attend to +different POS tags to align semantic operators with POS tags. Thus, the context +vector ct is written as: +ct = +Tx +� +j=1 +αtjhj +(4.3) +where the weights αtj are as follows: +αtj = +exp(a(st−1, hj)) +�Tx +k=1 exp(a(st−1, hk)) +(4.4) +The alignment model is the following single layer perceptron: +a(st−1, hj) = vT +a tanh(Wast−1 + Uahj) +(4.5) +where va, Wa, Ua are weights that should be trained. Finally, the function g in +equation 4.2 is used to predict the semantic operators and is simply models as +a linear layer with weights Wop acting on concatenation of previous predicted +semantic operator, context at time t, and previous hidden state of decoder. Thus: +g(yt−1, st, ct) = Wop(yt−1; ct; st−1) +(4.6) +4.2 +Experiments +Figure 6 shows the result of the first experiment. In this case, the embedding +dimension as well as hidden state dimension of the encoder is kept as small as 3 +since there is no need to increase the complexity of the model. Teacher forcing +has been used to accelerate the speed of training. The dimension for both encoder +and decoder embedding and hidden states in the second experiment in figure 7 is +increased, but no significant improvement in accuracy has been observed, which +once again reveals that POS tags do not need a big size vector to be represented. +There are just 13 semantic operators and therefore the complexity of embedding +7 + +Figure 6: experiment 1 +Figure 7: experiment 2 +and hidden states of the decoder is also kept small to reduce model complexity. A +scheduler is used in both experiments to reduce the learning rate every 10 epochs. +A pretrained word2vec model for POS tags could have been used but the present +paper learns the embeddings of both POS tags and semantic operators along with +the model jointly. +8 + +Model Loss +Model Accuracy +Train +Train +Validation +F 6L0 +Validation +1.76 +0.78 +0.77 - +1.75 +0.76 +1.74 +0.75 +0.74 - +1.73 +0.73 +1.72 +0.72 +0.0 +2.5 +5.0 +7.5 +10.0 +12.5 +15.0 +17.5 +0.0 +2.5 +5.0 +7.5 +10.0 +12.5 +15.0 +17.5Model Loss +Model Accuracy +1.81 +Train + Train +Validation + Validation +1.80 - +0.76 - +1.79 +0.74 +1.78 +1.77 +0.72 +1.76 +1.75 +0.70 +1.74 +0.68 - +1.73 +0 +5 +10 +15 +20 +25 +30 +0 +5 +10 +15 +20 +25 +30Table 1: Experiments +ex +opt +epochs +starting +lr +batch +size +teacher +forcing +encoder +emb- +Dim +encoder +hid- +Dim +decoder +emb- +Dim +decoder +hid- +Dim +1 +Adam +20 +1e-3 +10 +0.5 +3 +3 +3 +3 +2 +SGD +30 +1e-2 +5 +0.5 +5 +10 +4 +12 +5 +Applications +Two applications of the model in section 4 is given. The first application shows +how semantic operator prediction could be used for a more expressive CopyNet +Model by adding semantic operators as an extra feature. The second application +shows how these operators could be used to have more expressive scores in the +graph based approach which can be trained by max-margin loss or even a simple +cross entropy loss as is used in (Pasupat et al. 2019). +5.1 +Conditioning To Enhance CopyNet +Leveraging CopyNet idea (Gu et al. 2016) for supervised learning of QDMR is +straightforward and is done by many researchers. The semantic operator predic- +tion in the present paper can be used as an extra feature for CopyNet and It could +be implemented in different ways. The original formulation of copyNet in (Gu +et al. 2016) uses the following probability to generate a target word yt: +p(yt|st, yt−1, ct, M) = p(yt, g|st, yt−1, ct, M) + p(yt, c|st, yt−1, ct, M) +(5.1) +where M = {h1, . . . , hTS}, g is the generator-mode, c is the copy-mode and ct +is the context at time t. Now the result of the model in section 4 could be used +to condition on an extra expressive feature which is semantic operator alsopt at +decoder time step t. Thus, +p(yt|st, yt−1, ct, M, alsopt) = p(yt, g|st, yt−1, ct, M, alsopt)+p(yt, c|st, yt−1, ct, M, alsopt) +(5.2) +There are many ways to model p(yt, g|st, yt−1, ct, M, alsopt) but the new problem +is how to align the semantic operator prediction called by sopt′ with the decoder +9 + +time steps to model alsopt. Note that t′ in sopt′ refers to decoder for operator pre- +diction while t in alsopt refers to time step of the decoder in CopyNet model and +they should be aligned. One idea is to define two actions namely "use_current" ac- +tion and "use_next". This can be modeled by a softmax function followed by mul- +tilayer perceptron(MLP) to predict these two labels. The first label "use_current" +informs the decoder to just use the current prediction of semantic operator and +they are still aligned. The second label "use_next" expresses the fact that a mis- +alignment has occurred and it has to move the pointer one step forward to make +both sequences align. Thus the following MLP is used to model it: +action(t) = softmax(MLP(yt−1, sopt′)) +(5.3) +where yt−1 in equation 5.3 shows that the action is very sensitive to the words +that are produced by the CoyNet decoder model. It is also sensitive to the value +operator prediction at time step t′ of the latent model. Now, alsopt is obtained by +the following relation: +alsopt+1 = +� sopt′, +if action is current +sopt′+1, +if action is next +� +The simplest idea is adding a new loss coming conditioning also on sopt which +is latent variable with value from 13 operators. This can be imagined as an Ex- +pectation Maximization(EM) model that in the expectation step, the operator pre- +diction model of the present paper is calculated and in the maximization step the +parameter of CopyNet model are learnt. Training could be separated or end to +end. Thus, the following negative likelihood should be minimized. +Lenh = − 1 +N +N +� +k=1 +T +� +t=1 +log[p(y(k) +t |y(k) + 0 there exists f ∈ C(A)+⊗C(B)+ +such that 0 ≤ |g|−f ≤ δ ˆχ(1A⊗B). Since C(A)¯⊗C(B) is Archimedean, choose δ > 0 +such that |g| ∧ δ ˆχ(1A⊗B) ̸= |g|. Then f is nonzero. We have shown that T (f) ̸= 0 +when 0 ̸= f ∈ C(A) ⊗ C(B). Since T is a Riesz homomorphism, 0 < T (f) ≤ |T (g)|. +Therefore, T (g) ̸= 0, and T is a Riesz isomorphism. Consequently, C(A)¯⊗C(B) is +Riesz isomorphic to C(A ⊗ B). +□ +3. Applications +In this section, we use Theorem 2.1 to provide a solution for Fremlin’s problem +315Y(f) in [6]. The statement leads to an observation on Dedekind completeness +in the Fremlin tensor product of place functions and a statement on bands in the +Fremlin tensor product of infinite dimensional Archimedean Riesz spaces. +Problem 3.1. (Fremlin, 315Y(f) of [6]) Let A and B be Boolean algebras. A⊗B is +complete if and only if either A = {0} or B = {0} or A is finite and B is complete +or B is finite and A is complete. +Proof. If A = {0} or B = {0}, the result is trivial. Assume A and B are nontrivial +Boolean algebras. +Suppose A⊗B is complete. It follows from Theorems 2.1 and 1.13 that C(A⊗B) ∼= +C(A)¯⊗C(B) is Dedekind complete. By Proposition 3.6 of [8], C(A) and C(B) are +Dedekind complete. From Theorem 1.13, A and B are complete. It remains to show +that one of the Boolean algebras is finite. However, the Dedekind completeness of +C(A)¯⊗C(B) implies that C(A) ∼= c00(I) for a set I ⊆ C(A) or C(B) ∼= c00(J) for a +set J ⊆ C(B) (see Theorem 1.6). Since each Carath´eodory space of place functions +contains a unit, C(A) or C(B) is finite dimensional. Thus, A is finite or B is finite. +The sufficiency is proven analogously via Theorem 1.6. +□ +Corollary 3.2. Let A and B be nontrivial Boolean algebras. C(A)¯⊗C(B) is Dedekind +complete if and only if one of A or B is finite and the other is complete. +Recall that for an Archimedean Riesz space E, its collection of bands, denoted +B(E), forms a complete Boolean algebra. +Our last application shows that for +Archimedean Riesz spaces E and F, the set of bands in E ¯⊗F is rarely Boolean +isomorphic to B(E) ⊗ B(F). That is, if E and F are infinite dimensional, not every +band B of E ¯⊗F can be “decomposed” into the Fremlin tensor product of a band +in E and a band in F. + +ON THE BOOLEAN ALGEBRA TENSOR PRODUCT +9 +Corollary 3.3. Let E and F be infinite dimensional Archimedean Riesz spaces. +Then B(E) ⊗ B(F) is not Boolean isomorphic to B(E ¯⊗F). +Proof. By Lemma 1.9, neither B(E) nor B(F) is finite. Then B(E) ⊗ B(F) is not +complete by Problem 3.1. However, Theorem 1.7 states that the Boolean algebra +of bands is complete for any Archimedean Riesz space, so B(E ¯⊗F) is complete. +□ +References +1. C. D. Aliprantis and O. Burkinshaw, Positive operators, Springer, Dordrecht, 2006. +2. P. D. Allenby and C. C. A. Labuschagne, On the uniform density of C(X)⊗C(Y ) in C(X×Y ), +Indag. Math. (N.S.) 20 (2009), no. 1, 19–22. +3. G. Buskes, B. de Pagter, and A. van Rooij, The Loomis-Sikorski theorem revisited, Algebra +Universalis 58 (2008), no. 4, 413–426. +4. G. Buskes and L.P. Thorn, Two results on Fremlin’s Archimedean Riesz space tensor product, +arXiv:2206.06283 (July 2022). +5. D. H. Fremlin, Tensor products of Archimedean vector lattices, Amer. J. Math. 94 (1972), +777–798. +6. +, Measure theory. Vol. 3, Torres Fremlin, Colchester, 2004, Measure algebras, Cor- +rected second printing of the 2002 original. +7. Casper Goffman, Remarks on lattice ordered groups and vector lattices. I. Carath´eodory func- +tions, Trans. Amer. Math. Soc. 88 (1958), 107–120. +8. J. Grobler, Lattice tensor products in different categories of Riesz spaces, Research Gate (July +2022). +9. J´an Jakub´ık, On vector lattices of elementary Carath´eodory functions, Czechoslovak Math. +J. 55(130) (2005), no. 1, 223–236. +10. W. A. J. Luxemburg and A. C. Zaanen, Riesz spaces. Vol. I, North-Holland Publishing Co., +Amsterdam-London, 1971. +11. A. C. Zaanen, Introduction to operator theory in Riesz spaces, Springer-Verlag, Berlin, 1997. +Department of Mathematics, University of Mississippi, University, MS 38677 +Email address: mmbuskes@olemiss.edu +Department of Mathematics, University of Mississippi, University, MS 38677 +Email address: lthorn@go.olemiss.edu + diff --git a/bNFKT4oBgHgl3EQfoi4x/content/tmp_files/load_file.txt b/bNFKT4oBgHgl3EQfoi4x/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8cd81855c1555a9c089f9473cc55e5ca9690aac1 --- /dev/null +++ b/bNFKT4oBgHgl3EQfoi4x/content/tmp_files/load_file.txt @@ -0,0 +1,393 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf,len=392 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content='11866v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content='FA] 27 Jan 2023 ARXIV SUBMISSION Volume 00, Number 0, Pages 000–000 S 0002-9939(XX)0000-0 ON THE BOOLEAN ALGEBRA TENSOR PRODUCT VIA CARATH´EODORY SPACES OF PLACE FUNCTIONS GERARD BUSKES AND PAGE THORN Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' We show that the Carath´eodory space of place functions on the free product of two Boolean algebras is Riesz isomorphic with Fremlin’s Archimedean Riesz space tensor product of their respective Carath´eodory spaces of place functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' We provide a solution to Fremlin’s problem 315Y(f) in [6] con- cerning completeness in the free product of Boolean algebras by applying our results on the Archimedean Riesz space tensor product to Carath´eodory spaces of place functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Introduction and preliminary material Fremlin asserts in problem 315Y(f) of [6] that the Boolean algebra tensor product of two nontrivial Boolean algebras is complete if and only if one is finite and the other is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' In Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content='6 of [4], we prove that the Fremlin tensor product of two Dedekind complete Riesz spaces rarely is Dedekind complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' In fact, if the tensor product is Dedekind complete, then one of the two spaces is Riesz isomorphic to the set of all finite-valued functions on a subset of that space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' To connect 315Y(f) of [6] with Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content='6 of [4], we employ Carath´eodory’s Riesz space of place functions on a Boolean algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' The main result is Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content='1 with applications given in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' The necessary terms for Boolean algebras, the free product, Riesz spaces, and Carath´eodory spaces of place functions are provided in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' We reserve A, B for Boolean algebras and E, F, G for Archimedean Riesz spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Boolean algebras and their free product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' For Boolean algebras, see chapter 31 of [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Two elements x and y of a Boolean algebra are called disjoint if x∧y = 0, in which case we write x ⊥ y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Two subsets A and B of a Boolean algebra are called disjoint if x ⊥ y for every x ∈ A and y ∈ B, in which case we write A ⊥ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' We define the disjoint sum of two elements x and y in a Boolean algebra by x ⊕ y = (x ∧ y′) ∨ (x′ ∧ y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' A Boolean algebra is complete if every nonempty subset has a supremum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' (312F of [6]) Let A and B be Boolean algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' A function χ: A → B is said to be a Boolean homomorphism if for all x, y ∈ A, (i) χ(x ∧ y) = χ(x) ∧ χ(y);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Received by the editors January 27, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' 46A40 , 46M05, 06E99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Riesz space, Vector lattice, Boolean algebra, Tensor product, Free product, Dedekind complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' ©XXXX American Mathematical Society 1 2 GERARD BUSKES AND PAGE THORN (ii) χ(x ⊕ y) = χ(x) ⊕ χ(y);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' (iii) χ(1A) = 1B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' A bijective Boolean homomorphism is called a Boolean isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' If there exists an isomorphism χ: A → B, then the Boolean algebras A and B are said to be Boolean isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Proposition 312H of [6] proves additionally that every Boolean homomorphism preserves finite suprema, that is χ(x ∨ y) = χ(x) ∨ χ(y) for every x, y ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' The Stone space of a Boolean algebra A is the set Z of nonzero ring homomor- phisms from A to Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Set ˆa = {z : z ∈ Z, z(a) = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' By Stone’s Theorem (see, for instance, 311E of [6]), the canonical map a �→ ˆa : A → P(Z) is an injective ring homomorphism which we call the Stone representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' For more on Stone spaces, see [6] where Fremlin defines and utilizes Stone spaces to define the Boolean algebra tensor product, called the free product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' (Fremlin, 315I of [6]) (i) Let {Ai}i∈I be a family of Boolean algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' For each i ∈ I, let Zi be the Stone space of Ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Set Z = � i∈I Zi, with the product topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Then the free product of {Ai}i∈I is the algebra of open-and-closed sets in Z, denoted � i∈I Ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' (ii) For i ∈ I and a ∈ Ai, the set ˆa ⊆ Zi representing a is an open-and-closed subset of Zi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' because z �→ z(i): Z → Zi is continuous, ǫi(a) = {z : z(i) ∈ ˆa} is open-and-closed, so belongs to A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' In this context, ǫi : Ai → A is called the canonical map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' In the following theorem, we list the necessary material from 315J and 315K of [6] in the language of Fremlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Let {Ai}i∈I be a family of Boolean algebras, with free product A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' (i) The canonical map ǫi : Ai → A is a Boolean homomorphism for every i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' (ii) For any Boolean algebra B and any family {ϕi}i∈I such that ϕi is a Boolean homomorphism from Ai to B for every i, there is a unique Boolean homo- morphism ϕ: A → B such that ϕi = ϕ ◦ ǫi for each i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' (iii) Write C for the set of those members of A expressible in the form infj∈J ǫj(aj), where J ⊆ I is finite and aj ∈ Aj for every j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Then every member of A is expressible as the supremum of a disjoint finite subset of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' (iv) A = {0A} if and only if there is some i ∈ I such that Ai = {0Ai}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' (v) If Ai ̸= {0A} for every i ∈ I, then ǫi is injective for every i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' (vi) Let Ai ̸= {0A} for every i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' If J ⊆ I is finite and aj is a nonzero member of Aj for each j ∈ J, then infj∈J ǫj(aj) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Archimedean Riesz spaces and their tensor product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' See [11] for Archimedean Riesz spaces and [5] for Riesz bimorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' ON THE BOOLEAN ALGEBRA TENSOR PRODUCT 3 Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content='2 of [5]) Let E and F be Archimedean Riesz spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' There exists an Archimedean Riesz space G and a Riesz bimorphism ϕ: E × F → G such that whenever H is an Archimedean Riesz space and ψ: E × F → H is a Riesz bimorphism, there is a unique Riesz homomorphism T : G → H such that T ◦ ϕ = ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' G of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content='4 is the Archimedean Riesz space tensor product of E and F, denoted by E ¯⊗F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' The “universal property of E ¯⊗F” refers to the implication that any Archimedean Riesz space paired with a Riesz bimorphism satisfying Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content='4 is Riesz isomorphic to G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' The Riesz bimorphism ⊗: E × F → E ¯⊗F embeds the algebraic tensor product E ⊗ F into E ¯⊗F via ⊗(e, f) = e ⊗ f for all e ∈ E and f ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' We define a few terms needed for the statement of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Let E be an Archimedean Riesz space and let I be a nonempty set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' c00(I, E) is the set of all maps f : I → E for which S(f) = {x ∈ I : f(x) ̸= 0} is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' If E = R, then c00(I, E) is written c00(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Let f and g be elements of a Riesz space E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' The ideals generated by f and g are denoted by Ef and Eg respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' We denote the principal bands generated by f and g with [f] and [g] respectively (see, for instance, pg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' 30 of [11] for definitions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' A Riesz space is Dedekind complete if every bounded subset of E has a supremum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Every Dedekind complete Riesz space is Archimedean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' In [4], we characterized when the tensor product of two Dedekind complete Riesz spaces is Dedekind complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content='6 of [4]) Suppose E and F are Dedekind complete Riesz spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' The following are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' (1) Ex ¯⊗Fy is Dedekind complete for every x ∈ E+ and y ∈ F +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' (2) [Ex is finite dimensional for every x ∈ E+] or [Fy is finite dimensional for every y ∈ F +].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' (3) E ∼= c00(I) for a set I ⊆ E or F ∼= c00(J) for a set J ⊆ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' (4) E ¯⊗F ∼= c00(I, F) for a set I ⊆ E or E ¯⊗F ∼= c00(J, E) for a set J ⊆ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' (5) E ¯⊗F is Dedekind complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' As an intermediary between an Archimedean Riesz spaces and Boolean algebras, we consider Boolean algebras of bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' The following three statements are used in Section 3 and are given for the reader’s convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' (22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content='6, 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content='8 of [10]) Let E be a Riesz space and define B(E) = {B ⊆ E : B is a band}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' B(E) is an order complete distributive lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' B(E), partially ordered by inclusion, is a Boolean algebra if and only if E is Archimedean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Let E be a Riesz space and f, g ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Then |f| ∧ |g| = 0 implies [f] ⊥ [g].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Let |f| ∧ |g| = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Certainly, Ef ⊥ Eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Suppose h1 ∈ [f] and h2 ∈ [g].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Using the fact that Ef ⊥ Eg, it is straightforward to show that |h1| ∧ |h2| = 0 via 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content='1 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content='8 of [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Thus, [g1] ⊥ [g2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' □ 4 GERARD BUSKES AND PAGE THORN Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' If E is an infinite dimensional Archimedean Riesz space, then B(E) is not finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' By the contrapositive of Theorem 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content='10 in [10], there is an infinite subset of mutually disjoint nonzero elements in E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Thus, there are an infinite number of mutually disjoint bands in E by Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' □ Carath´eodory spaces of place functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' (pg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' 40 of [1]) Let E be a Riesz space and e ∈ E+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Then x ∈ E+ is said to be a component of e whenever x ∧ (e − x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' The collection of all components of e, denoted C(e), is a Boolean algebra under the partial ordering induced by E (pg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' 40 of [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' With e as a strong order unit (pg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' 51 of [11]), a connection between Archimedean Riesz spaces and Boolean algebras is described explicitly in the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content='1 of [3]) Let A be a Boolean algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' There exists an Archimedean Riesz space E with a strong unit e with the following properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' (i) There exists a Boolean isomorphism χ: A → C(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' (ii) E is the linear span of C(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' (E, χ) is unique up to isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' It is denoted by C(A) and is called the space of place functions on A in the sense of Carath´eodory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Let λi, γj ∈ R be nonzero;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' n, m ∈ N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' xi ∈ A be pairwise disjoint;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' and yj ∈ A be pairwise disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Two elements f = n � i=1 λiχ(xi) and g = m � j=1 γjχ(yj) are equivalent if �n i=1 xi = �m j=1 yj and if λi = γj whenever xi ∧ yj ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' C(A) is the set of all such equivalence classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Henceforth, we take f = �n i=1 λiχ(xi) to represent all elements of C(A) that are equivalent to f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' We define addition in C(A) in the style of Goffman in [7] and Jakubik in [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' For a different approach, see [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' For x, y ∈ A, let x −1 y be the complement of x ∧ y relative to x, that is, x ∧ (x ∧ y)′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Then addition in C(A) is defined by f + g = n � i=1 m � j=1 (λi + γj)χ(xi ∧ yj) + n � i=1 λiχ(xi −1 m � j=1 yj) + m � j=1 γjχ(yj −1 n � i=1 xi) where in the summation only those terms are taken into account in which λi+γj ̸= 0 and the elements xi ∧ yj, xi −1 � yj, and yj −1 � xi are non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' It is routine to verify that addition is well-defined in C(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Jakubik proves in [9] that the completeness of a Boolean algebra is equivalent to the Dedekind completeness of its Carath´eodory space of place functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' However, his propositions assume complete distributivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Since this work has no need for a Boolean algebra to be completely distributive, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content='13 is proven with credit to Propositions 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content='2(a) and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content='6 of [9] for its similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' (page 231 of [9]) Let Y be a sublattice of a lattice X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Y is said to be a regular sublattice of X if: (i) whenever x0 ∈ Y and ∅ ̸= X ⊆ Y such that x0 = supY X, then x0 = supX X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' and ON THE BOOLEAN ALGEBRA TENSOR PRODUCT 5 (ii) whenever x1 ∈ Y and ∅ ̸= X ⊆ Y such that x1 = infY X, then x1 = infX X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Let A be a Boolean algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' A is complete if and only if C(A) is Dedekind complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Assume that A is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Let D be a bounded subset of C(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Then there exists g ∈ C(A) such that g ≥ f for every f ∈ C(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Find λi ∈ R, n ∈ N, and xi ∈ A such that g = �n i=1 λiχ(xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Set x = x1 ∨ · · · ∨ xn and λ = max{λ1, · · · , λn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Then D ⊆ [0, λχ(x)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' By assumption, the interval [0, x] is complete in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' It follows from Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content='4 of [9] that A is a regular subset of C(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Then the interval [0, χ(x)] is complete as a subset of C(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' In particular, [0, λχ(x)] is complete, so sup(D) exists in C(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' To prove sufficiency, assume that C(A) is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Let χ: A → C(e) be the Boolean isomorphism from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Note that e = χ(1A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Let D be a subset of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Since C(A) is Dedekind complete, sup χ(D) exists in C(A)+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' For every x ∈ D, χ(x) is a component of χ(1A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Thus, sup χ(D) = 2 sup χ(D) ∧ χ(1A) so that 0 = sup χ(D) ∧ (χ(1A) − sup χ(D)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' By definition, sup χ(D) is a component of e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Let y = χ−1(sup χ(D)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Since χ is a Boolean isomorphism, y is an upper bound for D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Suppose there exists y′ such that x ≤ y′ < y for every x ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Then χ(y′) ≥ supx∈D χ(x) = χ(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Thus, χ(y′) = χ(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Since χ is one-to-one, y′ = y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Therefore, y = sup(D) exists in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' The Fremlin tensor product of Carath´eodory spaces of place functions In this section, we relate Boolean algebras A, B, and A⊗B to their Carath´eodory spaces of place functions C(A), C(B), and C(A ⊗ B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' The notation of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content='11 is used with the addition of subscripts to indicate which Boolean algebra is at work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' The symbols in (1), (2), and (3) will be used freely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' (1) χA : A → C(A), χB : B → C(B), and ˆχ: A⊗B → C(A⊗B) are the Boolean isomorphisms from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' (2) C(A), C(B), and C(A⊗B) have units χA(1A), χB(1B), and ˆχ(1A⊗B) re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' (3) ǫA : A → A ⊗ B and ǫB : B → A ⊗ B are the canonical Boolean homomor- phisms in Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' C(A)¯⊗C(B) and C(A ⊗ B) are Riesz isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Assume that A and B are nontrivial Boolean algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' For f ∈ C(A), there exist n ∈ N, pairwise disjoint xi ∈ A, and nonzero λi ∈ R such that f = �n i=1 λiχA(xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' For g ∈ C(B), there exist m ∈ N, pairwise disjoint uj ∈ B, and nonzero γj ∈ R so g = �m j=1 λjχB(uj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Define ψ: C(A) × C(B) → C(A ⊗ B) by ψ(f, g) =ψ \uf8eb \uf8ed n � i=1 λiχA(xi), m � j=1 γjχB(uj) \uf8f6 \uf8f8 = n � i=1 m � j=1 (λiγj)ˆχ(ǫA(xi) ∧ ǫB(uj)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' 6 GERARD BUSKES AND PAGE THORN It follows from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content='3 (iv) and (vi) that the definition of ψ is independent of the representations chosen for f and g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Let f1 = f and f2 = �p k=1 δkχA(yk) for nonzero δk ∈ R, p ∈ N and pairwise disjoint yk ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Recall that f1 + f2 is defined to be � i � k (λi + δk)χA(xi ∧ yk) + � i λiχA(xi−1 � k yk) + � k δkχA(yk−1 � i xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Claim: ψ is bilinear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' ψ(f1 + f2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' g) =ψ \uf8eb \uf8ed� i λiχA(xi) + � k δkχA(yk),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' � j γjχB(uj) \uf8f6 \uf8f8 = � i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content='j (λi + δk)γj ˆχ (ǫA(xi ∧ yk) ∧ ǫB(uj)) + � i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content='j (λiγj)ˆχ � ǫA(xi−1 � k yk) ∧ ǫB(uj) � + � k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content='j (δkγj)ˆχ � ǫA(yk−1 � i xi) ∧ ǫB(uj) � = � i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content='j �� k (λiγj)ˆχ (ǫA(xi ∧ yk) ∧ ǫB(uj)) + (λiγj)ˆχ � ǫA(xi−1 � k yk) ∧ ǫB(uj) �� + � k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content='j �� i (δkγj)ˆχ (ǫA(xi ∧ yk) ∧ ǫB(uj)) + (δkγj)ˆχ � ǫA(yk−1 � i xi) ∧ ǫB(uj) �� = � i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content='j (λiγj) � ˆχ �� k ǫA(xi ∧ yk) ∧ ǫB(uj) � + ˆχ � ǫA(xi−1 � k yk) ∧ ǫB(uj) �� + � k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content='j (δkγj) � ˆχ �� i ǫA(xi ∧ yk) ∧ ǫB(uj) � + ˆχ � ǫA(yk−1 � i xi) ∧ ǫB(uj) �� (∗) = � i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content='j (λiγj)ˆχ(ǫA(xi) ∧ ǫB(uj)) + � k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content='j (δkγj)ˆχ(ǫA(yk) ∧ ǫB(uj)) =ψ(f1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' g) + ψ(f2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' (∗) is justified because yk ⊥ yk′ for all k ̸= k′ and xi ⊥ xi′ for all i ̸= i′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Symmetri- cally, ψ(f, g1 + g2) = ψ(f, g1) + ψ(f, g2) for f ∈ C(A) and g1, g2 ∈ C(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' It follows from the definition of ψ that ψ(λf, g) = ψ(f, λg) = λψ(f, g) for every λ ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Claim: ψ is a Riesz bimorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Assume f1 ∧ f2 = 0 and g ∈ C(B)+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Using the same representations for f1, f2, and g as above, it follows that xi ⊥ yk for all i and k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Then since the maps ˆχ and ǫA ON THE BOOLEAN ALGEBRA TENSOR PRODUCT 7 are Boolean homomorphisms and {xi}n i=1, {yk}p k=1 are each pairwise disjoint, ψ(f1, g) ∧ ψ(f2, g) =ψ \uf8eb \uf8ed� i λiχA(xi), � j γjχB(uj) \uf8f6 \uf8f8 ∧ ψ \uf8eb \uf8ed� k δkχA(yk), � j γjχB(uj) \uf8f6 \uf8f8 = \uf8eb \uf8ed� i,j (λiγj)ˆχ(ǫA(xi) ∧ ǫB(uj)) \uf8f6 \uf8f8 ∧ \uf8eb \uf8ed� k,j (δkγj)ˆχ(ǫA(yk) ∧ ǫB(uj)) \uf8f6 \uf8f8 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Likewise if f ∈ C(A)+ and g1 ∧ g2 = 0 in C(B), then ψ(f, g1) ∧ ψ(f, g2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' By Theorem 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content='1 of [11], ψ is a Riesz bimorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' It follows from the universal property of the Fremlin tensor product that there is a unique Riesz homomorphism T : C(A)¯⊗C(B) → C(A ⊗ B) such that ψ = T ◦ ⊗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' C(A) × C(B) ψ � ⊗ � C(A)¯⊗C(B) T � C(A ⊗ B) Claim: T is a Riesz isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Step 1: T is onto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Let h ∈ C(A ⊗ B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Then h = �n i=1 λi ˆχ(ei) for some pairwise disjoint ei ∈ A ⊗ B, n ∈ N, and nonzero λi ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Fix i ∈ {1, · · · , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' By Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content='3 (iii), there exists a finite disjoint subset {ǫA(ak) ∧ ǫB(bk)}m k=1 (m ∈ N) of A ⊗ B such that ei = m � k=1 ǫA(ak) ∧ ǫB(bk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Then it follows from the definition of ψ that ˆχ(ei) =ˆχ � m � k=1 ǫA(ak) ∧ ǫB(bk) � = m � k=1 ˆχ(ǫA(ak) ∧ ǫB(bk)) = m � k=1 ψ(χA(ak), χB(bk)) = m � k=1 T ◦ ⊗(χA(ak), χB(bk)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Since T preserves finite suprema, ˆχ(ei) is in the image of T for every i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' It follows from the linearity of T that h is in the image of T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Step 2: T is one-to-one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Suppose f ∈ C(A) ⊗ C(B), the algebraic tensor product of C(A) and C(B), such that f is nonzero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Then for some n ∈ N, nonzero λk ∈ R, and nontrivial xk ∈ A, uk ∈ B such that f = n � k=1 λkχA(xk) ⊗ χB(uk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' 8 GERARD BUSKES AND PAGE THORN Since ǫA, ǫB, and ˆχ are injective Boolean isomorphisms, T (f) =T � n � k=1 λkχA(xk) ⊗ χB(uk) � = n � k=1 λkψ (χA(xk), χB(uk)) = n � k=1 λk ˆχ(ǫA(xk) ∧ ǫB(uj)) ̸=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Let g be a nonzero element of the Riesz space tensor product of C(A) and C(B), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' g ∈ C(A)¯⊗C(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content='2 of [2], for all δ > 0 there exists f ∈ C(A)+⊗C(B)+ such that 0 ≤ |g|−f ≤ δ ˆχ(1A⊗B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Since C(A)¯⊗C(B) is Archimedean, choose δ > 0 such that |g| ∧ δ ˆχ(1A⊗B) ̸= |g|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Then f is nonzero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' We have shown that T (f) ̸= 0 when 0 ̸= f ∈ C(A) ⊗ C(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Since T is a Riesz homomorphism, 0 < T (f) ≤ |T (g)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Therefore, T (g) ̸= 0, and T is a Riesz isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Consequently, C(A)¯⊗C(B) is Riesz isomorphic to C(A ⊗ B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Applications In this section, we use Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content='1 to provide a solution for Fremlin’s problem 315Y(f) in [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' The statement leads to an observation on Dedekind completeness in the Fremlin tensor product of place functions and a statement on bands in the Fremlin tensor product of infinite dimensional Archimedean Riesz spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Problem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' (Fremlin, 315Y(f) of [6]) Let A and B be Boolean algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' A⊗B is complete if and only if either A = {0} or B = {0} or A is finite and B is complete or B is finite and A is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' If A = {0} or B = {0}, the result is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Assume A and B are nontrivial Boolean algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Suppose A⊗B is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' It follows from Theorems 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content='1 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content='13 that C(A⊗B) ∼= C(A)¯⊗C(B) is Dedekind complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content='6 of [8], C(A) and C(B) are Dedekind complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' From Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content='13, A and B are complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' It remains to show that one of the Boolean algebras is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' However, the Dedekind completeness of C(A)¯⊗C(B) implies that C(A) ∼= c00(I) for a set I ⊆ C(A) or C(B) ∼= c00(J) for a set J ⊆ C(B) (see Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Since each Carath´eodory space of place functions contains a unit, C(A) or C(B) is finite dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Thus, A is finite or B is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' The sufficiency is proven analogously via Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' □ Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Let A and B be nontrivial Boolean algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' C(A)¯⊗C(B) is Dedekind complete if and only if one of A or B is finite and the other is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Recall that for an Archimedean Riesz space E, its collection of bands, denoted B(E), forms a complete Boolean algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Our last application shows that for Archimedean Riesz spaces E and F, the set of bands in E ¯⊗F is rarely Boolean isomorphic to B(E) ⊗ B(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' That is, if E and F are infinite dimensional, not every band B of E ¯⊗F can be “decomposed” into the Fremlin tensor product of a band in E and a band in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' ON THE BOOLEAN ALGEBRA TENSOR PRODUCT 9 Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Let E and F be infinite dimensional Archimedean Riesz spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Then B(E) ⊗ B(F) is not Boolean isomorphic to B(E ¯⊗F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' By Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content='9, neither B(E) nor B(F) is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Then B(E) ⊗ B(F) is not complete by Problem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' However, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content='7 states that the Boolean algebra of bands is complete for any Archimedean Riesz space, so B(E ¯⊗F) is complete.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Fremlin, Tensor products of Archimedean vector lattices, Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' 94 (1972), 777–798.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' , Measure theory.' metadata={'source': 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Lattice tensor products in different categories of Riesz spaces, Research Gate (July 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' J´an Jakub´ık, On vector lattices of elementary Carath´eodory functions, Czechoslovak Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' 55(130) (2005), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' 1, 223–236.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Luxemburg and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Zaanen, Riesz spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' I, North-Holland Publishing Co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=', Amsterdam-London, 1971.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Zaanen, Introduction to operator theory in Riesz spaces, Springer-Verlag, Berlin, 1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content=' Department of Mathematics, University of Mississippi, University, MS 38677 Email address: mmbuskes@olemiss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content='edu Department of Mathematics, University of Mississippi, University, MS 38677 Email address: lthorn@go.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content='olemiss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} +page_content='edu' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFKT4oBgHgl3EQfoi4x/content/2301.11866v1.pdf'} diff --git a/btE2T4oBgHgl3EQfwQhi/content/tmp_files/2301.04099v1.pdf.txt b/btE2T4oBgHgl3EQfwQhi/content/tmp_files/2301.04099v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..aa09ecea5ce41f16b1cf0af9b0d55ba80ad3932d --- /dev/null +++ b/btE2T4oBgHgl3EQfwQhi/content/tmp_files/2301.04099v1.pdf.txt @@ -0,0 +1,1648 @@ +(Anti-)Stokes Scattering on Kinks +Jarah Evslin1,2 ∗ and Hui Liu3,2,4 † +1) Institute of Modern Physics, NanChangLu 509, Lanzhou 730000, China +2) University of the Chinese Academy of Sciences, YuQuanLu 19A, Beijing 100049, China +3) School of Fundamental Physics and Mathematical Sciences, Hangzhou Institute for +Advanced Study, University of Chinese Academy of Sciences, Hangzhou 310024, China +4) Arnold Sommerfeld Center, Ludwig-Maximilians-Universit¨at, Theresienstraße 37, 80333 +M¨unchen, Germany +Abstract +At leading order, there are three inelastic scattering processes beginning with a quan- +tum kink and a fundamental meson. Meson multiplication, in which the final state +is a kink and two mesons, was treated recently. In this note we treat the other two, +(anti)-Stokes scattering, in which the kink’s shape mode is (de-)excited and the final +state contains one meson. In the case of a general scalar kink, we find analytic formulas +for the forward and backward scattering amplitudes and probabilities as functions of +the momentum of the incident meson. The general results are then specialized to the +kink of the φ4 double-well model. +1 +Introduction +Scalar theories in (1+1)-dimensions provide some of the simplest quantum field theory +models. If the scalar field is subjected to a degenerate potential, in addition to fundamental +meson excitations there will also be nonperturbative kinks. +The classical field theory of such models is already surprisingly rich. Most attention has +focused on kink-antikink scattering [1, 2, 3, 4]. Here, it has long been known [5] that the range +of initial relative speeds leading to distinct outcomes has a fractal structure of resonance +windows. It was once thought that this results from the internal excitation spectrum of the +kinks. Certainly they play a role [6, 7, 8]. However it was then found [9, 10] that such +windows appear even in models in which the kink has no internal excitations. Thus, it has +become clear that the interactions of kinks with the bulk dynamics are important [11, 12]. +∗jarah@impcas.ac.cn +†hui.liu@campus.lmu.de +1 +arXiv:2301.04099v1 [hep-th] 10 Jan 2023 + +This bulk dynamics is itself quite rich. There are spectral walls [13, 14] beyond which in- +ternal excitations of kinks escape into the continuum. While these have striking consequences +classically, in the quantum theory they are rather smooth [15]. Also, after a kink-antikink +collision, many kinds of modes are excited. In the end, only the longest-lived remain. Among +these, the oscillons [16, 17] survive for an amazingly long time. However, again the quantum +theory appears to be different. In the quantum theory, new decay channels open which +greatly reduce the oscillon lifetime [18]. +In summary, two critical pieces of the picture are missing. +The first is a systematic +understanding of interactions between kinks and bulk degrees of freedom. The second is an +understanding of the role played by quantum corrections, and whether they disappear in the +classical limit or, as the oscillon case and perhaps the spectral wall case seem to suggest, +radically affect the physics. +This motivates an understanding of interactions between kinks and elementary meson +quanta in the quantum theory. Such interactions are much simpler than kink-antikink inter- +actions. However, the classic literature on quantum kink-meson scattering has been limited +largely to finding effective Yukawa couplings between kinks and mesons [19, 20]. +Recently, a linearized perturbation theory for such models has been developed at one- +loop in Ref. [21] and beyond in Ref. [22]. It greatly simplifies calculations in the one-kink +sector, which consists of states with a kink and any finite number of mesons, with respect +to the traditional collective coordinate approach of Refs. [23, 24]. +Using this approach it was soon realized that, at leading order, there are precisely three +inelastic scattering processes which begin with a single kink and a single meson. The first +is meson multiplication, in which the meson is absorbed by the kink and two mesons are +emitted. This was recently studied in Ref. [25]. The other two are Stokes and anti-Stokes +scattering. Stokes scattering is a process where a meson scatters off of a ground state kink +while exciting its shape mode. Anti-Stokes scattering is a process where a meson scatters +off of an excited kink and deexcites its shape mode. +In the present note, we present the first-ever treatment of these two processes. After a +review of linearized kink perturbation theory in Sec. 2, we calculate the probability, as a +function of the momentum of the incoming meson, of Stokes and anti-Stokes scattering in +Secs. 3 and 4 respectively. Our results are specialized to the φ4 double-well model in Sec. 5. +2 + +2 +Linearized Kink Perturbation Theory +2.1 +The Main Idea +In Refs. [21, 22] a new, Hamiltonian formalism has been introduced for calculations in +the kink sector of a quantum theory of a scalar field φ(x) and its conjugate π(x) in 1+1 +dimensions. The kink sector is the Fock space of states consisting of a finite number of +fundamental mesons in addition to a single quantum kink. We will refer to the Fock space +of mesons in the absence of a kink as the vacuum sector. +The vacuum sector states can be constructed in perturbation theory. One decomposes the +field in a plane wave basis, constructing creation and annihilation operators. The vacuum is +defined as the state which is annihilated by all annihilation operators, and the vacuum sector +is generated by finite numbers of creation operators acting on the vacuum. Hamiltonian +eigenstates can be found perturbatively by solving the Hamiltonian eigenvalue problem. +This perturbative approach fails for the kink sector. This is evident already in classical +theory, where it results from the fact that large moments of the field do not tend to zero. +The kink sector corresponds to classical field configurations which are close to the classical +kink solution φ(x, t) = f(x). The higher moments of φ(x, t) − f(x) are small, and so one +expects a perturbative approach in φ(x, t) − f(x) to yield kink sector states. +Linearized kink perturbation theory is a formalism for doing this in quantum field theory. +A unitary displacement operator Df is constructed in the Schrodinger picture as +Df = Exp +� +−i +� +dxf(x)π(x) +� +. +(2.1) +We use Df as a passive transformation, renaming the coordinate system of the Hilbert space +and transforming the operators that act on them. More precisely, we define the kink frame +as the coordinate system on the Hilbert space in which the ket |ψ⟩ represents the state Df|ψ⟩ +as defined in the usual, defining frame. With this definition, it is easily shown that, in the +kink frame, energies are measured and time is evolved by the kink Hamiltonian H′ +H′ = D† +fHDf +(2.2) +where H is the original Hamiltonian, which defines the theory. +What have we gained? In the kink frame, it is the kink sector which is constructed +perturbatively using creation operators. Thus in the presence of a kink, the construction +of Hamiltonian eigenstates, form factors, and even amplitudes and probabilities for various +processes are reduced to perturbative problems in the kink frame. +3 + +What have we lost? We had to choose a particular kink solution f(x). In a translation- +invariant theory, there would be a moduli space of choices f(x−x0) for every real x0. Thus we +have lost manifest translation invariance. We must work locally, close to some base point in +moduli space. However, if we are interested in translation-invariant states, or more precisely +momentum eigenstates, which we will be in this paper1, then it is sufficient to understand +any region in moduli space to understand every region. And so this will not be a problem, +we simply work perturbatively in x0, and impose translation-invariance at will to simplify +expressions. +2.2 +The Details +While we expect this formalism to apply quite generally, so far we have only applied it to +Schrodinger picture Hamiltonians of the form +H = +� +dx : H(x) :a, +H(x) = π2(x) +2 ++ (∂xφ(x))2 +2 ++ V ( +√ +λφ(x)) +λ +. +(2.3) +Here V is a degenerate potential and φ(x, t) = f(x) is a solution of the classical equations +of motion which interpolates between two degenerate minima. +The notation ::a represents normal ordering of the creation and annihilation operators +for plane waves. It is defined at a mass scale m, which has two definitions +m2 = V (2)( +√ +λf(±∞)), +V (n)( +√ +λφ(x)) = ∂nV ( +√ +λφ(x)) +(∂ +√ +λφ(x))n +(2.4) +corresponding to the scalar mass at the two minima of the potential at infinity. If these dis- +agree, then quantum corrections break the degeneracy and the kink becomes an accelerating +false vacuum bubble wall [26]. We will not consider this case. +We find the eigenstates of the kink Hamiltonian H′ perturbatively in the coupling con- +stant λ. To do this we decompose all quantities in powers of λ. For example, the energy Q +of the kink ground state is decomposed into � +i Qi where each Qi is of order O(λi−1). Note +that Q0 is just the classical energy of the classical kink solution. +1This paper will be entirely in the center of mass frame of the kink and meson, and so all states will +be eigenstates of the total momentum operator with eigenvalue zero. Wave packets will be constructed +consisting of different momenta for the meson, recalling that the kink momenta will always be equal and +opposite. +Translation-invariance is with respect to simultaneous and equal translations of the kink and +mesons. +4 + +The kink Hamiltonian itself is decomposed into terms H′ +i with i factors of the fundamental +fields, when normal ordered. These include +H′ +0 = Q0, +H′ +1 = 0, +H′ +n>2 = λ +n +2 −1 +� +dxV (n)( +√ +λf(x)) +n! +: φn(x) :a . +(2.5) +The most important is H′ +2, as its eigenvectors are the first step in the perturbative +expansion for the kink sector states. Written in terms of x it is rather odd, resembling a free +Hamiltonian but with a position-dependent mass term. +To write it more transparently, we will introduce the kink’s normal modes g(x), defined +to be small, classical fluctuations about the kink, which solve the Sturm-Liouville equation +V (2)( +√ +λf(x))g(x) = ω2g(x) + g′′(x), +φ(x, t) = e−iωtg(x). +(2.6) +They are classified by their frequency ω. The real solution gB(x) with ωB = 0 is called the +zero mode. Any real mode gS(x) with 0 < ωS < m is called a shape mode. Above this lie +the continuum modes gk(x) with ωk = +√ +m2 + k2. We fix the conventions +ωk += +√ +m2 + k2, +g∗ +k(x) = g−k(x) +(2.7) +� +dx|gB(x)|2 += +1, +� +dxgk1(x)g∗ +k2(x) = 2πδ(k1 − k2), +� +dxgS1(x)g∗ +S2(x) = δS1S2. +The normal modes generate all bounded functions, and so, instead of plane waves, we +may use them to decompose the fields [27] +φ(x) += +φ0gB(x) + +� +� dk +2π +� +B‡ +k + B−k +2ωk +� +gk(x), +B‡ +k = +B† +k +(2ωk), +B‡ +S = +B† +S +(2ωS) +(2.8) +π(x) += +π0gB(x) + i +� +� dk +2π +� +ωkB‡ +k − B−k +2 +� +gk(x), +B−S = BS, +� +� dk +2π = +� dk +2π + +� +S +into operators φ0, π0, B and B‡. This provides a new basis of our operator algebra, and any +operator may be written in terms of these operators. The canonical commutation relations +satisfied by φ(x) and π(x) imply that these satisfy +[φ0, π0] = i, +� +BS1, B‡ +S2 +� += δS1S2, +� +Bk1, B‡ +k2 +� += 2πδ (k1 − k2) . +(2.9) +Finally we are ready to write H′ +2. It is [27] +H′ +2 = Q1 + Hfree , +Hfree = π2 +0 +2 + +� +S +ωSB‡ +SBS + +� dk +2πωkB‡ +kBk. +(2.10) +5 + +Here Q1 is the one-loop correction to the kink mass. The π2 +0 term is the kinetic energy of a +free quantum-mechanical particle of mass Q0 with position operator φ0/√Q0. This particle +is the center of mass of the kink. The other terms are quantum harmonic oscillators for +the shape modes S and continuum modes k. B‡ +S excites a shape mode, while B‡ +k excites a +continuum mode. +The kink ground state |0⟩ of the kink Hamiltonian H′ can be decomposed into contribu- +tions |0⟩i, of order O(λi/2). The first term in our semiclassical expansion, |0⟩0, is the vacuum +of H′ +2. It is the ground state of each term in (2.10), and so is completely characterized by +the conditions +π0|0⟩0 = Bk|0⟩0 = BS|0⟩0 = 0. +(2.11) +3 +Stokes Scattering +In a Stokes scattering event, one meson is absorbed by a ground state kink, one meson +is emitted and a shape mode is excited. The initial condition is therefore a superposition +|Φ⟩0 += +� dk1 +2π αk1|k1⟩0, +αk = +� +dxΦ(x)gk(x) +(3.1) +Φ(x) += +Exp +� +−(x − x0)2 +4σ2 ++ ixk0 +� +, +x0 ≪ − 1 +m, +1 +k0 +, 1 +m ≪ σ ≪ |x0| +of one-meson states +|k1⟩0 = B‡ +k1|0⟩0 +(3.2) +in the kink sector. Here the meson wave packet begins at x = x0, which is far to the left +of the kink, which is at x = 0. It moves to the right with momentum roughly equal to k0. +The final state consists of a meson and a kink whose shape mode is excited. It is therefore +a superposition of states of the form +|Sk2⟩0 = B‡ +SB‡ +k2|0⟩0. +(3.3) +At lowest order, O( +√ +λ), the only term in the kink Hamiltonian that can interpolate +between these states is +HI += +√ +λ +2 +� dk1 +2π +dk2 +2π +VS,k2,−k1 +ωk1 +B‡ +SB‡ +k2Bk1 +(3.4) +VS,k2,−k1 += +� +dxV (3)( +√ +λf(x))gS(x)gk2(x)g−k1(x). +6 + +At order O( +√ +λ), the corresponding terms in the time evolution operator are +e−it(H′ +2+HI) = e−itH′ +2 − i +� t +0 +dt1e−i(t−t1)H′ +2HIe−it1H′ +2 + O(λ). +(3.5) +We will drop the first term, as it will not contribute to the matrix elements below. Acting +this on a one-kink, one-meson state one finds Stokes scattering +e−iH′t|k1⟩0 +���� +O( +√ +λ) += −i +√ +λ +2ωk1 +� dk2 +2π VS,k2,−k1e− it +2 (ωk1+ωS+ωk2)sin +�� ωS+ωk2−ωk1 +2 +� +t +� +(ωS + ωk2 − ωk1)/2 |Sk2⟩0. +(3.6) +This process is on-shell when k1 = ±kS +I where we have defined +ωkS +I = ωk2 + ωS, +kS +I > 0. +(3.7) +At large times, we may use the identity +lim +t→∞ +sin +�� ωS+ωk2−ωk1 +2 +� +t +� +(ωS + ωk2 − ωk1)/2 = 2πδ(ωS + ωk2 − ωk1) = +�ωkS +I +kS +I +� � +2πδ(k1 − kS +I ) + 2πδ(k1 + kS +I ) +� +(3.8) +to perform the k2 integral. Folding this result into the wave packet (3.1), one finds the Stokes +scattered part of the state at large times t +e−iH′t|Φ⟩0 +���� +O( +√ +λ) += +−i +√ +λ +� dk1 +2π +αk1 +2ωk1 +� dk2 +2π VS,k2,−k1e− it +2 (ωk1+ωS+ωk2)sin +�� ωS+ωk2−ωk1 +2 +� +t +� +(ωS + ωk2 − ωk1)/2 |Sk2⟩0 += +−i +√ +λ +2 +� dk2 +2π +e +−iωkS +I +t +kS +I +� +αkS +I VS,k2,−kS +I + α−kS +I VS,k2,kS +I +� +|Sk2⟩0. +(3.9) +The meson wave packet begins far from the kink, where one may apply the asymptotic +form of the normal modes +gk(x) += +� +Bke−ikx + Ckeikx +if +x ≪ −1/m +Dke−ikx + Ekeikx +if +x ≫ 1/m +(3.10) +|Bk|2 + |Ck|2 += +|Dk|2 + |Ek|2 = 1, +B∗ +k = B−k, +C∗ +k = C−k, +D∗ +k = D−k, +E∗ +k = E−k +to evaluate the coefficients αk of the wave packet. As kS +I is defined to be positive and k0 is +chosen to be positive, in Eq. (3.9) only two cases appear +αkS +I += +2σ√π +� +BkS +I eix0(k0−kS +I )e−σ2(k0−kS +I )2 + CkS +I eix0(k0+kS +I )e−σ2(k0+kS +I )2� +(3.11) += +2σ√πBkS +I eix0(k0−kS +I )e−σ2(k0−kS +I )2 +7 + +and +α−kS +I += +2σ√π +� +B∗ +kS +I eix0(k0+kS +I )e−σ2(k0+kS +I )2 + C∗ +kS +I eix0(k0−kS +I )e−σ2(k0−kS +I )2� +(3.12) += +2σ√πC∗ +kS +I eix0(k0−kS +I )e−σ2(k0−kS +I )2. +Substituting these back into Eq. (3.9), one finds the relevant part of the state at large times +t +e−iH′t|Φ⟩0 +���� +O( +√ +λ) += +−iσ +√ +πλ +� dk2 +2π eix0(k0−kS +I )e−σ2(k0−kS +I )2e +−iωkS +I +t +� ˜VS,k2,−kS +I +kS +I +� +|Sk2⟩0 +˜VS,k2,−kS +I += +BkS +I VS,k2,−kS +I + C∗ +kS +I VS,k2,kS +I . +(3.13) +Note that in the case of a reflectionless kink, C = 0 and so +���˜V +��� = |V |. +Using the inner product +0⟨Sk1|Sk2⟩0 = 2πδ(k1 − k2) +4ωSωk2 +0⟨0|0⟩0 +(3.14) +we find the matrix elements +0⟨Sk2|e−iH′t|Φ⟩0 +0⟨0|0⟩0 += −iσ +√ +πλ +4ωSωk2kS +I +eix0(k0−kS +I )e−σ2(k0−kS +I )2e +−iωkS +I +t ˜VS,k2,−kS +I +(3.15) +which square to +���� +0⟨Sk2|e−iH′t|Φ⟩0 +0⟨0|0⟩0 +���� +2 += +σ2πλ +16ω2 +Sω2 +k2kS +I +2 +���˜VS,k2,−kS +I +��� +2 +e−2σ2(k0−kS +I )2 +(3.16) += +σπ3/2λ +16 +√ +2ω2 +Sω2 +k2kS +I +2 +���˜VS,k2,−kS +I +��� +2 +δ(kS +I − k0). +The last equality holds in the limit σ → ∞. +To calculate the Stokes scattering probability, we will need the projector P onto final +states with an excited kink and a single meson +P = +� +dk2Pdiff(k2), +Pdiff(k2) = 4ωSωk2 +2π +|Sk2⟩00⟨Sk2| +0⟨0|0⟩0 +. +(3.17) +Using the inner product +0⟨k1|k2⟩0 +0⟨0|0⟩0 += 2πδ(k1 − k2) +2ωk1 +(3.18) +8 + +one obtains the normalization of the initial state +0⟨Φ|Φ⟩0 +0⟨0|0⟩0 += +� +d2k +(2π)2αk1α∗ +k2 +0⟨k2|k1⟩0 +0⟨0|0⟩0 += +� dk +2π +|αk|2 +2ωk += +1 +2ωk0 +� dk +2π|αk|2 +(3.19) += +1 +2ωk0 +� dk +2π +� +dx +� +dygk(x)g∗ +k(y)Φ(x)Φ∗(y) += +1 +2ωk0 +� +dx|Φ(x)|2 = σ√π +√ +2ωk0 +where we used ωk ∼ ωk0 in the last step in the first line. +Both 0⟨k1|k2⟩0 and 0⟨0|0⟩0 are infinite, and so the previous expression is strictly speaking +not defined. In Ref. [28] we describe how such inner products may be calculated system- +atically, by dividing the numerator and denominator by the translation group. There are +corrections with respect to the naive manipulations above, as a result of the nondiagonal +action of the translation operator in the kink frame. However, these corrections are always +subleading by a power of +√ +λ and so do not affect our probability at O(λ). +Finally we may assemble all of these ingredients to write the total probability of Stokes +scattering at O(λ) +PS += +0⟨Φ|eiH′tPe−iH′t|Φ⟩0 +0⟨Φ|Φ⟩0 += +� dk2 +2π +4ωSωk2 +0⟨0|0⟩0 +��0⟨Sk2|e−iH′t|Φ⟩0 +��2 +0⟨Φ|Φ⟩0/0⟨0|0⟩0 +1 +0⟨0|0⟩0 +(3.20) += +� dk2 +2π 4ωSωk2 +σπ3/2λ +16 +√ +2ω2 +Sω2 +k2kS +I +2 +���˜VS,k2,−kS +I +��� +2 +δ(kS +I − k0) +� +σ√π +√ +2ωk0 +� += +πλωk0 +4ωS(ωk0 − ωS)k2 +0 +� dk2 +2π +���˜VS,k2,−kS +I +��� +2 +δ(kS +I − k0) += +λ +���˜VS,√ +(ωk0−ωS)2−m2,−k0 +��� +2 ++ +���˜VS,−√ +(ωk0−ωS)2−m2,−k0 +��� +2 +8ωSk0 +� +(ωk0 − ωS)2 − m2 +. +We see that the probability is the sum of two terms. The first is the probability that the +emitted meson travels in the same direction as the initial meson, while the second is the +probability that it travels in the opposite direction. We will see in an example below that +such reflection occurs even in the case of a reflectionless kink. +In the initial and final states (3.1) and (3.3), the meson travels at a constant velocity +k0/ωk0 when far from the kink. However an order O( +√ +λ) quantum correction to these states, +when evolved with respect to H′ +2, can in principle contribute to the amplitude at the same +order O( +√ +λ) as the leading term in the states when evolved with e−it(H′ +2+HI) at O( +√ +λ). Even +9 + +though our initial and final states (3.1) and (3.3) contain no such O( +√ +λ) correction, such a +correction would be created by the evolution e−itH′ as the meson travels far from a kink [25]. +As described in Ref. [25], one can include an order O( +√ +λ) quantum correction to the +initial and final states so that they are undeformed as they travel, while far from the kink. +Such states are eigenstates not of the kink Hamiltonian H′, but rather of the left and right +vacuum Hamiltonians, which are defined by expanding the defining Hamiltonian about the +vacua to the left and right of the kink. These quantum corrections arise from the three-meson +vertex far from the kink. Far from the kink, the mesons separately conserve momentum. +As a result, these processes are far off-shell, leading to a cloud of far off-shell mesons about +the initial and final mesons. One therefore expects that this cloud does not contribute to +the asymptotic probability of meson multiplication or Stokes scattering. In Ref. [25] it was +shown, in the case of meson multiplication, that this is indeed the case. The argument +proceeds identically here, as Stokes scattering is just meson multiplication in which one of +the created mesons is a bound state. Therefore we conclude that there are also no initial or +final state corrections here. +4 +Anti-Stokes Scattering +In anti-Stokes scattering, the kink begins with an excited shape mode and an approaching +meson wave packet. The initial state is thus +|Φ⟩0 = +� dk1 +2π αk1 |Sk1⟩0 , +|Sk1⟩0 = B‡ +SB‡ +k1|0⟩0. +(4.1) +The final state consists of a meson packet and a deexcited kink, and so is in the space of +states spanned by |k2⟩0. +At O( +√ +λ), the only term which interpolates between these two states is +HI = +√ +λ +4ωS +� dk1 +2π +dk2 +2π +VS,k2,−k1 +ωk1 +B‡ +k2BSBk1, +HI|Sk1⟩0 = +√ +λ +4ωS +� dk2 +2π +VS,k2,−k1 +ωk1 +|k2⟩0 . +(4.2) +At leading order, a finite time evolution then yields +e−iH′t|Sk1⟩0 +���� +O( +√ +λ) += −i +√ +λ +4ωSωk1 +� dk2 +2π VS,k2,−k1e− it +2 (ωk1+ωS+ωk2)sin +�� ωk1+ωS−ωk2 +2 +� +t +� +(ωk1 + ωS − ωk2)/2 |k2⟩0. (4.3) +This process is only on shell if k2 = ±kaS +I +where we now define kaS +I +differently from the +case of Stokes scattering in Sec. 3 +ωkaS +I = ωk2 − ωS, +kaS +I > 0. +(4.4) +10 + +At large times, only the on-shell k2 values contribute as +lim +t→∞ +sin +�� ωk1+ωS−ωk2 +2 +� +t +� +(ωk1 + ωS − ωk2)/2 = +�ωkaS +I +kaS +I +� � +2πδ(k1 − kaS +I ) + 2πδ(k1 + kaS +I ) +� +. +(4.5) +Substituting this limit into Eq. (4.3) and folding the result into our initial wave packet (4.1) +we find the anti-Stokes scattered part of the state at time t +e−iH′t|Φ⟩0 +���� +O( +√ +λ) += +−i +√ +λ +4ωS +� dk1 +2π +αk1 +ωk1 +� dk2 +2π VS,k2,−k1e− it +2 (ωk1+ωS+ωk2)sin +�� ωk1+ωS−ωk2 +2 +� +t +� +(ωk1 + ωS − ωk2)/2 |k2⟩0 += +−i +√ +λ +4ωS +� dk2 +2π e−iωk2t +� 1 +kaS +I +� � +αkaS +I VS,k2,−kaS +I + α−kaS +I VS,k2,kaS +I +� +|k2⟩0 += +−iσ +√ +πλ +2ωS +� dk2 +2π eix0(k0−kaS +I )e−σ2(k0−kaS +I )2e−iωk2t +� ˜VS,k2,−kaS +I +kaS +I +� +|k2⟩0 +(4.6) +which is summarized by the matrix elements +0⟨k2|e−iH′t|Φ⟩0 +0⟨0|0⟩0 += −iσ +√ +πλ +4ωSωk2kaS +I +eix0(k0−kaS +I )e−σ2(k0−kaS +I )2e−iωk2t ˜VS,k2,−kaS +I . +(4.7) +In the limit σ → ∞, in which the initial meson wave packet is monochromatic, this reduces +to +���� +0⟨k2|e−iH′t|Φ⟩0 +0⟨0|0⟩0 +���� +2 += +σ2πλ +16ω2 +Sω2 +k2kaS +I +2 +���˜VS,k2,−kaS +I +��� +2 +e−2σ2(k0−kaS +I )2 +(4.8) += +σπ3/2λ +16 +√ +2ω2 +Sω2 +k2kaS +I +2 +���˜VS,k2,−kaS +I +��� +2 +δ(kaS +I − k0). +We want to calculate the probability that the final state has one ground state kink and +one meson. Such states are preserved by the projector +P = +� +dk2Pdiff(k2), +Pdiff(k2) = 1 +2π +2ωk2 +0⟨0|0⟩0 +|k2⟩00⟨k2|. +(4.9) +Including the correction factor for the norm of the initial state +0⟨Φ|Φ⟩0 +0⟨0|0⟩0 += +� +d2k +(2π)2αk1α∗ +k2 +0⟨Sk2|Sk1⟩0 +0⟨0|0⟩0 += +� dk +2π +|αk|2 +4ωSωk += +1 +4ωSωk0 +� dk +2π|αk|2 (4.10) += +1 +4ωSωk0 +� dk +2π +� +dx +� +dygk(x)g∗ +k(y)Φ(x)Φ∗(y) += +1 +4ωSωk0 +� +dx|Φ(x)|2 = +σ√π +2 +√ +2ωSωk0 +11 + +where we again used ωk ∼ ωk0 in the last step of the first line, we find that the total +probability of anti-Stokes scattering is +PaS += +0⟨Φ|eiH′tPe−iH′t|Φ⟩0 +0⟨Φ|Φ⟩0 += +� dk2 +2π +2ωk2 +0⟨0|0⟩0 +��0⟨k2|e−iH′t|Φ⟩0 +��2 +0⟨Φ|Φ⟩0/0⟨0|0⟩0 +1 +0⟨0|0⟩0 +(4.11) += +� dk2 +2π 2ωk2 +σπ3/2λ +16 +√ +2ω2 +Sω2 +k2kaS +I +2 +���˜VS,k2,−kaS +I +��� +2 +δ(kaS +I − k0) +� +σ√π +2 +√ +2ωSωk0 +� += +πλωk0 +4ωS(ωk0 + ωS)k2 +0 +� dk2 +2π +���˜VS,k2,−kaS +I +��� +2 +δ(kaS +I − k0) += +λ +���˜VS,√ +(ωk0+ωS)2−m2,−k0 +��� +2 ++ +���˜VS,−√ +(ωk0+ωS)2−m2,−k0 +��� +2 +8ωSk0 +� +(ωk0 + ωS)2 − m2 +. +Again the first term is the probability that the outgoing meson travels in the same direction +as the incoming meson. +5 +Example: φ4 Double-Well Model +5.1 +Analytic Results +Consider the φ4 double-well model, which is defined by the potential +V ( +√ +λφ(x)) = λφ2(x) +4 +�√ +λφ(x) − +√ +2m +�2 +. +(5.1) +It has a single shape mode, with frequency +ωS = +√ +3β, +β = m +2 . +(5.2) +The normal modes are +gk(x) += +e−ikx +ωk +� +k2 + β2 +� +k2 − 2β2 + 3β2sech2(βx) − 3iβktanh(βx) +� +(5.3) +gS(x) += +� +3β +2 tanh(βx)sech(βx), +gB(x) = +√3β +2 +sech2(βx) +leading to +VSk1k2 = π3 +√ +3 +8 +� +17β4 − (ω2 +k1 − ω2 +k2)2� +(β2 + k2 +1 + k2 +2) + 8β2k2 +1k2 +2 +β3/2ωk1ωk2 +� +β2 + k2 +1 +� +β2 + k2 +2 +sech +�π(k1 + k2) +2β +� +(5.4) +12 + +and so +���˜VS,±√ +(ωk0−ωS)2−m2,−k0 +��� += +���VS,±√ +(ωk0−ωS)2−m2,−k0 +��� +(5.5) += |(−10β2 + 3ωSωk0 − 3k2 +0) (k2 +0 − ωSωk0 + 2β2) + (k2 +0 − 2ωSωk0 + 3β2) k2 +0| +(ωk0 − ωS)ωk0 +� +ω2 +k0 − 2ωSωk0 +� +β2 + k2 +0 +×3 +� +3βπsech +� +π(± +� +k2 +0 − 2ωSωk0 + 3β2 − k0) +2β +� += 6 +� +3βπ +√ωk0(ωk0 − ωS) +√ωk0 − 2ωS +� +β2 + k2 +0 +sech +� +π(± +� +k2 +0 − 2ωSωk0 + 3β2 − k0) +2β +� +. +The probability of Stokes scattering is then +PS += +ωk0(ωk0 − ωS)2 +(ωk0 − 2ωS)(β2 + k2 +0)k0 +� +k2 +0 − 2ωSωk0 + 3β2 +×9 +√ +3π2λ +2 +� +sech2 +� +π( +� +k2 +0 − 2ωSωk0 + 3β2 − k0) +2β +� ++ sech2 +� +π( +� +k2 +0 − 2ωSωk0 + 3β2 + k0) +2β +�� +. +(5.6) +The first sech term is the probability that the outgoing meson continues in the same +direction as the initial meson, while the second is the probability that it travels in the +opposite direction. The ratio of these two possibilities is just the ratio of the two sech terms. +At this order, the energy shift in the meson sector is exactly ωS. However the total +momentum of the mesons is not conserved. Far from the kink, the initial meson momentum +is k0 and the final momentum is +� +k2 +0 − 2ωSωk0 + 3β2. The arguments of the sech terms +are the momentum transfers between the mesons and the kink in the case of forward and +backward scattering. +13 + +Similarly, in the case of anti-Stokes scattering +���˜VS,±√ +(ωk0+ωS)2−m2,−k0 +��� += +���VS,±√ +(ωk0+ωS)2−m2,−k0 +��� +(5.7) += |(−3k2 +0 − 3ωSωk0 − 10β2) (k2 +0 + ωSωk0 + 2β2) + (k2 +0 + 2ωSωk0 + 3β2)k2 +0| +ωk0(ωk0 + ωS) +� +ω2 +k0 + 2ωSωk0 +� +β2 + k2 +0 +×3 +� +3βπsech +� +� +π +�� +k2 +0 + 2ωSωk0 + 3β2 ± k0 +� +2β +� +� += 6 +� +3βπ +√ωk0(ωk0 + ωS) +√ωk0 + 2ωS +� +β2 + k2 +0 +sech +� +π(± +� +k2 +0 + 2ωSωk0 + 3β2 − k0) +2β +� +leading to a probability of +PaS += +ωk0(ωk0 + ωS)2 +(ωk0 + 2ωS)(β2 + k2 +0)k0 +� +k2 +0 + 2ωSωk0 + 3β2 +×9 +√ +3π2λ +2 +� +sech2 +� +π( +� +k2 +0 + 2ωSωk0 + 3β2 − k0) +2β +� ++ sech2 +� +π( +� +k2 +0 + 2ωSωk0 + 3β2 + k0) +2β +�� +. +(5.8) +5.2 +Numerical Results +The probabilities depend on the dimensionless coupling λ/m2 as well as the dimensionless +momentum k0/m. We have fixed our units such that the meson mass, far from a kink, +is m = 1. The probabilities of Stokes scattering on a ground state kink and anti-Stokes +scattering on an excited kink are plotted in Figs. 1 and 2 respectively. +Note that Stokes scattering is only energetically allowed for a sufficiently high initial +momentum, whereas the probability of anti-Stokes scattering diverges at small momenta. +Of course, once the probability, not divided by λ, is of order unity, higher order corrections +dominate. In both cases, close to the threshold, backward and forward scattering become +equally probable. +In Fig. 3 we compare the total probabilities of these processes to that of the only other +inelastic process allowed at this order, meson multiplication [25]. This is the process in +which a kink and a meson collide, yielding a kink and two mesons. While the probabilities +of Stokes and anti-Stokes scattering tend to zero for large initial momenta, the probability of +14 + +1.575 +1.580 +1.585 +1.590 +1.595 +1.600 +1 +2 +3 +4 +5 +6 +7 +k0 +PS(k0)/λ +2.0 +2.5 +3.0 +3.5 +4.0 +4.5 +5.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +k0 +PS(k0)/λ +Figure 1: The forward (red), backward (blue) and total (black) probabilities PS(k0) of Stokes +scattering, with m = 1. +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +2 +4 +6 +8 +10 +k0 +PaS(k0)/λ +1 +2 +3 +4 +5 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +k0 +PaS(k0)/λ +Figure 2: The forward (red), backward (blue) and total (black) probabilities PaS(k0) of +anti-Stokes scattering, with m = 1. +15 + +meson multiplication tends to a constant. In particular, we see that (anti)Stokes scattering +dominates for low initial meson momenta, while meson multiplication dominates at higher +momenta, with a cross-over when the initial momentum is about twice the meson mass. +2 +4 +6 +8 +10 +12 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +k0 +P(k0)/λ +Figure 3: The total probability of meson multiplication (black) from Ref. [25], plotted against +the probability of Stokes (red) and anti-Stokes (blue) scattering. +6 +Remarks +At order O(λ), the inelastic scattering of a quantum kink and fundamental meson is now +fully understood. There are three allowed processes. First, in meson multiplication, the +meson may split in two. Second, if the kink is in its ground state, then when the meson +interacts it may excite a shape mode. Finally, if a shape mode is initially excited, then when +the meson interacts it may de-excite the shape mode. The first interaction dominates at +high energies, while the others become very large near their low energy thresholds. +We have always begun with an eigenstate of the free Hamiltonian, and measured the state +in an eigenstate of the free Hamiltonian. This involves matrix elements which are formally +infinite, as one must integrate over all possible positions of the center of mass in the compact +space. However, the same matrix elements appear in the numerator and denominator, and +so fortunately they cancel. In a companion paper [29] we treat such ratios more carefully, +dividing by the translation symmetry so that the numerator and denominator are both finite. +We find that indeed there are corrections to the results obtained via a naive cancellation. +However these corrections are suppressed by a power of λ, and so are not relevant here. If +16 + +one wishes to compute loop corrections, however, the corrections found in Ref. [29] must be +included as they enter at the same order. +While we maintain that kink-meson scattering is of intrinsic interest, it also contributes to +our understanding of the interactions of kinks with their environment. In the linear regime, +we expect this interaction to be dominated by just the processes described above. Shortly +beyond the linear regime, on the other hand, there will be processes which are of higher +order in the amplitude of the radiation, such as meson fusion [30, 31, 32]. These become +more relevant as one transitions to the classical regime. In the near future we would like to +understand such higher order processes in the quantum theory. +Acknowledgement +JE is supported by NSFC MianShang grants 11875296 and 11675223. HL acknowledges the +support from CAS-DAAD Joint Fellowship Programme for Doctoral students of UCAS. +References +[1] Y. Zhong, X. L. Du, Z. C. Jiang, Y. X. Liu and Y. Q. 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D 77 (2008), 125012 doi:10.1103/PhysRevD.77.125012 +[arXiv:0802.0080 [hep-th]]. +20 + diff --git a/btE2T4oBgHgl3EQfwQhi/content/tmp_files/load_file.txt b/btE2T4oBgHgl3EQfwQhi/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b17822f04c5859eb05dc846de81a44b7b69953e9 --- /dev/null +++ b/btE2T4oBgHgl3EQfwQhi/content/tmp_files/load_file.txt @@ -0,0 +1,700 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf,len=699 +page_content='(Anti-)Stokes Scattering on Kinks Jarah Evslin1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='2 ∗ and Hui Liu3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} 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Fundamental Physics and Mathematical Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' Hangzhou Institute for Advanced Study,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' University of Chinese Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' Hangzhou 310024,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' China 4) Arnold Sommerfeld Center,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' Ludwig-Maximilians-Universit¨at,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' Theresienstraße 37,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' 80333 M¨unchen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' Germany Abstract At leading order,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' there are three inelastic scattering processes beginning with a quan- tum kink and a fundamental meson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' Meson multiplication, in which the final state is a kink and two mesons, was treated recently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' In this note we treat the other two, (anti)-Stokes scattering, in which the kink’s shape mode is (de-)excited and the final state contains one meson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' In the case of a general scalar kink, we find analytic formulas for the forward and backward scattering amplitudes and probabilities as functions of the momentum of the incident meson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' The general results are then specialized to the kink of the φ4 double-well model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' 1 Introduction Scalar theories in (1+1)-dimensions provide some of the simplest quantum field theory models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' If the scalar field is subjected to a degenerate potential, in addition to fundamental meson excitations there will also be nonperturbative kinks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' The classical field theory of such models is already surprisingly rich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' Most attention has focused on kink-antikink scattering [1, 2, 3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' Here, it has long been known [5] that the range of initial relative speeds leading to distinct outcomes has a fractal structure of resonance windows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' It was once thought that this results from the internal excitation spectrum of the kinks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' Certainly they play a role [6, 7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' However it was then found [9, 10] that such windows appear even in models in which the kink has no internal excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' Thus, it has become clear that the interactions of kinks with the bulk dynamics are important [11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' ∗jarah@impcas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='cn †hui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='liu@campus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='lmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='de 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='04099v1 [hep-th] 10 Jan 2023 This bulk dynamics is itself quite rich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' There are spectral walls [13, 14] beyond which in- ternal excitations of kinks escape into the continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' While these have striking consequences classically, in the quantum theory they are rather smooth [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' Also, after a kink-antikink collision, many kinds of modes are excited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' In the end, only the longest-lived remain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' Among these, the oscillons [16, 17] survive for an amazingly long time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' However, again the quantum theory appears to be different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' In the quantum theory, new decay channels open which greatly reduce the oscillon lifetime [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' In summary, two critical pieces of the picture are missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' The first is a systematic understanding of interactions between kinks and bulk degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' The second is an understanding of the role played by quantum corrections, and whether they disappear in the classical limit or, as the oscillon case and perhaps the spectral wall case seem to suggest, radically affect the physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' This motivates an understanding of interactions between kinks and elementary meson quanta in the quantum theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' Such interactions are much simpler than kink-antikink inter- actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' However, the classic literature on quantum kink-meson scattering has been limited largely to finding effective Yukawa couplings between kinks and mesons [19, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' Recently, a linearized perturbation theory for such models has been developed at one- loop in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' [21] and beyond in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' It greatly simplifies calculations in the one-kink sector, which consists of states with a kink and any finite number of mesons, with respect to the traditional collective coordinate approach of Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' [23, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' Using this approach it was soon realized that, at leading order, there are precisely three inelastic scattering processes which begin with a single kink and a single meson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' The first is meson multiplication, in which the meson is absorbed by the kink and two mesons are emitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' This was recently studied in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' The other two are Stokes and anti-Stokes scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' Stokes scattering is a process where a meson scatters off of a ground state kink while exciting its shape mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' Anti-Stokes scattering is a process where a meson scatters off of an excited kink and deexcites its shape mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' In the present note, we present the first-ever treatment of these two processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' After a review of linearized kink perturbation theory in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' 2, we calculate the probability, as a function of the momentum of the incoming meson, of Stokes and anti-Stokes scattering in Secs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' 3 and 4 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' Our results are specialized to the φ4 double-well model in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' 2 2 Linearized Kink Perturbation Theory 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='1 The Main Idea In Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' [21, 22] a new, Hamiltonian formalism has been introduced for calculations in the kink sector of a quantum theory of a scalar field φ(x) and its conjugate π(x) in 1+1 dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' The kink sector is the Fock space of states consisting of a finite number of fundamental mesons in addition to a single quantum kink.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' We will refer to the Fock space of mesons in the absence of a kink as the vacuum sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' The vacuum sector states can be constructed in perturbation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' One decomposes the field in a plane wave basis, constructing creation and annihilation operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' The vacuum is defined as the state which is annihilated by all annihilation operators, and the vacuum sector is generated by finite numbers of creation operators acting on the vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' Hamiltonian eigenstates can be found perturbatively by solving the Hamiltonian eigenvalue problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' This perturbative approach fails for the kink sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' This is evident already in classical theory, where it results from the fact that large moments of the field do not tend to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' The kink sector corresponds to classical field configurations which are close to the classical kink solution φ(x, t) = f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' The higher moments of φ(x, t) − f(x) are small, and so one expects a perturbative approach in φ(x, t) − f(x) to yield kink sector states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' Linearized kink perturbation theory is a formalism for doing this in quantum field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' A unitary displacement operator Df is constructed in the Schrodinger picture as Df = Exp � −i � dxf(x)π(x) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='1) We use Df as a passive transformation, renaming the coordinate system of the Hilbert space and transforming the operators that act on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' More precisely, we define the kink frame as the coordinate system on the Hilbert space in which the ket |ψ⟩ represents the state Df|ψ⟩ as defined in the usual, defining frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' With this definition, it is easily shown that, in the kink frame, energies are measured and time is evolved by the kink Hamiltonian H′ H′ = D† fHDf (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='2) where H is the original Hamiltonian, which defines the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' What have we gained?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' In the kink frame, it is the kink sector which is constructed perturbatively using creation operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' Thus in the presence of a kink, the construction of Hamiltonian eigenstates, form factors, and even amplitudes and probabilities for various processes are reduced to perturbative problems in the kink frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' 3 What have we lost?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' We had to choose a particular kink solution f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' In a translation- invariant theory, there would be a moduli space of choices f(x−x0) for every real x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' Thus we have lost manifest translation invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' We must work locally, close to some base point in moduli space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' However, if we are interested in translation-invariant states, or more precisely momentum eigenstates, which we will be in this paper1, then it is sufficient to understand any region in moduli space to understand every region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' And so this will not be a problem, we simply work perturbatively in x0, and impose translation-invariance at will to simplify expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='2 The Details While we expect this formalism to apply quite generally, so far we have only applied it to Schrodinger picture Hamiltonians of the form H = � dx : H(x) :a, H(x) = π2(x) 2 + (∂xφ(x))2 2 + V ( √ λφ(x)) λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='3) Here V is a degenerate potential and φ(x, t) = f(x) is a solution of the classical equations of motion which interpolates between two degenerate minima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' The notation ::a represents normal ordering of the creation and annihilation operators for plane waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' It is defined at a mass scale m, which has two definitions m2 = V (2)( √ λf(±∞)), V (n)( √ λφ(x)) = ∂nV ( √ λφ(x)) (∂ √ λφ(x))n (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='4) corresponding to the scalar mass at the two minima of the potential at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' If these dis- agree, then quantum corrections break the degeneracy and the kink becomes an accelerating false vacuum bubble wall [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' We will not consider this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' We find the eigenstates of the kink Hamiltonian H′ perturbatively in the coupling con- stant λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' To do this we decompose all quantities in powers of λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' For example, the energy Q of the kink ground state is decomposed into � i Qi where each Qi is of order O(λi−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' Note that Q0 is just the classical energy of the classical kink solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' 1This paper will be entirely in the center of mass frame of the kink and meson, and so all states will be eigenstates of the total momentum operator with eigenvalue zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' Wave packets will be constructed consisting of different momenta for the meson, recalling that the kink momenta will always be equal and opposite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' Translation-invariance is with respect to simultaneous and equal translations of the kink and mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' 4 The kink Hamiltonian itself is decomposed into terms H′ i with i factors of the fundamental fields, when normal ordered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' These include H′ 0 = Q0, H′ 1 = 0, H′ n>2 = λ n 2 −1 � dxV (n)( √ λf(x)) n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' : φn(x) :a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='5) The most important is H′ 2, as its eigenvectors are the first step in the perturbative expansion for the kink sector states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' Written in terms of x it is rather odd, resembling a free Hamiltonian but with a position-dependent mass term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' To write it more transparently, we will introduce the kink’s normal modes g(x), defined to be small, classical fluctuations about the kink, which solve the Sturm-Liouville equation V (2)( √ λf(x))g(x) = ω2g(x) + g′′(x), φ(x, t) = e−iωtg(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='6) They are classified by their frequency ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' The real solution gB(x) with ωB = 0 is called the zero mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' Any real mode gS(x) with 0 < ωS < m is called a shape mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' Above this lie the continuum modes gk(x) with ωk = √ m2 + k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' We fix the conventions ωk = √ m2 + k2, g∗ k(x) = g−k(x) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='7) � dx|gB(x)|2 = 1, � dxgk1(x)g∗ k2(x) = 2πδ(k1 − k2), � dxgS1(x)g∗ S2(x) = δS1S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' The normal modes generate all bounded functions, and so, instead of plane waves, we may use them to decompose the fields [27] φ(x) = φ0gB(x) + � � dk 2π � B‡ k + B−k 2ωk � gk(x), B‡ k = B† k (2ωk), B‡ S = B† S (2ωS) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='8) π(x) = π0gB(x) + i � � dk 2π � ωkB‡ k − B−k 2 � gk(x), B−S = BS, � � dk 2π = � dk 2π + � S into operators φ0, π0, B and B‡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' This provides a new basis of our operator algebra, and any operator may be written in terms of these operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' The canonical commutation relations satisfied by φ(x) and π(x) imply that these satisfy [φ0, π0] = i, � BS1, B‡ S2 � = δS1S2, � Bk1, B‡ k2 � = 2πδ (k1 − k2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='9) Finally we are ready to write H′ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' It is [27] H′ 2 = Q1 + Hfree , Hfree = π2 0 2 + � S ωSB‡ SBS + � dk 2πωkB‡ kBk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='10) 5 Here Q1 is the one-loop correction to the kink mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' The π2 0 term is the kinetic energy of a free quantum-mechanical particle of mass Q0 with position operator φ0/√Q0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' This particle is the center of mass of the kink.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' The other terms are quantum harmonic oscillators for the shape modes S and continuum modes k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' B‡ S excites a shape mode, while B‡ k excites a continuum mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' The kink ground state |0⟩ of the kink Hamiltonian H′ can be decomposed into contribu- tions |0⟩i, of order O(λi/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' The first term in our semiclassical expansion, |0⟩0, is the vacuum of H′ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' It is the ground state of each term in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='10), and so is completely characterized by the conditions π0|0⟩0 = Bk|0⟩0 = BS|0⟩0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='11) 3 Stokes Scattering In a Stokes scattering event, one meson is absorbed by a ground state kink, one meson is emitted and a shape mode is excited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' The initial condition is therefore a superposition |Φ⟩0 = � dk1 2π αk1|k1⟩0, αk = � dxΦ(x)gk(x) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='1) Φ(x) = Exp � −(x − x0)2 4σ2 + ixk0 � , x0 ≪ − 1 m, 1 k0 , 1 m ≪ σ ≪ |x0| of one-meson states |k1⟩0 = B‡ k1|0⟩0 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='2) in the kink sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' Here the meson wave packet begins at x = x0, which is far to the left of the kink, which is at x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' It moves to the right with momentum roughly equal to k0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' The final state consists of a meson and a kink whose shape mode is excited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' It is therefore a superposition of states of the form |Sk2⟩0 = B‡ SB‡ k2|0⟩0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='3) At lowest order, O( √ λ), the only term in the kink Hamiltonian that can interpolate between these states is HI = √ λ 2 � dk1 2π dk2 2π VS,k2,−k1 ωk1 B‡ SB‡ k2Bk1 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='4) VS,k2,−k1 = � dxV (3)( √ λf(x))gS(x)gk2(x)g−k1(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' 6 At order O( √ λ), the corresponding terms in the time evolution operator are e−it(H′ 2+HI) = e−itH′ 2 − i � t 0 dt1e−i(t−t1)H′ 2HIe−it1H′ 2 + O(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='5) We will drop the first term, as it will not contribute to the matrix elements below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' Acting this on a one-kink, one-meson state one finds Stokes scattering e−iH′t|k1⟩0 ���� O( √ λ) = −i √ λ 2ωk1 � dk2 2π VS,k2,−k1e− it 2 (ωk1+ωS+ωk2)sin �� ωS+ωk2−ωk1 2 � t � (ωS + ωk2 − ωk1)/2 |Sk2⟩0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='6) This process is on-shell when k1 = ±kS I where we have defined ωkS I = ωk2 + ωS, kS I > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='7) At large times, we may use the identity lim t→∞ sin �� ωS+ωk2−ωk1 2 � t � (ωS + ωk2 − ωk1)/2 = 2πδ(ωS + ωk2 − ωk1) = �ωkS I kS I � � 2πδ(k1 − kS I ) + 2πδ(k1 + kS I ) � (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='8) to perform the k2 integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' Folding this result into the wave packet (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='1), one finds the Stokes scattered part of the state at large times t e−iH′t|Φ⟩0 ���� O( √ λ) = −i √ λ � dk1 2π αk1 2ωk1 � dk2 2π VS,k2,−k1e− it 2 (ωk1+ωS+ωk2)sin �� ωS+ωk2−ωk1 2 � t � (ωS + ωk2 − ωk1)/2 |Sk2⟩0 = −i √ λ 2 � dk2 2π e −iωkS I t kS I � αkS I VS,k2,−kS I + α−kS I VS,k2,kS I � |Sk2⟩0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='9) The meson wave packet begins far from the kink, where one may apply the asymptotic form of the normal modes gk(x) = � Bke−ikx + Ckeikx if x ≪ −1/m Dke−ikx + Ekeikx if x ≫ 1/m (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='10) |Bk|2 + |Ck|2 = |Dk|2 + |Ek|2 = 1, B∗ k = B−k, C∗ k = C−k, D∗ k = D−k, E∗ k = E−k to evaluate the coefficients αk of the wave packet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' As kS I is defined to be positive and k0 is chosen to be positive, in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='9) only two cases appear αkS I = 2σ√π � BkS I eix0(k0−kS I )e−σ2(k0−kS I )2 + CkS I eix0(k0+kS I )e−σ2(k0+kS I )2� (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='11) = 2σ√πBkS I eix0(k0−kS I )e−σ2(k0−kS I )2 7 and α−kS I = 2σ√π � B∗ kS I eix0(k0+kS I )e−σ2(k0+kS I )2 + C∗ kS I eix0(k0−kS I )e−σ2(k0−kS I )2� (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='12) = 2σ√πC∗ kS I eix0(k0−kS I )e−σ2(k0−kS I )2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' Substituting these back into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='9), one finds the relevant part of the state at large times t e−iH′t|Φ⟩0 ���� O( √ λ) = −iσ √ πλ � dk2 2π eix0(k0−kS I )e−σ2(k0−kS I )2e −iωkS I t � ˜VS,k2,−kS I kS I � |Sk2⟩0 ˜VS,k2,−kS I = BkS I VS,k2,−kS I + C∗ kS I VS,k2,kS I .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='13) Note that in the case of a reflectionless kink, C = 0 and so ���˜V ��� = |V |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' Using the inner product 0⟨Sk1|Sk2⟩0 = 2πδ(k1 − k2) 4ωSωk2 0⟨0|0⟩0 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='14) we find the matrix elements 0⟨Sk2|e−iH′t|Φ⟩0 0⟨0|0⟩0 = −iσ √ πλ 4ωSωk2kS I eix0(k0−kS I )e−σ2(k0−kS I )2e −iωkS I t ˜VS,k2,−kS I (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='15) which square to ���� 0⟨Sk2|e−iH′t|Φ⟩0 0⟨0|0⟩0 ���� 2 = σ2πλ 16ω2 Sω2 k2kS I 2 ���˜VS,k2,−kS I ��� 2 e−2σ2(k0−kS I )2 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='16) = σπ3/2λ 16 √ 2ω2 Sω2 k2kS I 2 ���˜VS,k2,−kS I ��� 2 δ(kS I − k0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' The last equality holds in the limit σ → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' To calculate the Stokes scattering probability, we will need the projector P onto final states with an excited kink and a single meson P = � dk2Pdiff(k2), Pdiff(k2) = 4ωSωk2 2π |Sk2⟩00⟨Sk2| 0⟨0|0⟩0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='17) Using the inner product 0⟨k1|k2⟩0 0⟨0|0⟩0 = 2πδ(k1 − k2) 2ωk1 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='18) 8 one obtains the normalization of the initial state 0⟨Φ|Φ⟩0 0⟨0|0⟩0 = � d2k (2π)2αk1α∗ k2 0⟨k2|k1⟩0 0⟨0|0⟩0 = � dk 2π |αk|2 2ωk = 1 2ωk0 � dk 2π|αk|2 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='19) = 1 2ωk0 � dk 2π � dx � dygk(x)g∗ k(y)Φ(x)Φ∗(y) = 1 2ωk0 � dx|Φ(x)|2 = σ√π √ 2ωk0 where we used ωk ∼ ωk0 in the last step in the first line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' Both 0⟨k1|k2⟩0 and 0⟨0|0⟩0 are infinite, and so the previous expression is strictly speaking not defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' [28] we describe how such inner products may be calculated system- atically, by dividing the numerator and denominator by the translation group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' There are corrections with respect to the naive manipulations above, as a result of the nondiagonal action of the translation operator in the kink frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' However, these corrections are always subleading by a power of √ λ and so do not affect our probability at O(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' Finally we may assemble all of these ingredients to write the total probability of Stokes scattering at O(λ) PS = 0⟨Φ|eiH′tPe−iH′t|Φ⟩0 0⟨Φ|Φ⟩0 = � dk2 2π 4ωSωk2 0⟨0|0⟩0 ��0⟨Sk2|e−iH′t|Φ⟩0 ��2 0⟨Φ|Φ⟩0/0⟨0|0⟩0 1 0⟨0|0⟩0 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='20) = � dk2 2π 4ωSωk2 σπ3/2λ 16 √ 2ω2 Sω2 k2kS I 2 ���˜VS,k2,−kS I ��� 2 δ(kS I − k0) � σ√π √ 2ωk0 � = πλωk0 4ωS(ωk0 − ωS)k2 0 � dk2 2π ���˜VS,k2,−kS I ��� 2 δ(kS I − k0) = λ ���˜VS,√ (ωk0−ωS)2−m2,−k0 ��� 2 + ���˜VS,−√ (ωk0−ωS)2−m2,−k0 ��� 2 8ωSk0 � (ωk0 − ωS)2 − m2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' We see that the probability is the sum of two terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' The first is the probability that the emitted meson travels in the same direction as the initial meson, while the second is the probability that it travels in the opposite direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' We will see in an example below that such reflection occurs even in the case of a reflectionless kink.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' In the initial and final states (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='1) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='3), the meson travels at a constant velocity k0/ωk0 when far from the kink.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' However an order O( √ λ) quantum correction to these states, when evolved with respect to H′ 2, can in principle contribute to the amplitude at the same order O( √ λ) as the leading term in the states when evolved with e−it(H′ 2+HI) at O( √ λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' Even 9 though our initial and final states (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='1) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='3) contain no such O( √ λ) correction, such a correction would be created by the evolution e−itH′ as the meson travels far from a kink [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' As described in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' [25], one can include an order O( √ λ) quantum correction to the initial and final states so that they are undeformed as they travel, while far from the kink.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' Such states are eigenstates not of the kink Hamiltonian H′, but rather of the left and right vacuum Hamiltonians, which are defined by expanding the defining Hamiltonian about the vacua to the left and right of the kink.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' These quantum corrections arise from the three-meson vertex far from the kink.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' Far from the kink, the mesons separately conserve momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' As a result, these processes are far off-shell, leading to a cloud of far off-shell mesons about the initial and final mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' One therefore expects that this cloud does not contribute to the asymptotic probability of meson multiplication or Stokes scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' [25] it was shown, in the case of meson multiplication, that this is indeed the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' The argument proceeds identically here, as Stokes scattering is just meson multiplication in which one of the created mesons is a bound state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' Therefore we conclude that there are also no initial or final state corrections here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' 4 Anti-Stokes Scattering In anti-Stokes scattering, the kink begins with an excited shape mode and an approaching meson wave packet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' The initial state is thus |Φ⟩0 = � dk1 2π αk1 |Sk1⟩0 , |Sk1⟩0 = B‡ SB‡ k1|0⟩0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='1) The final state consists of a meson packet and a deexcited kink, and so is in the space of states spanned by |k2⟩0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' At O( √ λ), the only term which interpolates between these two states is HI = √ λ 4ωS � dk1 2π dk2 2π VS,k2,−k1 ωk1 B‡ k2BSBk1, HI|Sk1⟩0 = √ λ 4ωS � dk2 2π VS,k2,−k1 ωk1 |k2⟩0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='2) At leading order, a finite time evolution then yields e−iH′t|Sk1⟩0 ���� O( √ λ) = −i √ λ 4ωSωk1 � dk2 2π VS,k2,−k1e− it 2 (ωk1+ωS+ωk2)sin �� ωk1+ωS−ωk2 2 � t � (ωk1 + ωS − ωk2)/2 |k2⟩0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='3) This process is only on shell if k2 = ±kaS I where we now define kaS I differently from the case of Stokes scattering in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' 3 ωkaS I = ωk2 − ωS, kaS I > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='4) 10 At large times, only the on-shell k2 values contribute as lim t→∞ sin �� ωk1+ωS−ωk2 2 � t � (ωk1 + ωS − ωk2)/2 = �ωkaS I kaS I � � 2πδ(k1 − kaS I ) + 2πδ(k1 + kaS I ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='5) Substituting this limit into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='3) and folding the result into our initial wave packet (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='1) we find the anti-Stokes scattered part of the state at time t e−iH′t|Φ⟩0 ���� O( √ λ) = −i √ λ 4ωS � dk1 2π αk1 ωk1 � dk2 2π VS,k2,−k1e− it 2 (ωk1+ωS+ωk2)sin �� ωk1+ωS−ωk2 2 � t � (ωk1 + ωS − ωk2)/2 |k2⟩0 = −i √ λ 4ωS � dk2 2π e−iωk2t � 1 kaS I � � αkaS I VS,k2,−kaS I + α−kaS I VS,k2,kaS I � |k2⟩0 = −iσ √ πλ 2ωS � dk2 2π eix0(k0−kaS I )e−σ2(k0−kaS I )2e−iωk2t � ˜VS,k2,−kaS I kaS I � |k2⟩0 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='6) which is summarized by the matrix elements 0⟨k2|e−iH′t|Φ⟩0 0⟨0|0⟩0 = −iσ √ πλ 4ωSωk2kaS I eix0(k0−kaS I )e−σ2(k0−kaS I )2e−iωk2t ˜VS,k2,−kaS I .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='7) In the limit σ → ∞, in which the initial meson wave packet is monochromatic, this reduces to ���� 0⟨k2|e−iH′t|Φ⟩0 0⟨0|0⟩0 ���� 2 = σ2πλ 16ω2 Sω2 k2kaS I 2 ���˜VS,k2,−kaS I ��� 2 e−2σ2(k0−kaS I )2 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='8) = σπ3/2λ 16 √ 2ω2 Sω2 k2kaS I 2 ���˜VS,k2,−kaS I ��� 2 δ(kaS I − k0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' We want to calculate the probability that the final state has one ground state kink and one meson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' Such states are preserved by the projector P = � dk2Pdiff(k2), Pdiff(k2) = 1 2π 2ωk2 0⟨0|0⟩0 |k2⟩00⟨k2|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='9) Including the correction factor for the norm of the initial state 0⟨Φ|Φ⟩0 0⟨0|0⟩0 = � d2k (2π)2αk1α∗ k2 0⟨Sk2|Sk1⟩0 0⟨0|0⟩0 = � dk 2π |αk|2 4ωSωk = 1 4ωSωk0 � dk 2π|αk|2 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='10) = 1 4ωSωk0 � dk 2π � dx � dygk(x)g∗ k(y)Φ(x)Φ∗(y) = 1 4ωSωk0 � dx|Φ(x)|2 = σ√π 2 √ 2ωSωk0 11 where we again used ωk ∼ ωk0 in the last step of the first line, we find that the total probability of anti-Stokes scattering is PaS = 0⟨Φ|eiH′tPe−iH′t|Φ⟩0 0⟨Φ|Φ⟩0 = � dk2 2π 2ωk2 0⟨0|0⟩0 ��0⟨k2|e−iH′t|Φ⟩0 ��2 0⟨Φ|Φ⟩0/0⟨0|0⟩0 1 0⟨0|0⟩0 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='11) = � dk2 2π 2ωk2 σπ3/2λ 16 √ 2ω2 Sω2 k2kaS I 2 ���˜VS,k2,−kaS I ��� 2 δ(kaS I − k0) � σ√π 2 √ 2ωSωk0 � = πλωk0 4ωS(ωk0 + ωS)k2 0 � dk2 2π ���˜VS,k2,−kaS I ��� 2 δ(kaS I − k0) = λ ���˜VS,√ (ωk0+ωS)2−m2,−k0 ��� 2 + ���˜VS,−√ (ωk0+ωS)2−m2,−k0 ��� 2 8ωSk0 � (ωk0 + ωS)2 − m2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' Again the first term is the probability that the outgoing meson travels in the same direction as the incoming meson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' 5 Example: φ4 Double-Well Model 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='1 Analytic Results Consider the φ4 double-well model, which is defined by the potential V ( √ λφ(x)) = λφ2(x) 4 �√ λφ(x) − √ 2m �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='1) It has a single shape mode, with frequency ωS = √ 3β, β = m 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='2) The normal modes are gk(x) = e−ikx ωk � k2 + β2 � k2 − 2β2 + 3β2sech2(βx) − 3iβktanh(βx) � (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='3) gS(x) = � 3β 2 tanh(βx)sech(βx), gB(x) = √3β 2 sech2(βx) leading to VSk1k2 = π3 √ 3 8 � 17β4 − (ω2 k1 − ω2 k2)2� (β2 + k2 1 + k2 2) + 8β2k2 1k2 2 β3/2ωk1ωk2 � β2 + k2 1 � β2 + k2 2 sech �π(k1 + k2) 2β � (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='4) 12 and so ���˜VS,±√ (ωk0−ωS)2−m2,−k0 ��� = ���VS,±√ (ωk0−ωS)2−m2,−k0 ��� (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='5) = |(−10β2 + 3ωSωk0 − 3k2 0) (k2 0 − ωSωk0 + 2β2) + (k2 0 − 2ωSωk0 + 3β2) k2 0| (ωk0 − ωS)ωk0 � ω2 k0 − 2ωSωk0 � β2 + k2 0 ×3 � 3βπsech � π(± � k2 0 − 2ωSωk0 + 3β2 − k0) 2β � = 6 � 3βπ √ωk0(ωk0 − ωS) √ωk0 − 2ωS � β2 + k2 0 sech � π(± � k2 0 − 2ωSωk0 + 3β2 − k0) 2β � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' The probability of Stokes scattering is then PS = ωk0(ωk0 − ωS)2 (ωk0 − 2ωS)(β2 + k2 0)k0 � k2 0 − 2ωSωk0 + 3β2 ×9 √ 3π2λ 2 � sech2 � π( � k2 0 − 2ωSωk0 + 3β2 − k0) 2β � + sech2 � π( � k2 0 − 2ωSωk0 + 3β2 + k0) 2β �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='6) The first sech term is the probability that the outgoing meson continues in the same direction as the initial meson, while the second is the probability that it travels in the opposite direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' The ratio of these two possibilities is just the ratio of the two sech terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' At this order, the energy shift in the meson sector is exactly ωS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' However the total momentum of the mesons is not conserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' Far from the kink, the initial meson momentum is k0 and the final momentum is � k2 0 − 2ωSωk0 + 3β2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' The arguments of the sech terms are the momentum transfers between the mesons and the kink in the case of forward and backward scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' 13 Similarly, in the case of anti-Stokes scattering ���˜VS,±√ (ωk0+ωS)2−m2,−k0 ��� = ���VS,±√ (ωk0+ωS)2−m2,−k0 ��� (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='7) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='= |(−3k2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='0 − 3ωSωk0 − 10β2) (k2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='0 + ωSωk0 + 2β2) + (k2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='0 + 2ωSωk0 + 3β2)k2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='0| ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='ωk0(ωk0 + ωS) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='ω2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='k0 + 2ωSωk0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='β2 + k2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='×3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='3βπsech ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='π ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='k2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='0 + 2ωSωk0 + 3β2 ± k0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='2β ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='= 6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='3βπ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='√ωk0(ωk0 + ωS) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='√ωk0 + 2ωS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='β2 + k2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='sech ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='π(± ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='k2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='0 + 2ωSωk0 + 3β2 − k0) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='2β ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='leading to a probability of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='PaS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='ωk0(ωk0 + ωS)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='(ωk0 + 2ωS)(β2 + k2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='0)k0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='k2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='0 + 2ωSωk0 + 3β2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='×9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='3π2λ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='sech2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='π( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='k2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='0 + 2ωSωk0 + 3β2 − k0) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='2β ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='+ sech2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='π( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='k2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='0 + 2ωSωk0 + 3β2 + k0) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='2β ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='8) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='2 Numerical Results The probabilities depend on the dimensionless coupling λ/m2 as well as the dimensionless momentum k0/m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' We have fixed our units such that the meson mass, far from a kink, is m = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' The probabilities of Stokes scattering on a ground state kink and anti-Stokes scattering on an excited kink are plotted in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' 1 and 2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' Note that Stokes scattering is only energetically allowed for a sufficiently high initial momentum, whereas the probability of anti-Stokes scattering diverges at small momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' Of course, once the probability, not divided by λ, is of order unity, higher order corrections dominate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' In both cases, close to the threshold, backward and forward scattering become equally probable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' 3 we compare the total probabilities of these processes to that of the only other inelastic process allowed at this order, meson multiplication [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' This is the process in which a kink and a meson collide, yielding a kink and two mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' While the probabilities of Stokes and anti-Stokes scattering tend to zero for large initial momenta, the probability of 14 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='575 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='580 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='585 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='590 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='595 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='600 1 2 3 4 5 6 7 k0 PS(k0)/λ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='7 k0 PS(k0)/λ Figure 1: The forward (red), backward (blue) and total (black) probabilities PS(k0) of Stokes scattering, with m = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='5 2 4 6 8 10 k0 PaS(k0)/λ 1 2 3 4 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='2 k0 PaS(k0)/λ Figure 2: The forward (red), backward (blue) and total (black) probabilities PaS(k0) of anti-Stokes scattering, with m = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' 15 meson multiplication tends to a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' In particular, we see that (anti)Stokes scattering dominates for low initial meson momenta, while meson multiplication dominates at higher momenta, with a cross-over when the initial momentum is about twice the meson mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' 2 4 6 8 10 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content='2 k0 P(k0)/λ Figure 3: The total probability of meson multiplication (black) from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' [25], plotted against the probability of Stokes (red) and anti-Stokes (blue) scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' 6 Remarks At order O(λ), the inelastic scattering of a quantum kink and fundamental meson is now fully understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' There are three allowed processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' First, in meson multiplication, the meson may split in two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' Second, if the kink is in its ground state, then when the meson interacts it may excite a shape mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' Finally, if a shape mode is initially excited, then when the meson interacts it may de-excite the shape mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' The first interaction dominates at high energies, while the others become very large near their low energy thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' We have always begun with an eigenstate of the free Hamiltonian, and measured the state in an eigenstate of the free Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' This involves matrix elements which are formally infinite, as one must integrate over all possible positions of the center of mass in the compact space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' However, the same matrix elements appear in the numerator and denominator, and so fortunately they cancel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' In a companion paper [29] we treat such ratios more carefully, dividing by the translation symmetry so that the numerator and denominator are both finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' We find that indeed there are corrections to the results obtained via a naive cancellation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' However these corrections are suppressed by a power of λ, and so are not relevant here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' If 16 one wishes to compute loop corrections, however, the corrections found in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' [29] must be included as they enter at the same order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' While we maintain that kink-meson scattering is of intrinsic interest, it also contributes to our understanding of the interactions of kinks with their environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' In the linear regime, we expect this interaction to be dominated by just the processes described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' Shortly beyond the linear regime, on the other hand, there will be processes which are of higher order in the amplitude of the radiation, such as meson fusion [30, 31, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' These become more relevant as one transitions to the classical regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' In the near future we would like to understand such higher order processes in the quantum theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' Acknowledgement JE is supported by NSFC MianShang grants 11875296 and 11675223.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} +page_content=' HL acknowledges the support from CAS-DAAD Joint Fellowship Programme for Doctoral students of UCAS.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfwQhi/content/2301.04099v1.pdf'} diff --git a/c9E2T4oBgHgl3EQfwwh7/vector_store/index.pkl b/c9E2T4oBgHgl3EQfwwh7/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..00b497e3aa2f73c9e68d23dce77e559ee25a2e97 --- /dev/null +++ b/c9E2T4oBgHgl3EQfwwh7/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:d2eda9bcb50b5819e9ba5367725b7fa1a7ed7f6c205be481961585f228b58e5d +size 420191 diff --git a/cNFJT4oBgHgl3EQf9y04/content/2301.11689v1.pdf b/cNFJT4oBgHgl3EQf9y04/content/2301.11689v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..81afa742b3387e9f1c8337802c5f22b1a56256fb --- /dev/null +++ b/cNFJT4oBgHgl3EQf9y04/content/2301.11689v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:a7e1edecfff98a4e6a49c871107768844e1a54538e62ec6174a3fa69b425eda4 +size 727361 diff --git 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Salento, Italy, +‡Halmstad University, Sweden, §Autonomous University of Barcelona, Spain +Abstract—We consider the problem of ESPRIT-oriented pre- +coder design for beamspace angle-of-departure (AoD) estimation +in downlink mmWave multiple-input single-output communi- +cations. Standard precoders (i.e., directional/sum beams) yield +poor performance in AoD estimation, while Cram´er-Rao bound- +optimized precoders undermine the so-called shift invariance +property (SIP) of ESPRIT. To tackle this issue, the problem +of designing ESPRIT-oriented precoders is formulated to jointly +optimize over the precoding matrix and the SIP-restoring matrix +of ESPRIT. We develop an alternating optimization approach +that updates these two matrices under unit-modulus constraints +for analog beamforming architectures. Simulation results demon- +strate the validity of the proposed approach while providing +valuable insights on the beampatterns of the ESPRIT-oriented +precoders. +Index Terms—Precoder design, beamspace ESPRIT, channel +estimation, mmWave communications. +I. INTRODUCTION +Positioning in 5G relies to a large extent on the use of +mmWave frequencies, with their ample bandwidth and large +antenna arrays [1]–[3]. Large bandwidths offer high delay +resolution but provide limited opportunities for optimization, +as base stations (BSs) must use non-overlapping subcarriers +for multi-BS positioning solutions. Large antenna arrays yield +high angle resolution, as well as the ability to shape signals +in the spatial domain, e.g., for interference control, but also +for optimizing positioning performance [4]. Harnessing the +improved resolution and also exploiting optimized spatial +designs enhance the performance of the channel estimation +routine, which detects the number of paths, and for each +path estimates the geometric parameters (i.e., time-of-arrival +(ToA), angle-of-arrival (AoA), angle-of-departure (AoD)) [5]. +As channel estimation is a joint function among communi- +cation, positioning, and sensing, it is important to develop +methods that are both accurate and of moderate complexity [6], +especially for integrated sensing and communication (ISAC) +systems towards 6G multi-functional wireless networks [7]. +In a general pilot-based channel estimation setup, the opti- +mal channel parameters are the maximum a posteriori (MAP) +estimates given the received signal sequence. However, opti- +mization methods employed in MAP estimation can involve +heavy computations. On the other hand, it is notable that +mmWave channels are usually sparse, due to a limited number +of multipath propagation arriving at the receiver with relatively +strong path gains. As a result, sparsity-inspired low-complexity +channel estimation methods are developed [8]–[13]. Among +them, the estimation of signal parameters via rotational invari- +ance techniques (ESPRIT)-based channel estimation methods +have been widely studied, due to their good trade-off between +estimation performance and complexity [11]–[13]. Recently, +ESPRIT-based approaches have been applied to the beamspace, +which is attractive since analogue and/or digital beamforming +structures are employed in most massive MIMO mmWave +Fig. 1. mmWave MISO downlink scenario where the UE aims to estimate the +AoDs of multiple paths using high-resolution beamspace ESPRIT methods. +systems [13]–[15]. However, to apply beamspace ESPRIT +methods, precoders are required to hold the shift invariance +property (SIP). Examples of such precoding matrices include +the discrete Fourier transform (DFT) beams [13], [16] and the +directional beams [14]. When the SIP does not hold for the +precoding matrix, an approximation will be applied during +the derivation of the beamspace ESPRIT methods, leading +to performance degradations [14]. In addition, research on +Cram´er-Rao bound (CRB)-optimized precoder design suggests +that the optimal precoding matrix usually does not hold the +SIP [4], [17], [18]. In other words, there is an inevitable +performance loss when low-complexity ESPRIT methods are +employed with CRB-optimized precoders. +In this paper, we investigate the problem of ESPRIT- +oriented precoder design for AoD estimation in mmWave +communications, targeting a near-optimal precoding scheme +in terms of accuracy while enjoying the low-complexity and +high-resolution ESPRIT methods for channel estimation. Our +specific contributions are as follows: +• We formulate the problem of ESPRIT-oriented precoder +design as a beampattern synthesis problem that considers +joint optimization of the precoding matrix and the SIP- +restoring matrix of ESPRIT. +• We propose an alternating optimization strategy that +updates the precoder and the SIP-restoring matrix se- +quentially under the unit-modulus constraint on individual +precoder elements, suitable for phase-only beamforming +architectures. +• Through simulation results, we provide important in- +sights into the beampatterns of the resulting ESPRIT- +oriented precoders and demonstrate the effectiveness of +the proposed design approach in ESPRIT-based channel +estimation. +II. SYSTEM MODEL AND PROBLEM DESCRIPTION +A. System Model +We consider a mmWave MISO downlink (DL) flat-fading +communications scenario with an NTx-antenna BS and a +single-antenna user equipment (UE), as shown in Fig. 1. +arXiv:2301.01585v1 [eess.SP] 4 Jan 2023 + +y +个 +UE +BSConsidering the presence of L paths1, the received signal at +the UE at transmission instance m and snapshot n is given by +ym,n = +√ +P +L−1 +� +ℓ=0 +αℓ,n aT(θℓ)fmsm,n + zm,n +(1) +for m = 1, . . . , M and n = 1, . . . , N, where M and N +denote, respectively, the number of transmissions and the +number of snapshots2. In (1), P denotes the transmit power, +[a(θ)]k = ej2π d +λ k sin θ, k = 0, . . . , NTx − 1, is the steering +vector for the BS TX array, λ = c/fc is the wavelength with c +and fc denoting the speed of propagation and carrier frequency, +respectively, d is the array element spacing, fm ∈ CNTx×1 de- +notes the BS precoder at time m, αℓ,n and θℓ are the complex +channel gain and AoD of the ℓth path for the nth snapshot, +respectively, sm,n is the pilot symbol, and zm,n ∼ CN(0, σ2) +is additive white Gaussian noise (AWGN) with power σ2. For +simplicity, we set sm,n = 1, ∀m, n. +Aggregating the observations (1) over M transmissions, we +have the received signal at the nth snapshot +yn = +√ +PFTVαn + zn , +(2) +where yn ≜ [y1,n · · · yM,n]T ∈ CM×1, F ≜ [f1 · · · fM] ∈ +CNTx×M is the precoding matrix satisfying tr +� +FFH� += +M, +V +≜ +[a(θ0) · · · a(θL−1)] +∈ +CNTx×L, +αn +≜ +[α0,n · · · αL−1,n]T ∈ CL×1, and zn ∼ CN(0, σ2I) represents +the AWGN component. +B. Problem Description +In the considered mmWave scenario, the UE aims to esti- +mate the AoDs θ = [θ0 · · · θL−1]T using beamspace ESPRIT +[16], [21] from the beamspace observations {yn}N +n=1 in (2). +The problem of interest is to design the BS precoding matrix +F to maximize the accuracy of estimation of θ at the UE while +at the same time trying to preserve as much as possible the +SIP required by ESPRIT-based estimation [16]. +III. ESPRIT-ORIENTED PRECODER DESIGN +In this section, we provide a review of beamspace ESPRIT +and revisit the SIP, which enforces a certain structure on +the precoder. Based on this structure and using an ESPRIT- +unaware baseline precoder Fbase (which will be introduced +later in Sec. IV-A), we formulate a novel precoder design +problem that jointly optimizes beampattern synthesis accuracy +(with respect to Fbase) and ESPRIT SIP error (i.e., the level +of degradation of SIP), leading to near-optimal performance +for ESPRIT-based estimators. +A. Review of Beamspace ESPRIT +From (2), we compute the covariance matrix +R = 1 +N +N +� +n=1 +ynyH +n = PFTV +� 1 +N +N +� +n=1 +αnαH +n +� +VHF∗ + σ2I , +(3) +where +V +is +a +Vandermonde +matrix +which +holds +the +SIP, +satisfying +J1V += +J2VΦH +where +J1 = +� +INTx−1, 0(NTx−1)×1 +� +∈ R(NTx−1)×NTx and J2 = +1We consider a mmWave tracking scenario [17], [19], [20] with known +L. +2Here, snapshots may correspond to, for instance, different subcarriers of +an orthogonal frequency-division multiplexing (OFDM) system. In this case, it +is reasonable to assume that the channel gains αℓ,n change across snapshots, +but the AoDs θℓ remain constant. +� +0(NTx−1)×1, INTx−1 +� +∈ R(NTx−1)×NTx are selection ma- +trices, and Φ = Diag([[a(θ0)]1, [a(θ1)]1, · · · , [a(θL−1)]1]T). +In (3), we assume that +1 +N +�N +n=1 αnαH +n is a diagonal matrix +(i.e., paths are decorrelated), meaning that the dimension of +the signal subspace is L. In the precoded case, it has been +shown in [14], [16] that if the matrix F holds the SIP, i.e., +J1F = J2FΛ +(4) +for some non-singular Λ ∈ CM×M, we can restore the SIP +from C = FTV, by finding a non-null matrix Q such that +QCΦ = QΛTC , +(5) +where Q ∈ CM×M satisfies +Q[FT eM +ΛTFTe1] = 0, +(6) +and em ∈ RNTx×1 is the m-th column of the identity matrix +INTx. From (6), Q can be obtained as Q = IM −�1 +i=0 qiqH +i , +where q0, q1 ∈ CM×1 are orthonormal column vectors span- +ning the subspace corresponding to [FT eM +ΛTFTe1] ∈ +CM×2 [14]. Since perfect SIP cannot always be guaranteed3 +in (4), one can resort to the least-squares (LS) solution to find +an approximate Λ [14]: +�ΛLS = arg min +Λ +∥J1F − J2FΛ∥2 +F +(7) += +� +FHJH +2 J2F +�−1 FHJH +2 J1F , +(8) +where ∥·∥F denotes the Frobenius norm. +Given an estimate of the covariance matrix R, the signal +subspace matrix Us ∈ CM×L can be obtained through the +SVD (or truncated SVD) operation. Since both C and Us +span the same signal subspace, we have C = UsT, where +T ∈ CL×L is a non-singular matrix. Using the SIP of C in (5), +we further obtain QUsΠ = QΛTUs where Π = TΦT−1. +The diagonal elements in Φ will be used to estimate the +AoD of each path. The beamspace ESPRIT approach can be +summarized as follows: +• Find Λ and Q for given F. +• Obtain an estimate of R as �R using multiple snapshots. +• Perform SVD (or truncated SVD) on �R to obtain Us. +• Obtain the least-square (LS) solution of Π as �Π = +(QUs)†QΛTUs, where (·)† +denotes Moore-Penrose +pseudo-inverse. +• Perform eigenvalue decomposition on �Π to obtain an +estimate of Φ, and retrieve the corresponding AoDs. +B. ESPRIT-Oriented Precoder Design with SIP Considera- +tions +We formulate the problem of ESPRIT-oriented precoder +design as a beampattern synthesis via joint optimization of +F and Λ, starting from a desired beampattern created by an +ESPRIT-unconstrained precoder Fbase as baseline. The goal is +to minimize the weighted average of the beampattern synthesis +error and the ESPRIT SIP error, quantified by the error of the +LS solution in (7): +min +F,Λ +��B − ATF +��2 +F +� +�� +� +beampattern synthesis +accuracy ++ η ∥J1F − J2FΛ∥2 +F +� +�� +� +SIP approximation +error +(9a) +s.t. |[F]n,m| = 1, ∀n, m , +(9b) +3Perfect SIP holds for DFT beams and directional/sum beams (i.e., +steering vectors). + +Algorithm 1 ESPRIT-Oriented Precoder Design via Joint +Optimization of F and Λ in (9) +1: Input: Baseline precoder Fbase, transmit steering matrix +A, selection matrices J1 and J2, SIP error weight η, +convergence threshold ϵ. +2: Output: ESPRIT-oriented precoder F, SIP-restoring ma- +trix Λ. +3: Initialization: +4: Initialize the precoder as F = Fbase. +5: Initialize the SIP-restoring matrix as Λ = �ΛLS via (8). +6: Alternating Optimization Iterations: +7: repeat +8: +Update F in (12) via [22, Alg. 1]. +9: +Update Λ via (8). +10: until the objective (9a) converges. +where B = ATFbase ∈ CNgrid×M represents the desired +beampattern corresponding to Fbase at Ngrid angular grid +points {θi}Ngrid +i=1 , A = +� +a(θ1) · · · a(θNgrid) +� +∈ CNTx×Ngrid +is the transmit steering matrix evaluated at the specified grid +locations, and η is a predefined weight on the SIP error, chosen +to provide a suitable trade-off between beampattern synthesis +accuracy and SIP approximation error. In addition, the con- +straint (9b) is imposed to ensure compatibility with phase-only +beamforming architectures [22] (e.g., analog passive arrays +[23]). In the case of phase-amplitude beamforming (e.g., via +active phased arrays [23]), the problem becomes the special +case of (9) without the constraint (9b). +C. Alternating Optimization to Solve (9) +The problem (9) is non-convex due to (i) the non-convexity +of (9a) in the joint variable F and Λ, and (ii) the unit-modulus +constraint in (9b). To tackle (9), we resort to an alternating +optimization method that updates F and Λ in an iterative +fashion. +1) Optimize F for fixed Λ: Using the vectorization property +of the Kronecker product, the objective function (9a) can be +rewritten as +g(f) +(10) += +��b − +� +IM ⊗ AT � +f +��2 +2 + η +��� +IM ⊗ J1 − ΛT ⊗ J2 +� +f +��2 +2 , +where f ≜ vec (F) and b ≜ vec (B). Defining +Q ≜ +� +IM ⊗ AT �H � +IM ⊗ AT � +(11) ++ η +� +IM ⊗ J1 − ΛT ⊗ J2 +�H � +IM ⊗ J1 − ΛT ⊗ J2 +� +, +p ≜ +� +IM ⊗ AT �H b , +the problem (9) for fixed Λ can be expressed as +min +f +f HQf − 2ℜ +� +pHf +� +(12) +s.t. |fn| = 1, ∀n . +The problem (12) can be solved using gradient projections +iterations in [22, Alg. 1]. +2) Optimize Λ for fixed F: The subproblem of (9) for fixed +F is exactly the LS problem defined in (7), whose solution is +provided in (8). +The overall algorithm to solve (9) via alternating optimiza- +tion of F and Λ is summarized in Algorithm 1. +-10 +-8 +-6 +-4 +-2 +0 +2 +4 +6 +8 +10 +-20 +-15 +-10 +-5 +0 +5 +10 +15 +Fig. 2. +Illustration of the beampatterns corresponding to the sum and +difference beams in (13), steered towards the AoD θ = 0◦. The sum beam +provides the obvious benefit of maximizing the SNR towards the desired angle, +while the difference beam improves AoD accuracy in the small neighborhood +around the targeted angle via its sharp curvature, which enables small angular +deviations to induce large amplitude changes. +IV. SIMULATION RESULTS +To evaluate the performance of the proposed ESPRIT- +oriented precoder design approach in Algorithm 1, we perform +numerical simulations using a mmWave setup with fc = +28 GHz, NTx = 64 and d = λ/2. The signal-to-noise ratio +(SNR) of the ℓth path is defined as SNRℓ = P|αℓ|2/σ2. +In the following parts, we first present our approach for +creating baseline precoders and provide illustrative examples +on beampatterns associated to ESPRIT-oriented precoders to +gain insights into how ESPRIT SIP considerations change +the shape of the beampatterns. Then, we evaluate the AoD +estimation performance of the designed precoders. +A. Baseline Construction via Codebook-Based Approach +Following the idea in [4], we propose to construct the base- +line precoder Fbase via a codebook-based approach. Suppose +that the BS has a coarse a-priori information on the AoDs θ +in the form of uncertainty intervals, e.g., obtained via tracking +routines [4], [24], [25]. Let Uℓ = [θmin,ℓ, θmax,ℓ] denote the +uncertainty interval for the AoD of the ℓth path and {θℓ,i}Nℓ +i=1 +the uniformly spaced AoDs covering Uℓ, where the grid size +Nℓ is dictated by the 3 dB beamwidth angular spacing [26]. +Accordingly, we define the codebook [4] +Fbase ≜ +� +Fsum γFdiff� +, +(13) +where +Fsum ≜ +� +Fsum +0 +· · · Fsum +L−1 +� +, +(14) +Fdiff ≜ +� +Fdiff +0 +· · · Fdiff +L−1 +� +, +(15) +Fsum +ℓ +≜ [a∗(θℓ,1) · · · a∗(θℓ,Nℓ)] , +(16) +Fdiff +ℓ +≜ [.a∗(θℓ,1) · · · .a∗(θℓ,Nℓ)] , +(17) +for ℓ = 0, . . . , L − 1, with .a(θ) ≜ ∂a(θ)/∂θ. Here, Fsum +and Fdiff correspond to sum (directional) and difference +(derivative) beams commonly employed in monopulse radar +processing for accurate AoD estimation [27]. Similar to radar, +a combined use of these beams is shown to be optimal for +positioning, as well [4]. In (13), γ represents the predefined +weighting factor of the difference beams with respect to the +sum beams, which is set to γ = 0.01 in simulations, and +Fsum and Fdiff are normalized to have the same Frobenius +norm before applying γ. To provide visualization and physical +intuition, Fig. 2 shows the beampatterns of the sum and +difference beams. + +-20 +-18 +-16 +-14 +-12 +-10 +-8 +-6 +-4 +-2 +0 +-20 +-10 +0 +10 +20 +(a) +-20 +-18 +-16 +-14 +-12 +-10 +-8 +-6 +-4 +-2 +0 +-20 +-10 +0 +10 +20 +(b) +-20 +-18 +-16 +-14 +-12 +-10 +-8 +-6 +-4 +-2 +0 +-20 +-10 +0 +10 +20 +(c) +Fig. 3. +The beampatterns of the ESPRIT-oriented precoders obtained via +Algorithm 1 for varying η values, where the baseline precoder Fbase is set +to the difference beam with θ = −10◦. +B. Illustrative Examples for ESPRIT-Oriented Precoders +Fig. 3 shows the beampatterns of the ESPRIT-oriented +precoders for three different η values in Algorithm 1 by +using the difference beam as the baseline. In Fig. 4, the +corresponding SIP errors in (9a) are plotted with respect to η. +For small η, the ESPRIT-oriented precoder has a beampattern +very close to that of the difference beam since Algorithm 1 +places more emphasis on beampattern synthesis accuracy than +on SIP approximation error, as seen from (9a). As η increases, +SIP gains more emphasis, meaning that the resulting beam +approaches the sum beam, for which the SIP is perfectly +satisfied, as discussed in Sec. III-A. This leads us to the +following important observation. +Observation 1: Phase-only ESPRIT-oriented precoder con- +verges from difference beam towards sum beam as η in- +creases. +To provide further insights, we show in Fig. 5 the phase dif- +ferences across the antenna elements of the ESPRIT-oriented +precoder for various η values. For small η, the ESPRIT- +oriented precoder is close to the difference beam, which has +a phase jump at the center of the array. The phase difference +100 +101 +102 +103 +104 +105 +106 +0 +0.01 +0.02 +0.03 +0.04 +0.05 +0.06 +0.07 +Fig. 4. SIP error in (9a) with respect to the penalty parameter η, where Fbase +is the difference beam with θ = −10◦. +0 +10 +20 +30 +40 +50 +60 +50 +100 +150 +200 +Fig. 5. Phase changes of the ESPRIT-oriented precoder obtained via Algo- +rithm 1 across the antenna elements for varying η values, where the baseline +precoder is taken as the difference beam with θ = −10◦. +profile becomes more smooth as η increases due to the SIP +requirement, which causes the resulting beam to converge to +the sum beam (which has uniform phase increments). Thus, +the second important observation regarding ESPRIT-oriented +precoders is stated as follows. +Observation 2: ESPRIT SIP requirement enforces uniform +phase increments across antenna elements. +C. Evaluation of AoD Estimation Performance +To evaluate the AoD estimation performance of the ESPRIT- +oriented precoders designed via Algorithm 1, we investigate +the accuracy quantified through the root mean-squared error +(RMSE) of θ, i.e., +RMSEθ = +� +E +����θ − θ +��2��1/2 , +(18) +where �θ = [�θ0 · · · �θL−1]T represents the estimate of θ from +y in (2). To obtain �θ, we apply 1-D beamspace ESPRIT [16] +described in Sec. III-A on the observations y in (2). We run +100 Monte Carlo trials with 50 snapshots each to construct the +covariance matrix for ESPRIT at each trial. The channel gains +αℓ are generated randomly across the snapshots by multiplying +a fixed gain (determined based on SNRℓ) with a random zero- +mean complex Gaussian coefficient with standard deviation 10. +In addition, based on the results in Sec. IV-B, we set η = 105 +in Algorithm 1. For performance benchmarking, we consider +the following precoders: +• Sum: The precoder Fsum in (14), which by definition +contains only unit-amplitude elements (i.e., steering vec- +tors), leading to phase-only beamforming without further +optimization. +• Sum-Diff: The precoder Fbase in (13), optimized to have +unit-amplitude elements by using [22, Alg. 1], which +corresponds to a single F update step in Algorithm 1. + +• Sum-Diff, ESPRIT-Or.: The precoder obtained via the +proposed ESPRIT-oriented precoder design algorithm in +Algorithm 1. +All the precoders are normalized to have the same Frobenius +norm ∥F∥F so that the total transmit power in (2) remains the +same among the different strategies for fair comparison. +We first consider a single-path scenario with θ0 = 20◦ and +U0 = [17◦, 23◦]. Fig. 6 shows the RMSEs obtained by the +considered precoding strategies as a function of the SNR, also +in comparison with the CRB4. It can be observed that the +ESPRIT-oriented precoder provides noticeable improvement +over the ESPRIT-unaware conventional sum-diff precoder at +low SNRs, indicating the effectiveness of the proposed design +strategy in Algorithm 1. However, the conventional sum pre- +coder outperforms the ESPRIT-oriented design at low SNRs, +while the RMSEs of all the precoders converge to the CRB as +the SNR increases. This suggests that although Algorithm 1 +succeeds in improving the performance, the sum precoder +appears to be the best choice in this specific scenario. +Next, we consider a different setting with θ0 = 70◦ and +U0 = [67◦, 73◦], whose results are reported in Fig. 7. We +observe that the proposed ESPRIT-oriented design significantly +outperforms both the traditional sum precoder and sum-diff +precoder in the medium and high SNR regimes, closing the +gap to the CRB. Comparing Fig. 6 and Fig. 7, it is seen that +performance gains provided by Algorithm 1 depend on the +AoD of the path. To further investigate this point, we plot in +Fig. 8 the RMSE with respect to the path AoD for a fixed +SNR of 20 dB with varying degrees of angular uncertainty. +A common observation is that for all AoDs, the ESPRIT- +oriented sum-diff precoder outperforms the standard sum-diff +precoder, which does not consider the ESPRIT SIP conditions, +suggesting that Algorithm 1 can provide considerable accuracy +gains in ESPRIT-based estimation. For ±1◦ uncertainty, the +ESPRIT-oriented precoder achieves lower RMSE than the sum +precoder for θ ∈ [−60◦, 60◦] in agreement with [4], while the +trend becomes the opposite outside this interval. Looking at +the ±3◦ uncertainty case, the sum precoder performs slightly +better than the ESPRIT-oriented one around θ = 0◦, while the +latter can significantly outperform the former at the end-fire +of the array, i.e., when the absolute value of the AoD is above +60◦. Furthermore, for the ±5◦ uncertainty case, the proposed +ESPRIT-oriented design provides substantial gains over the +sum precoder for almost the entire range of AoD values, which +further evidences the effectiveness of the proposed algorithm. +Finally, we investigate the RMSE performances for a two- +path scenario with θ = [20◦, 70◦], U0 = [17◦, 23◦], U1 = +[67◦, 73◦], and SNR = [20, 0] dB. Fig. 9 plots the RMSE +with respect to the SNR of the second path, where the SNRs +of both paths are changed simultaneously while keeping their +difference fixed. It is observed that the proposed ESPRIT-based +design achieves higher accuracy than the benchmark schemes +in the medium and high SNR regimes. The gap to the CRB +can be attributed to the intrinsic suboptimality of ESPRIT +[28], [29] and to imperfect decorrelation of the paths in the +estimated correlation matrix �R. +V. CONCLUDING REMARKS +In this paper, we have studied the problem of mmWave +precoder design tailored specifically to ESPRIT-based channel +4Since the CRBs belonging to the different precoders are very close to +each other, we only show the CRB corresponding to Fsum for the sake of +figure readability. +-30 +-20 +-10 +0 +10 +20 +30 +10-3 +10-2 +10-1 +100 +101 +102 +Fig. 6. ESPRIT RMSEs obtained by the considered precoders with respect +to SNR for a single-path scenario, where θ0 = 20◦ and U0 = [17◦, 23◦]. +-30 +-20 +-10 +0 +10 +20 +30 +10-3 +10-2 +10-1 +100 +101 +102 +Fig. 7. ESPRIT RMSEs obtained by the considered precoders with respect +to SNR for a single-path scenario, where θ0 = 70◦ and U0 = [67◦, 73◦]. +estimation. Considering the fact that standard precoders (i.e., +sum beam) fail to achieve satisfactory performance in AoD +estimation and that CRB-optimized precoders (sum-diff beam) +destroy the SIP of ESPRIT, leading to large degradations in +ESPRIT accuracy, we have developed a novel ESPRIT-oriented +precoder design approach that jointly optimizes the precoder +and the SIP-restoring matrix used in ESPRIT. Simulation +results have provided valuable insights into how the SIP +requirement impacts the beampattern of the ESPRIT-oriented +precoders and shown the effectiveness of the proposed design +strategy. As future work, similar design principles can be +employed to extend the current study to higher dimensions, +i.e., 2-D uniform rectangular arrays (URAs) at both the BS +and the UE sides, possibly with OFDM transmission, leading +to ESPRIT-oriented precoder and combiner designs for 5-D +channel estimation (AoD, AoA and delay) [14]. +ACKNOWLEDGMENT +This work was supported, in part, by the European Com- +mission through the H2020 project Hexa-X (Grant Agree- +ment no. 101015956), the MSCA-IF grant 888913 (OTFS- +RADCOM), ICREA Academia Program, and Spanish R+D +project PID2020-118984GB-I00. +REFERENCES +[1] S. Bartoletti et al., “Positioning and sensing for vehicular safety appli- +cations in 5G and beyond,” IEEE Communications Magazine, vol. 59, +no. 11, pp. 15–21, 2021. +[2] S. Dwivedi et al., “Positioning in 5G networks,” IEEE Communications +Magazine, vol. 59, no. 11, pp. 38–44, 2021. +[3] A. Fascista et al., “Low-complexity accurate mmwave positioning for +single-antenna users based on angle-of-departure and adaptive beam- +forming,” in IEEE International Conference on Acoustics, Speech and +Signal Processing (ICASSP), 2020, pp. 4866–4870. + +-80 +-60 +-40 +-20 +0 +20 +40 +60 +80 +10-3 +10-2 +10-1 +100 +101 +102 +(a) +-80 +-60 +-40 +-20 +0 +20 +40 +60 +80 +10-2 +100 +102 +(b) +-80 +-60 +-40 +-20 +0 +20 +40 +60 +80 +10-3 +10-2 +10-1 +100 +101 +102 +(c) +Fig. 8. ESPRIT RMSEs obtained by the considered precoders with respect +to the path AoD for a single-path scenario with (a) ±1, (b) ±3, and (c) ±5 +degrees of angular uncertainty for SNR = 20 dB. +-30 +-20 +-10 +0 +10 +20 +30 +10-2 +10-1 +100 +101 +102 +Fig. 9. ESPRIT RMSEs obtained by the considered precoders with respect +to SNR for a two-path scenario, where θ = [20◦, 70◦] with ±3 degrees of +uncertainty for both paths. +[4] M. F. 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Jiang et al., “Beamspace multidimensional ESPRIT approaches +for simultaneous localization and communications,” 2021. [Online]. +Available: https://arxiv.org/abs/2111.07450 +[15] F. Wen et al., “Tensor decomposition based beamspace ESPRIT for +millimeter wave mimo channel estimation,” in IEEE GLOBECOM, Abu +Dhabi, United Arab Emirates, 2018. +[16] G. Xu et al., “Beamspace ESPRIT,” IEEE Transactions on Signal +Processing, vol. 42, no. 2, pp. 349–356, 1994. +[17] N. Garcia et al., “Optimal precoders for tracking the AoD and AoA +of a mmWave path,” IEEE Transactions on Signal Processing, vol. 66, +no. 21, pp. 5718–5729, Nov 2018. +[18] A. Fascista et al., “RIS-aided joint localization and synchronization +with a single-antenna receiver: Beamforming design and low-complexity +estimation,” IEEE Journal of Selected Topics in Signal Processing, +vol. 16, no. 5, pp. 1141–1156, 2022. +[19] D. 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Li et al., “Performance analysis for DOA estimation algorithms: +unification, simplification, and observations,” IEEE Transactions on +Aerospace and Electronic Systems, vol. 29, no. 4, pp. 1170–1184, 1993. + diff --git a/ctAzT4oBgHgl3EQfnv1S/content/tmp_files/load_file.txt b/ctAzT4oBgHgl3EQfnv1S/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1a532c85565c91ddde5603e2a546eb98754c73f0 --- /dev/null +++ b/ctAzT4oBgHgl3EQfnv1S/content/tmp_files/load_file.txt @@ -0,0 +1,425 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf,len=424 +page_content='ESPRIT-Oriented Precoder Design for mmWave Channel Estimation Musa Furkan Keskin∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' Alessio Fascista†,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' Fan Jiang‡,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' Angelo Coluccia†,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' Gonzalo Seco-Granados§,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' Henk Wymeersch∗ ∗Chalmers University of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' Sweden,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' †University of Salento,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' Italy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' ‡Halmstad University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' Sweden,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' §Autonomous University of Barcelona,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' Spain Abstract—We consider the problem of ESPRIT-oriented pre- coder design for beamspace angle-of-departure (AoD) estimation in downlink mmWave multiple-input single-output communi- cations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' Standard precoders (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=', directional/sum beams) yield poor performance in AoD estimation, while Cram´er-Rao bound- optimized precoders undermine the so-called shift invariance property (SIP) of ESPRIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' To tackle this issue, the problem of designing ESPRIT-oriented precoders is formulated to jointly optimize over the precoding matrix and the SIP-restoring matrix of ESPRIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' We develop an alternating optimization approach that updates these two matrices under unit-modulus constraints for analog beamforming architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' Simulation results demon- strate the validity of the proposed approach while providing valuable insights on the beampatterns of the ESPRIT-oriented precoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' Index Terms—Precoder design, beamspace ESPRIT, channel estimation, mmWave communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' INTRODUCTION Positioning in 5G relies to a large extent on the use of mmWave frequencies, with their ample bandwidth and large antenna arrays [1]–[3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' Large bandwidths offer high delay resolution but provide limited opportunities for optimization, as base stations (BSs) must use non-overlapping subcarriers for multi-BS positioning solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' Large antenna arrays yield high angle resolution, as well as the ability to shape signals in the spatial domain, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=', for interference control, but also for optimizing positioning performance [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' Harnessing the improved resolution and also exploiting optimized spatial designs enhance the performance of the channel estimation routine, which detects the number of paths, and for each path estimates the geometric parameters (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=', time-of-arrival (ToA), angle-of-arrival (AoA), angle-of-departure (AoD)) [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' As channel estimation is a joint function among communi- cation, positioning, and sensing, it is important to develop methods that are both accurate and of moderate complexity [6], especially for integrated sensing and communication (ISAC) systems towards 6G multi-functional wireless networks [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' In a general pilot-based channel estimation setup, the opti- mal channel parameters are the maximum a posteriori (MAP) estimates given the received signal sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' However, opti- mization methods employed in MAP estimation can involve heavy computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' On the other hand, it is notable that mmWave channels are usually sparse, due to a limited number of multipath propagation arriving at the receiver with relatively strong path gains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' As a result, sparsity-inspired low-complexity channel estimation methods are developed [8]–[13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' Among them, the estimation of signal parameters via rotational invari- ance techniques (ESPRIT)-based channel estimation methods have been widely studied, due to their good trade-off between estimation performance and complexity [11]–[13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' Recently, ESPRIT-based approaches have been applied to the beamspace, which is attractive since analogue and/or digital beamforming structures are employed in most massive MIMO mmWave Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' mmWave MISO downlink scenario where the UE aims to estimate the AoDs of multiple paths using high-resolution beamspace ESPRIT methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' systems [13]–[15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' However, to apply beamspace ESPRIT methods, precoders are required to hold the shift invariance property (SIP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' Examples of such precoding matrices include the discrete Fourier transform (DFT) beams [13], [16] and the directional beams [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' When the SIP does not hold for the precoding matrix, an approximation will be applied during the derivation of the beamspace ESPRIT methods, leading to performance degradations [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' In addition, research on Cram´er-Rao bound (CRB)-optimized precoder design suggests that the optimal precoding matrix usually does not hold the SIP [4], [17], [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' In other words, there is an inevitable performance loss when low-complexity ESPRIT methods are employed with CRB-optimized precoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' In this paper, we investigate the problem of ESPRIT- oriented precoder design for AoD estimation in mmWave communications, targeting a near-optimal precoding scheme in terms of accuracy while enjoying the low-complexity and high-resolution ESPRIT methods for channel estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' Our specific contributions are as follows: We formulate the problem of ESPRIT-oriented precoder design as a beampattern synthesis problem that considers joint optimization of the precoding matrix and the SIP- restoring matrix of ESPRIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' We propose an alternating optimization strategy that updates the precoder and the SIP-restoring matrix se- quentially under the unit-modulus constraint on individual precoder elements, suitable for phase-only beamforming architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' Through simulation results, we provide important in- sights into the beampatterns of the resulting ESPRIT- oriented precoders and demonstrate the effectiveness of the proposed design approach in ESPRIT-based channel estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' SYSTEM MODEL AND PROBLEM DESCRIPTION A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' System Model We consider a mmWave MISO downlink (DL) flat-fading communications scenario with an NTx-antenna BS and a single-antenna user equipment (UE), as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content='01585v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content='SP] 4 Jan 2023 y 个 UE BSConsidering the presence of L paths1, the received signal at the UE at transmission instance m and snapshot n is given by ym,n = √ P L−1 � ℓ=0 αℓ,n aT(θℓ)fmsm,n + zm,n (1) for m = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' , M and n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' , N, where M and N denote, respectively, the number of transmissions and the number of snapshots2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' In (1), P denotes the transmit power, [a(θ)]k = ej2π d λ k sin θ, k = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' , NTx − 1, is the steering vector for the BS TX array, λ = c/fc is the wavelength with c and fc denoting the speed of propagation and carrier frequency, respectively, d is the array element spacing, fm ∈ CNTx×1 de- notes the BS precoder at time m, αℓ,n and θℓ are the complex channel gain and AoD of the ℓth path for the nth snapshot, respectively, sm,n is the pilot symbol, and zm,n ∼ CN(0, σ2) is additive white Gaussian noise (AWGN) with power σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' For simplicity, we set sm,n = 1, ∀m, n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' Aggregating the observations (1) over M transmissions, we have the received signal at the nth snapshot yn = √ PFTVαn + zn , (2) where yn ≜ [y1,n · · · yM,n]T ∈ CM×1, F ≜ [f1 · · · fM] ∈ CNTx×M is the precoding matrix satisfying tr � FFH� = M, V ≜ [a(θ0) · · · a(θL−1)] ∈ CNTx×L, αn ≜ [α0,n · · · αL−1,n]T ∈ CL×1, and zn ∼ CN(0, σ2I) represents the AWGN component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' Problem Description In the considered mmWave scenario, the UE aims to esti- mate the AoDs θ = [θ0 · · · θL−1]T using beamspace ESPRIT [16], [21] from the beamspace observations {yn}N n=1 in (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' The problem of interest is to design the BS precoding matrix F to maximize the accuracy of estimation of θ at the UE while at the same time trying to preserve as much as possible the SIP required by ESPRIT-based estimation [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' ESPRIT-ORIENTED PRECODER DESIGN In this section, we provide a review of beamspace ESPRIT and revisit the SIP, which enforces a certain structure on the precoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' Based on this structure and using an ESPRIT- unaware baseline precoder Fbase (which will be introduced later in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' IV-A), we formulate a novel precoder design problem that jointly optimizes beampattern synthesis accuracy (with respect to Fbase) and ESPRIT SIP error (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=', the level of degradation of SIP), leading to near-optimal performance for ESPRIT-based estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' Review of Beamspace ESPRIT From (2), we compute the covariance matrix R = 1 N N � n=1 ynyH n = PFTV � 1 N N � n=1 αnαH n � VHF∗ + σ2I , (3) where V is a Vandermonde matrix which holds the SIP, satisfying J1V = J2VΦH where J1 = � INTx−1, 0(NTx−1)×1 � ∈ R(NTx−1)×NTx and J2 = 1We consider a mmWave tracking scenario [17], [19], [20] with known L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' 2Here, snapshots may correspond to, for instance, different subcarriers of an orthogonal frequency-division multiplexing (OFDM) system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' In this case, it is reasonable to assume that the channel gains αℓ,n change across snapshots, but the AoDs θℓ remain constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' � 0(NTx−1)×1, INTx−1 � ∈ R(NTx−1)×NTx are selection ma- trices, and Φ = Diag([[a(θ0)]1, [a(θ1)]1, · · · , [a(θL−1)]1]T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' In (3), we assume that 1 N �N n=1 αnαH n is a diagonal matrix (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=', paths are decorrelated), meaning that the dimension of the signal subspace is L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' In the precoded case, it has been shown in [14], [16] that if the matrix F holds the SIP, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=', J1F = J2FΛ (4) for some non-singular Λ ∈ CM×M, we can restore the SIP from C = FTV, by finding a non-null matrix Q such that QCΦ = QΛTC , (5) where Q ∈ CM×M satisfies Q[FT eM ΛTFTe1] = 0, (6) and em ∈ RNTx×1 is the m-th column of the identity matrix INTx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' From (6), Q can be obtained as Q = IM −�1 i=0 qiqH i , where q0, q1 ∈ CM×1 are orthonormal column vectors span- ning the subspace corresponding to [FT eM ΛTFTe1] ∈ CM×2 [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' Since perfect SIP cannot always be guaranteed3 in (4), one can resort to the least-squares (LS) solution to find an approximate Λ [14]: �ΛLS = arg min Λ ∥J1F − J2FΛ∥2 F (7) = � FHJH 2 J2F �−1 FHJH 2 J1F , (8) where ∥·∥F denotes the Frobenius norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' Given an estimate of the covariance matrix R, the signal subspace matrix Us ∈ CM×L can be obtained through the SVD (or truncated SVD) operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' Since both C and Us span the same signal subspace, we have C = UsT, where T ∈ CL×L is a non-singular matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' Using the SIP of C in (5), we further obtain QUsΠ = QΛTUs where Π = TΦT−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' The diagonal elements in Φ will be used to estimate the AoD of each path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' The beamspace ESPRIT approach can be summarized as follows: Find Λ and Q for given F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' Obtain an estimate of R as �R using multiple snapshots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' Perform SVD (or truncated SVD) on �R to obtain Us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' Obtain the least-square (LS) solution of Π as �Π = (QUs)†QΛTUs, where (·)† denotes Moore-Penrose pseudo-inverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' Perform eigenvalue decomposition on �Π to obtain an estimate of Φ, and retrieve the corresponding AoDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' ESPRIT-Oriented Precoder Design with SIP Considera- tions We formulate the problem of ESPRIT-oriented precoder design as a beampattern synthesis via joint optimization of F and Λ, starting from a desired beampattern created by an ESPRIT-unconstrained precoder Fbase as baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' The goal is to minimize the weighted average of the beampattern synthesis error and the ESPRIT SIP error, quantified by the error of the LS solution in (7): min F,Λ ��B − ATF ��2 F � �� � beampattern synthesis accuracy + η ∥J1F − J2FΛ∥2 F � �� � SIP approximation error (9a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' |[F]n,m| = 1, ∀n, m , (9b) 3Perfect SIP holds for DFT beams and directional/sum beams (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=', steering vectors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' Algorithm 1 ESPRIT-Oriented Precoder Design via Joint Optimization of F and Λ in (9) 1: Input: Baseline precoder Fbase, transmit steering matrix A, selection matrices J1 and J2, SIP error weight η, convergence threshold ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' 2: Output: ESPRIT-oriented precoder F, SIP-restoring ma- trix Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' 3: Initialization: 4: Initialize the precoder as F = Fbase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' 5: Initialize the SIP-restoring matrix as Λ = �ΛLS via (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' 6: Alternating Optimization Iterations: 7: repeat 8: Update F in (12) via [22, Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' 9: Update Λ via (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' 10: until the objective (9a) converges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' where B = ATFbase ∈ CNgrid×M represents the desired beampattern corresponding to Fbase at Ngrid angular grid points {θi}Ngrid i=1 , A = � a(θ1) · · · a(θNgrid) � ∈ CNTx×Ngrid is the transmit steering matrix evaluated at the specified grid locations, and η is a predefined weight on the SIP error, chosen to provide a suitable trade-off between beampattern synthesis accuracy and SIP approximation error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' In addition, the con- straint (9b) is imposed to ensure compatibility with phase-only beamforming architectures [22] (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=', analog passive arrays [23]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' In the case of phase-amplitude beamforming (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=', via active phased arrays [23]), the problem becomes the special case of (9) without the constraint (9b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' Alternating Optimization to Solve (9) The problem (9) is non-convex due to (i) the non-convexity of (9a) in the joint variable F and Λ, and (ii) the unit-modulus constraint in (9b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' To tackle (9), we resort to an alternating optimization method that updates F and Λ in an iterative fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' 1) Optimize F for fixed Λ: Using the vectorization property of the Kronecker product, the objective function (9a) can be rewritten as g(f) (10) = ��b − � IM ⊗ AT � f ��2 2 + η ��� IM ⊗ J1 − ΛT ⊗ J2 � f ��2 2 , where f ≜ vec (F) and b ≜ vec (B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' Defining Q ≜ � IM ⊗ AT �H � IM ⊗ AT � (11) + η � IM ⊗ J1 − ΛT ⊗ J2 �H � IM ⊗ J1 − ΛT ⊗ J2 � , p ≜ � IM ⊗ AT �H b , the problem (9) for fixed Λ can be expressed as min f f HQf − 2ℜ � pHf � (12) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' |fn| = 1, ∀n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' The problem (12) can be solved using gradient projections iterations in [22, Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' 2) Optimize Λ for fixed F: The subproblem of (9) for fixed F is exactly the LS problem defined in (7), whose solution is provided in (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' The overall algorithm to solve (9) via alternating optimiza- tion of F and Λ is summarized in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' 10 8 6 4 2 0 2 4 6 8 10 20 15 10 5 0 5 10 15 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' Illustration of the beampatterns corresponding to the sum and difference beams in (13), steered towards the AoD θ = 0◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' The sum beam provides the obvious benefit of maximizing the SNR towards the desired angle, while the difference beam improves AoD accuracy in the small neighborhood around the targeted angle via its sharp curvature, which enables small angular deviations to induce large amplitude changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' SIMULATION RESULTS To evaluate the performance of the proposed ESPRIT- oriented precoder design approach in Algorithm 1, we perform numerical simulations using a mmWave setup with fc = 28 GHz, NTx = 64 and d = λ/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' The signal-to-noise ratio (SNR) of the ℓth path is defined as SNRℓ = P|αℓ|2/σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' In the following parts, we first present our approach for creating baseline precoders and provide illustrative examples on beampatterns associated to ESPRIT-oriented precoders to gain insights into how ESPRIT SIP considerations change the shape of the beampatterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' Then, we evaluate the AoD estimation performance of the designed precoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' Baseline Construction via Codebook-Based Approach Following the idea in [4], we propose to construct the base- line precoder Fbase via a codebook-based approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' Suppose that the BS has a coarse a-priori information on the AoDs θ in the form of uncertainty intervals, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=', obtained via tracking routines [4], [24], [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' Let Uℓ = [θmin,ℓ, θmax,ℓ] denote the uncertainty interval for the AoD of the ℓth path and {θℓ,i}Nℓ i=1 the uniformly spaced AoDs covering Uℓ, where the grid size Nℓ is dictated by the 3 dB beamwidth angular spacing [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' Accordingly, we define the codebook [4] Fbase ≜ � Fsum γFdiff� , (13) where Fsum ≜ � Fsum 0 · · Fsum L−1 � , (14) Fdiff ≜ � Fdiff 0 · · Fdiff L−1 � , (15) Fsum ℓ ≜ [a∗(θℓ,1) · · · a∗(θℓ,Nℓ)] , (16) Fdiff ℓ ≜ [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content='a∗(θℓ,1) · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content='a∗(θℓ,Nℓ)] , (17) for ℓ = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' , L − 1, with .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content='a(θ) ≜ ∂a(θ)/∂θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' Here, Fsum and Fdiff correspond to sum (directional) and difference (derivative) beams commonly employed in monopulse radar processing for accurate AoD estimation [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' Similar to radar, a combined use of these beams is shown to be optimal for positioning, as well [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' In (13), γ represents the predefined weighting factor of the difference beams with respect to the sum beams, which is set to γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content='01 in simulations, and Fsum and Fdiff are normalized to have the same Frobenius norm before applying γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' To provide visualization and physical intuition, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' 2 shows the beampatterns of the sum and difference beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' 20 18 16 14 12 10 8 6 4 2 0 20 10 0 10 20 (a) 20 18 16 14 12 10 8 6 4 2 0 20 10 0 10 20 (b) 20 18 16 14 12 10 8 6 4 2 0 20 10 0 10 20 (c) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' The beampatterns of the ESPRIT-oriented precoders obtained via Algorithm 1 for varying η values, where the baseline precoder Fbase is set to the difference beam with θ = −10◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' Illustrative Examples for ESPRIT-Oriented Precoders Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' 3 shows the beampatterns of the ESPRIT-oriented precoders for three different η values in Algorithm 1 by using the difference beam as the baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' 4, the corresponding SIP errors in (9a) are plotted with respect to η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' For small η, the ESPRIT-oriented precoder has a beampattern very close to that of the difference beam since Algorithm 1 places more emphasis on beampattern synthesis accuracy than on SIP approximation error, as seen from (9a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' As η increases, SIP gains more emphasis, meaning that the resulting beam approaches the sum beam, for which the SIP is perfectly satisfied, as discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' III-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' This leads us to the following important observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' Observation 1: Phase-only ESPRIT-oriented precoder con- verges from difference beam towards sum beam as η in- creases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' To provide further insights, we show in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' 5 the phase dif- ferences across the antenna elements of the ESPRIT-oriented precoder for various η values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' For small η, the ESPRIT- oriented precoder is close to the difference beam, which has a phase jump at the center of the array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' The phase difference 100 101 102 103 104 105 106 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content='07 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' SIP error in (9a) with respect to the penalty parameter η, where Fbase is the difference beam with θ = −10◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' 0 10 20 30 40 50 60 50 100 150 200 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' Phase changes of the ESPRIT-oriented precoder obtained via Algo- rithm 1 across the antenna elements for varying η values, where the baseline precoder is taken as the difference beam with θ = −10◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' profile becomes more smooth as η increases due to the SIP requirement, which causes the resulting beam to converge to the sum beam (which has uniform phase increments).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' Thus, the second important observation regarding ESPRIT-oriented precoders is stated as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' Observation 2: ESPRIT SIP requirement enforces uniform phase increments across antenna elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' Evaluation of AoD Estimation Performance To evaluate the AoD estimation performance of the ESPRIT- oriented precoders designed via Algorithm 1, we investigate the accuracy quantified through the root mean-squared error (RMSE) of θ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=', RMSEθ = � E ����θ − θ ��2��1/2 , (18) where �θ = [�θ0 · · · �θL−1]T represents the estimate of θ from y in (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' To obtain �θ, we apply 1-D beamspace ESPRIT [16] described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' III-A on the observations y in (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' We run 100 Monte Carlo trials with 50 snapshots each to construct the covariance matrix for ESPRIT at each trial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' The channel gains αℓ are generated randomly across the snapshots by multiplying a fixed gain (determined based on SNRℓ) with a random zero- mean complex Gaussian coefficient with standard deviation 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' In addition, based on the results in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' IV-B, we set η = 105 in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' For performance benchmarking, we consider the following precoders: Sum: The precoder Fsum in (14), which by definition contains only unit-amplitude elements (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=', steering vec- tors), leading to phase-only beamforming without further optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' Sum-Diff: The precoder Fbase in (13), optimized to have unit-amplitude elements by using [22, Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' 1], which corresponds to a single F update step in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' Sum-Diff, ESPRIT-Or.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' : The precoder obtained via the proposed ESPRIT-oriented precoder design algorithm in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' All the precoders are normalized to have the same Frobenius norm ∥F∥F so that the total transmit power in (2) remains the same among the different strategies for fair comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' We first consider a single-path scenario with θ0 = 20◦ and U0 = [17◦, 23◦].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' 6 shows the RMSEs obtained by the considered precoding strategies as a function of the SNR, also in comparison with the CRB4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' It can be observed that the ESPRIT-oriented precoder provides noticeable improvement over the ESPRIT-unaware conventional sum-diff precoder at low SNRs, indicating the effectiveness of the proposed design strategy in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' However, the conventional sum pre- coder outperforms the ESPRIT-oriented design at low SNRs, while the RMSEs of all the precoders converge to the CRB as the SNR increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' This suggests that although Algorithm 1 succeeds in improving the performance, the sum precoder appears to be the best choice in this specific scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' Next, we consider a different setting with θ0 = 70◦ and U0 = [67◦, 73◦], whose results are reported in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' We observe that the proposed ESPRIT-oriented design significantly outperforms both the traditional sum precoder and sum-diff precoder in the medium and high SNR regimes, closing the gap to the CRB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' Comparing Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' 6 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' 7, it is seen that performance gains provided by Algorithm 1 depend on the AoD of the path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' To further investigate this point, we plot in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' 8 the RMSE with respect to the path AoD for a fixed SNR of 20 dB with varying degrees of angular uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' A common observation is that for all AoDs, the ESPRIT- oriented sum-diff precoder outperforms the standard sum-diff precoder, which does not consider the ESPRIT SIP conditions, suggesting that Algorithm 1 can provide considerable accuracy gains in ESPRIT-based estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' For ±1◦ uncertainty, the ESPRIT-oriented precoder achieves lower RMSE than the sum precoder for θ ∈ [−60◦, 60◦] in agreement with [4], while the trend becomes the opposite outside this interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' Looking at the ±3◦ uncertainty case, the sum precoder performs slightly better than the ESPRIT-oriented one around θ = 0◦, while the latter can significantly outperform the former at the end-fire of the array, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=', when the absolute value of the AoD is above 60◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' Furthermore, for the ±5◦ uncertainty case, the proposed ESPRIT-oriented design provides substantial gains over the sum precoder for almost the entire range of AoD values, which further evidences the effectiveness of the proposed algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' Finally, we investigate the RMSE performances for a two- path scenario with θ = [20◦, 70◦], U0 = [17◦, 23◦], U1 = [67◦, 73◦], and SNR = [20, 0] dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' 9 plots the RMSE with respect to the SNR of the second path, where the SNRs of both paths are changed simultaneously while keeping their difference fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' It is observed that the proposed ESPRIT-based design achieves higher accuracy than the benchmark schemes in the medium and high SNR regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' The gap to the CRB can be attributed to the intrinsic suboptimality of ESPRIT [28], [29] and to imperfect decorrelation of the paths in the estimated correlation matrix �R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' CONCLUDING REMARKS In this paper, we have studied the problem of mmWave precoder design tailored specifically to ESPRIT-based channel 4Since the CRBs belonging to the different precoders are very close to each other, we only show the CRB corresponding to Fsum for the sake of figure readability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' 30 20 10 0 10 20 30 10-3 10-2 10-1 100 101 102 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' ESPRIT RMSEs obtained by the considered precoders with respect to SNR for a single-path scenario, where θ0 = 20◦ and U0 = [17◦, 23◦].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' 30 20 10 0 10 20 30 10-3 10-2 10-1 100 101 102 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' ESPRIT RMSEs obtained by the considered precoders with respect to SNR for a single-path scenario, where θ0 = 70◦ and U0 = [67◦, 73◦].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' Considering the fact that standard precoders (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=', sum beam) fail to achieve satisfactory performance in AoD estimation and that CRB-optimized precoders (sum-diff beam) destroy the SIP of ESPRIT, leading to large degradations in ESPRIT accuracy, we have developed a novel ESPRIT-oriented precoder design approach that jointly optimizes the precoder and the SIP-restoring matrix used in ESPRIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' Simulation results have provided valuable insights into how the SIP requirement impacts the beampattern of the ESPRIT-oriented precoders and shown the effectiveness of the proposed design strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' As future work, similar design principles can be employed to extend the current study to higher dimensions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=', 2-D uniform rectangular arrays (URAs) at both the BS and the UE sides, possibly with OFDM transmission, leading to ESPRIT-oriented precoder and combiner designs for 5-D channel estimation (AoD, AoA and delay) [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfnv1S/content/2301.01585v1.pdf'} +page_content=' ACKNOWLEDGMENT This work was supported, in part, by the European Com- mission through the H2020 project Hexa-X (Grant Agree- ment no.' metadata={'source': 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0000000000000000000000000000000000000000..54d4a004518039a86c24da6e4031b29476fbfe5a --- /dev/null +++ b/edFJT4oBgHgl3EQfTCwr/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:9ec7a17883e4fccc6564aafbd0ebfa1af40035860ec3fb4602ade6056fa603f0 +size 209194 diff --git a/ftA0T4oBgHgl3EQfHv83/content/tmp_files/2301.02064v1.pdf.txt b/ftA0T4oBgHgl3EQfHv83/content/tmp_files/2301.02064v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..d1c30a6ba8025bcdbb86a80607e6f458840222eb --- /dev/null +++ b/ftA0T4oBgHgl3EQfHv83/content/tmp_files/2301.02064v1.pdf.txt @@ -0,0 +1,1699 @@ +MS-DINO: Efficient Distributed Training of Vision Transformer Foundation Model in +Medical Domain through Masked Sampling +Sangjoon Parka, Ik-Jae Leeb, Jun Won Kimb,∗, Jong Chul Yec,∗∗ +aDepartment of Bio and Brain Engineering, Korea Advanced Institute of Science and Technology (KAIST), Daejeon, South Korea +bDepartment of Radiation Oncology, Gangnam Severance Hospital, Seoul, South Korea +cKim Jaechul Graduate School of AI, Korea Advanced Institute of Science and Technology (KAIST), Daejeon, South Korea +A R T I C L E I N F O +2000 MSC: +41A05, +41A10, +65D05, +65D17 +Keywords: Distributed learning, Self-su- +pervised learning, Vision Transformer, +Random permutation, +Computed To- +mography, Chest X-ray +A B S T R A C T +In spite of the recent success of deep learning in the medical domain, the problem of +data scarcity in the medical domain gets aggravated due to privacy and data ownership +issues. Distributed learning approaches including federated learning have been studied +to alleviate the problems, but they suffer from cumbersome communication overheads +and weakness in privacy protection. To address this, here we propose a self-supervised +masked sampling distillation method for vision transformer that can be performed with- +out continuous communication but still enhance privacy using a vision transformer- +specific encryption method. The effectiveness of our method is demonstrated with ex- +tensive experiments on two medical domain data and two different downstream tasks, +showing superior performances than those obtained with the existing distributed learn- +ing strategy as well as the fine-tuning only baseline. As the self-supervised model built +with the proposed method is capable of having a general semantic understanding of the +modality, we demonstrate its potential as a task-agnostic foundation model for various +medical tasks, widening the applicability in the medical domain. +© 2023 +1. Introduction +The success of deep learning (DL) has made it de facto stan- +dard in developing artificial intelligence (AI) powered medical +tools (Lee et al., 2017; Ting et al., 2018; Giger, 2018; Pesapane +et al., 2018), but the data and label-driven nature of the deep +learning requires imperative collaboration between multiple in- +stitutions. However, strict legal or institutional regulations often +forbid the free sharing of data derived from patients due to pri- +vacy concerns (Edemekong et al., 2018; Hoofnagle et al., 2019). +Usually, the de-identified data can be shared only between the +collaborators under formal consents, which can be one of the +most cumbersome obstacles in AI research. Distributed learn- +ing methods like federated learning (FL) (Koneˇcn`y et al., 2016) +and split learning (SL) (Vepakomma et al., 2018) have been in- +troduced to cope with this problem by alleviating the data gov- +ernance and ownership issues. +In the FL, the goal is to obtain a model on the server-side +while training data remain unmoved over the edge devices of +∗Co-corresonding author: Tel.: +82-2-2019-3152; fax: +82-2-2019-4855; +∗∗Co-corresonding author: Tel.: +82-42-350-4320; fax: +82-42-350-4310; +e-mail: junwon@yuhs.ac, jong.ye@kaist.ac.kr (Jong Chul Ye) +multiple clients. +In detail, the central server distributes the +global model to each client, and the clients perform training +iterations with their data in parallel, to return the results of lo- +cal computations to the server. The server then aggregates and +averages local updates, and distributes again the updated global +model. This process is repeatedly performed until the model +converges. While the FL has resolved the issues of data shar- +ing, it does not fully guarantee privacy. Specifically, data can +leak by reconstructing the private data used in training with the +inversion attack which uses the stolen gradients of local model +updates from insecure aggregation (Geiping et al., 2020). In +addition, it imposes heavy computational loads on the edge de- +vices of the clients as the most of computations and updates of +the model are performed in client-side devices (Li et al., 2020b; +Mammen, 2021). +In the SL, on the other hand, the entire model is split into +several sub-networks trained separately on the server-side and +client-side. Specifically, the first sub-network performs a for- +ward pass on the client-side device and sends the smashed fea- +tures to the second sub-network located on the server-side. The +server then performs forward propagation with these features +to pass back the subsequent features to the third sub-network +on the client side. The third sub-network on the client side can +arXiv:2301.02064v1 [cs.CV] 5 Jan 2023 + +2 +S. Park and JC Ye et al. +yield the outcome of the model, and the loss can be calculated +with the client-side label. Backpropagation through split sub- +networks on clients and server-sides is performed in the exact +opposite manner to the forward pass. Allocating small-sized +sub-networks to train on the client-side device reduces the com- +putational load of local clients that usually lack resources in +practical implementation. In addition, the SL offers model pri- +vacy by inserting black-box to clients and server, preventing +both from having access to the full network. However, there +remain privacy concerns like the hijacking of transmitted fea- +tures to be inverted to the original data (Gawron and Stubbings, +2022). +Besides these limitations, these methods may impose sub- +stantial communication overheads in practical implementation. +For instance, FL requires the entire model, which is usually +large-sized, to be aggregated and distributed between the server +and clients. Meanwhile, the feature and gradients from the split +subnetwork should be continuously interchanged in a relay- +based manner in the SL, enforcing the clients to be connected +during the entire training process (Li et al., 2020b). +Recently, a pure attention-based DL model named Vision +Transformer (ViT) (Dosovitskiy et al., 2020) has been intro- +duced to the vision community and has become a core compo- +nent of vision research. The ViT has several desirable prop- +erties thanks to its simple but powerful attention architecture, +and recent efforts to understand the properties of the ViT have +found that it has more shape-biased nature like human and is +less susceptible to perturbation like occlusion or random patch +permutation (Naseer et al., 2021). In addition, several recent +works on self-supervised learning have reported that the ViT- +based model can benefit more from the various self-supervised +learning schemes like learning semantic meaning with knowl- +edge distillation (Caron et al., 2021) or masked patch prediction +(Bao et al., 2021; He et al., 2022). +Inspired by the properties of ViT, here we present a novel +distributed self-supervised learning strategy to build a founda- +tion model that does not require continuous communication be- +tween server and clients. In detail, our insight comes from the +permutation invariant properties of self-attention can be utilized +to offer the encryption by feature-space random permutation +(Park and Ye, 2022). On top of this, we exploit another im- +portant property of ViT resulting from its patch-based image +processing, which enables the random masked sampling-based +self-supervised learning to train the foundation model solely on +the server-side, eliminating the need for continuous communi- +cation. +Our contributions can be highlighted as follows: +• We tackle the cumbersome problems of distributed self- +supervised learning by introducing a novel method that ef- +fectively leverages the properties of ViT. +• We conducte extensive experiments to show the superior- +ity of the proposed method on two medical domain data +and tasks, i.e., organ-at-risk (OAR) segmentation in com- +puted tomography (CT) and tuberculosis diagnosis in chest +X-ray (CXR). +• We experimentally prove the infeasibility of a privacy at- +tack on the proposed methods, implying that the proposed +method can offer better privacy compared with the existing +approaches. +2. Related Works +2.1. Self-supervised Vision Transformer +Self-supervised learning is gaining traction in the vision +community due to its outstanding successes in recent years, re- +ducing the gap with supervised learning (Jing and Tian, 2020; +Liu et al., 2021). A line of works on self-supervised learn- +ing leverages an approach to train the model by discriminating +the differently augmented versions of the given image, com- +monly called contrastive learning (Jaiswal et al., 2020). In a pi- +oneering work, Caron et al. (2021) proposed a contrastive learn- +ing method that allows the ViT to use contrastive learning via +teacher-student knowledge distillation, called distillation with- +out a label (DINO), eliminating the need for cumbersome neg- +ative samples required in traditional contrastive learning meth- +ods (Chen et al., 2020b; Grill et al., 2020; Zbontar et al., 2021). +In this method, the model learns the task-agnostic semantic +meaning of the image through local-to-global correspondence +with a random multi-crop strategy. When visualizing the self- +attention of the ViT model with DINO, the instance-level visual +semantics were well attended by the model, without any super- +vision for instance segmentation. This property was only ob- +servable in ViT, and better performances in both linear probes +and fine-tuning were observed in ViT compared to the CNN- +based model. +Since the ViT model is designed to be a pure patch-based +attention model, another strong self-supervised learning tech- +nique, random masked patch prediction, can be utilized for ViT- +based models, which resembles the masked language modeling +for Bidirectional Encoder Representations from Transformers +(BERT) pre-training in the field of natural language process- +ing (NLP) (Devlin et al., 2018). Bao et al. (2021) proposed a +Bidirectional Encoder representation from Image Transformers +(BEiT) that learns to predict the masked patch with discrete vi- +sual tokens obtained with the discrete tokenizer, demonstrating +that the same strategy in BERT can also be leveraged in vision +tasks. Instead of predicting discrete tokens, He et al. (2022) +proposed a rather simple strategy of learning directly from pre- +dicting pixels within the masked patches called masked au- +toencoder (MAE), by adopting computation efficient encoder- +decoder design. These random masking-based learning strate- +gies are specially designed for the ViT that process the im- +age in a patch-wise manner and are not suitable for a CNN- +based model that leverages the shared convolution kernels stride +across the image. +2.2. Federated Split Task-Agnostic Learning with Permutating +Pure ViT (p-FeSTA) +Inspired by the modular configuration of the ViT model that +can be divided into the embedder head, transformer body, and +task-specific tail (Chen et al., 2020a), the Federated Split Task- +agnostic (FeSTA) learning has been proposed to maximally ex- +ploit the distinct strengths of the FL and SL methods (Park + +S. Park and JC Ye et al. +3 +Fig. 1. Overall framework. First, the participating clients transmit the patch features encrypted with the arbitrary patch embedder and the feature-space +permutation module to the server. Then, the MS-DINO training is performed on a server-side device to make the foundation model to have a general +semantic understanding of the modality. Finally, the trained foundation model can be accessed by authorized clients to be used for various downstream +tasks. +et al., 2021), and to improve the performances of the individual +tasks with the collaboration between the participating clients +with different tasks. In more recent work, Federated Split Task- +Agnostic Learning with Permutating Pure ViT (p-FeSTA) has +been introduced to surmount the drawbacks of the FeSTA (Park +and Ye, 2022) by leveraging the permutation-invariant property +of the ViT by adopting random patch permutation to provide +better privacy as well as fewer communication overheads. The +main motivation of the p-FeSTA method is to reduce the com- +munication between the server and clients as well as enhance +the privacy with the permutation module in the feature space, +which is possible via the permutation invariant property of the +transformer encoder layers, the main component of ViT. Specif- +ically, the model can be trained with the patch features permu- +tated in the feature space with the novel feature-space permuta- +tion module, which provides privacy by keeping the malicious +attackers from faithfully reconstructing the private data from the +intermediate features. These permutated features can be safely +saved in the server-side memory and used throughout the en- +tire learning process, easing the burden to approximately half +of the FeSTA. However, this method is also not free of limita- +tions that restrict the generalized application. First, continuous +communication between the server and clients is mandatory for +model training as the labels are required for the update of the +shared transformer body. Second, multi-task learning was pos- +sible only with the clients simultaneously participating in dis- +tributed learning who want to perform relevant tasks. +3. Proposed Framework +The proposed self-supervised learning method, dubbed +Masked Sampling Distillation with No Label (MS-DINO), is +composed of three steps. +First, the patch features from all +images were extracted by an arbitrary patch embedder along +with another arbitrary position embedding, and their sequence +is randomly permutated with feature-space permutation mod- +ule. Then, the encrypted patch features are transmitted to the +server and stored. The server-side computational device uses +these encrypted features throughout the entire learning process +of the foundation model with no further communication with +the clients. Finally, the trained foundation model can be ac- +cessed by the authorized clients for the application of down- +stream tasks, offering performance superior to the fine-tuning- +only baselines. The overall framework is proposed in Fig. 1. +3.1. Foundation model learning +Fig. 2 illustrates the foundation model learning process with +the MS-DINO method. Similar to the approach proposed in +the p-FeSTA, given the image data of client x, the intermediate +patch features f are extracted from the arbitrary patch feature +extractor F, and permutated with the feature-space permutation +module permute, which can be defined as f = permute(F(x)). +Then, the encrypted features are transmitted from each client c +to the server. This process is performed at the beginning of the +learning process, and the remaining processes are performed +solely on the server-side, eliminating the need for both further + +Server +Local clients 1 ... C +D Encrypted features preparation +2) MS-DINO training +Permutation +Saved +module +Masked +Sampling +Feature set +(Large) +Teacher +: +Encrypted +Feature embed +features + Match +Distillation +: +: +Masked +Sampling +(Small) +口 +Student +: +③ Fine-tuning +Client 1 +Task- +Foundation +Segmentation +specific +Task-agnostic +model +module +Foundation +model +Client C +Task- +Foundation +Classification +specific + Upload +model +module +: Download4 +S. Park and JC Ye et al. +Fig. 2. Comparison of (A) the Distillation with No Labels (DINO) trained with federated learning (FL) and (B) the Masked Sampling Distillation with +No Labels (MS-DINO). Compared with the original DINO method where the model gets general semantic understanding via local-to-global correspon- +dence between the student and the momentum teacher, the model learns smaller number-to-larger number correspondence between the student and the +momentum teacher in the MS-DINO method, enabling the training with randomly permutated patch features. +communication and computation overheads in client-sides. In +the process of the feature extraction and encryption, we not only +extracted features from the original image x but also from an- +other randomly augmented version of the image ˆx to provide +diversity in data, resulting in two different encrypted features +data per image as below. +f = permutate(F(x)), +ˆf = permutate(F(ˆx)) +(1) +As the features are encrypted with both the patch embedder +unknown to the server along with the unknown position embed +and the permutation module in the feature space, it is infeasible +for the server to faithfully reconstruct the private data with the +transmitted feature. The detailed formulation and experiments +of the privacy protection will be discussed in Section 3.3 and +6.5. +With the encrypted features from all participating clients, +the server performs random masked sampling-based self- +supervised learning, by substituting the local-to-global of the +original DINO correspondence learning strategy with small- +to-large patches correspondence (Fig. 2). +As illustrated in +Fig. 2(A), the motivation of DINO is to teach the model the vi- +sual semantics of the image with the teacher-student knowledge +distillation, by providing the global views with larger crops to +the teacher and the multiple local views with smaller crops to +the student. We adopted and modified this key concept into +random masked sampling, by substituting the global views as +the larger number and the local views as the smaller number of +patch features sampled as illustrated in Fig. 2(B). In detail, for +the permutated feature f obtained from an image, the majority +of the permutated patch features are randomly sampled to make +the feature fl, and a relatively smaller number of the patches are +randomly sampled resulting in the feature fs. Like DINO, fl is + +■ +(A) +Client-side: Random cropping +Server-side: model averaging +Global views (Large) +Got +Averaging +embedder +Projection +teacher +Momentum +Patch +Teacher +C +: +Nc +Get +Go. +N +c=1 +Distribute +Random +Match i +:EMA +Averaging +cropping +■ +student +！！ +c +embedder +1 +Projection +Student +Patch +N +Ps +Transformer +c=1 +Gos +！ +Ges +Local views (Small) +Server-side: Random mask distillation +(B) +Random sampling (Large) +Client-side: encrypted feature set preparation +c +Projection +Permutation +Momentum +module +N +Pos +→PT +Embed +Teacher +1 +C=1 +i× Nc= +Got +Patch +: No further ! + embedder + transfer +EMA +Match i +！！ +1 +■ +..... +1 +Projection +Student +■ +Transformer +Gos +Saved Feature set +■ +Random sampling (Small)S. Park and JC Ye et al. +5 +fed to the teacher, while both fl and fs are fed to the student. +Then, the student is optimized to match the prediction of the +momentum teacher with relatively small information about the +image, in line with the local-to-global correspondence strategy +of the DINO. +Supposing the original feature fo, large sampled feature fl, +small sampled features fs, the set that contains fo, fl, and N +differently sampled fs can be defined as V = {fo, fl, f 1 +s , ...f N +s }. +Given the teacher and student models as Gθt and Gθs, the cross- +entropy loss as L, the student is trained to mimic the teacher’s +prediction with following optimization problem: +min +Gθs +� +f∈{fo,fl} +� +f ′∈V +f ′� f +L(Gθs(f ′),Gθt( f)) +(2) +During the learning process, the momentum teacher model is +updated with an exponential moving average (EMA) of the stu- +dent’s update, where λ follows a cosine scheduling: +Gθt = λGθt + (1 − λ)Gθs +(3) +The algorithm for the encrypted feature set preparation and the +foundation model learning with MS-DINO are formally pre- +sented in the algorithm 1. +Given the limited data availability for a single client, this +foundation model may merit improved generalization perfor- +mance, which is further investigated in Section 6.2 and 6.3. +3.2. Fine-tuning for Tasks of Interest +As depicted in Fig. 1, the authorized clients can access the +trained foundation model for their purposes. For instance, if +clients seek to train an OAR segmentation model for radiother- +apy planning, they can leverage the foundation model to en- +hance the generalization performance, since it holds the general +ability to attend to visual semantics within the image. +Specifically, for client, c, supposing the pre-trained foun- +dation model backbone as G and task-specific layer like de- +coder as Hc, data and label for fine-tuning as xc and yc, task- +specific loss function as Lc, the following optimization problem +is solved for the fine-tuning: +min +G,H +� +i=1 +Lc(Hc(G(xc)), yc) +(4) +3.3. Protecting Privacy with Feature-space Permutation Mod- +ule +Protecting privacy by randomly permutating patch features +in the feature space has been investigated in the previous work +(Park and Ye, 2022). As the embedded features from the clients +are saved on the server-side, privacy concerns may arise if the +“honest-but-curious” server or the malicious attacker who has +hijacked the features during the transmission may try to revert +the features into the original data. +Specifically, suppose that the encrypted features with permu- +tation are hijacked during the communication and there are suf- +ficient public data in the same domain available for the attacker. +Then, the attacker needs to solve two problems simultaneously: +(1) training the attacker-side feature extract that can embed the +Algorithm 1 Proposed MS-DINO algorithm +/* Run on Client c */ +1 Function ClientMain: +2 +Client initialize with arbitrary feature embedder F +3 +for data x ∈ {1, 2, . . . X} +4 +{x, x′} ← Augment(x) +5 +f = permute(F(x)) // original feature +6 +f ′ = permute(F(x′)) // augmented feature +7 +feature set f = {{ f1, f ′ +1}, {f2, f ′ +2} . . . {fX, f ′ +X}} ← {f, f ′} +8 +return f +/* Run on Main Server */ +9 Function ServerMain: +10 +Server initializes student Gθs and teacher Gθt +for clients c ∈ {1, 2, . . .C} do in parallel +11 +fc ← ClientMain(c) +Memory = {f1, f2, . . . fc} ← fc +// Save all features fc in Memory +12 +for epoch e ∈ Memory +13 +for features f, f ′ ∈ {1, 2, . . . E} +// Run MS-DINO learning in server-side device +14 +LDINO = � +f∈{fo,fl} +� +f ′∈V +f ′� f +L(Gθs( f ′),Gθt( f)) +15 +θs ← θs − η +N +∂LDINO +∂θs +// update student model +16 +θt = λθt+(1−λ)θs // EMA update of teacher model +17 +for clients c ∈ {1, 2, . . .C} do in parallel +18 +Distribute Gθs to authorized client +// Distribute foundation model to clients +image into the feature space the same as that of the unpermu- +tated hijacked feature, and (2) training the jigsaw solver to un- +permutate the encrypted features into unpermutated ones in the +feature space of hijacked features, not in the image space. +More specifically, we denote the permutated and the origi- +nal features embedded by the attacker-side model ˆF as ˜fpub and +fpub, respectively, the number of attacker-side images (e.g. pub- +licly available CT or CXR images) and the hijacked encrypted +features as m, and the permutated features hijacked by the at- +tacker during communication as ˜fpriv which is embedded by an +arbitrary client-side feature embedder F. +Then, the attacker-side model ˆF, discriminator D, and the de- +coder G can be trained by optimizing the following two learning +objectives: +min +ˆF max +D +m +� +i=1 +n +� +j=1 +[log(1 − D(J( ˜f (i) +priv))) + log D(J( ˜f (j) +pub))] +(5) +min +G +m +� +i=1 +Ldecoder(G(J( ˜f (i) +pub)), x(i) +pub) +(6) +where Ldecoder denotes reconstruction loss for decoder. Mean- +while, the second optimization problem for the jigsaw solve J + +6 +S. Park and JC Ye et al. +Table 1. Data for foundation model learning +Modality +Client #1 +Client #2 +Client #3 +Client #4 +CT +7,993 +8,063 +8,182 +7,973 +CXR +8,422 +8,424 +8,422 +8,422 +can be formulated as follow: +min +J +m +� +i=1 +Ljigsaw(J( ˜f (i) +pub), f (i) +pub) +(7) +where Ljigsaw denotes similarity loss in the feature space. +Note that to simultaneously solve the first two equations, +Eq. (5) and Eq. (6), the exact jigsaw solver should be unrav- +eled, which can be obtained if the Eq. (7) is successfully solved. +However, the jigsaw solver should be trained and utilized in the +same feature space as the hijacked encrypted feature, which ne- +cessitates knowing the correct solution for the attacker model ˆF +to embed the same feature space, and this is conversely the tar- +get of the optimization problem Eq. (5). Combined, optimiza- +tion of Eq. (5), Eq. (6), and Eq. (7) requires to already have +each other’s solutions, indicating that the problems are under- +determined and practically hard to solve. +Section 6.5 provides experimental results to support our as- +sertion. +4. Datasets +We investigated the applicability of the proposed method in +two different domain data and two different tasks, OAR seg- +mentation with CT for radiotherapy planning and tuberculosis +diagnosis with CXR. +4.1. Datasets for foundation model learning +For the foundation model learning, +two open-sourced +datasets, CT scans of the head and neck cancer patients from +The Cancer Imaging Archive (TCIA) Head and Neck Squa- +mous Cell Cancer (HNSCC) data (Grossberg A et al., 2020; +Grossberg et al., 2018; Elhalawani et al., 2017; Clark et al., +2013) and the CXR images from the Radiological Society of +North America (RSNA) pneumonia detection challenge data +(of North America, 2018) were used. +To simulate the application in clinical collaboration, we sim- +ulated the collaboration of four clients with different data di- +visions, emulating the collaboration between four institutions. +We divided the dataset into several subsets, defining each subset +as the data of each client as shown in Table 1. Among the CT +scans of a total of 619 patients from the HNSCC data, we used +the CT scans of 200 patients, 7,993 CT slices from 50 patients +(client #1), 8,063 from 50 patients (client #2), 8,182 from 50 +patients (client #3) and 7,973 from 50 patients (client #4) were +assigned for each client. Similarly, among a total of 33,690 +images from the RSNA pneumonia detection challenge dataset, +8,422 (client #1), 8,424 (client #2), 8,422 (client #3), and 8,422 +(client #4) images were assigned for each client (Table 1. +Table 2. Data for fine-tuning downstream task +Task +Description +Fine-tune +Test +Full +Limited +CT segmentation +OARs +2,911 +608 +521 +CXR classification +Normal +4,127 +327 +92 +Tuberculosis +1,135 +335 +46 +4.2. Datasets for fine-tuning downstream task +In practical implementation, the trained foundation model +will be accessed and fine-tuned for the client’s task of interest. +Therefore, we used the datasets containing both data and la- +bels for the downstream tasks, namely OAR segmentation with +CT and tuberculosis diagnosis with CXR. Given the problem +of limited data and label availability frequently with a single +client, we performed the experiments in two settings: data- +abundant (full) and data-insufficient (limited) settings. +For the downstream OAR segmentation task with CT, the +Medical Image Computing and Computer Assisted Intervention +(MICCAI) 2015 head and neck challenge dataset (Computing +and Interventions, 2015) was used as the fine-tuning dataset for +organ-at-risk segmentation. As the dataset contains a total of +2,911 CT slices and labels from 38 patients, all patients’ data +were utilized for the data-abundant setting, while the 8 patients’ +data containing 608 CT slices were used as the data-insufficient +setting. For the downstream tuberculosis diagnosis task with +CXR, 1,135 tuberculosis and 4,127 normal cases from two data +sources, the Montgomery County (MC) dataset (Jaeger et al., +2014) and TBX 11K dataset (Liu et al., 2020) were collected +for data-abundant setting. For data-insufficient setting, only the +MC dataset containing 335 tuberculosis and 327 normal cases +was used (Table 2). +4.3. Datasets for evaluation +To evaluate the benefit of the foundation model, we used two +datasets for each task. For evaluation of the segmentation per- +formance, we used the CT and region-of-interest (ROI) data +collected and delineated by the board-certified radiation oncol- +ogists from the local institution (Gangnam Severance Hospital) +to externally validate the performance of the developed model. +From 2007 to 2021, data from 44 head and neck cancer patients +were collected, and seven patient data containing all ROIs were +used for the evaluation. For evaluation of the classification per- +formance, we used a publicly accessible dataset (India tuber- +culosis dataset) (radder), which can be regarded as the external +validation that is collected from the data source different from +the training and fine-tuning. +5. Implementation Details +5.1. Details of Model development +The CT images underwent preprocessing by cropping the +center area of 224 × 224 from a total size of 512 × 512. The +upper and lower window of the Hounsfield unit (HU) were set +to -200 and 200, respectively, in consideration of the ranges + +S. Park and JC Ye et al. +7 +of HU of the OARs of interest. As the data were from differ- +ent sources, the pixel spacing was adjusted to match between +datasets. The CXR images were preprocessed with histogram +equalization Gaussian blurring and normalization, and finally +resized to 224 × 224. +As the arbitrary feature embedder, the patch embedder of the +DINO model pre-trained on ImageNet was used. The embed- +ded features were permutated with the feature-space permuta- +tion module proposed in Park and Ye (2022), yielding the en- +crypted features for foundation model learning. For the body +part of the Transformer, the transformer encoder of ViT small +having 6 heads, 12 layers, and a patch size of 8 was used, which +was initialized with the DINO self-supervised learning weights +on the ImageNet. +We used the same-sized teacher and student models, and +the multi-masked sampling strategy was adopted instead of the +multi-cropping strategy following the original implementation +of Caron et al. (2021). As a global view, the smaller number +of patch features were masked, resulting in the ratio of 0.9 - +1.0 and 0.8 - 1.0 being sampled for CT and CXR, respectively. +For multiple local views, the larger number of patch features +were masked, sampling with a ratio of 0.3 - 0.5 and 0.3 - 0.4 +for CT and CT. The same configuration with the original work +of DINO was adopted for comparison, with crop sizes of 0.4 +- 1.0 and 0.05 - 0.4 for global and local views, respectively. +Considering the relatively small dataset size and complexity of +the medical image, we reduced the dimensionality of the DINO +head output from 65,536 to 8,192. +For the foundation model learning with the proposed MS- +DINO, Adam W optimizer (Loshchilov and Hutter, 2017) was +used along with a cosine decay scheduler with a maximum +learning rate of 0.00004 with a batch size of 8. The model was +trained in the server-side device for 5 epochs. For training the +foundation model with the DINO method via FL, the same op- +timizer, scheduler, and learning rate was used with a batch size +of 4 per client, and the model was trained for 10,500 federated +rounds in the client-side devices to match the total number of +updates to MS-DINO learning in the server-side. For FL, both +the student and teacher model parameters were averaged every +100 rounds. +For the downstream OAR segmentation task, the foundation +ViT model and the UperNet (Xiao et al., 2018) decoder were +used as the encoder and the decoder, respectively, following the +implementation in Bao et al. (2021); He et al. (2022). As the +UNet for comparison, standard UNet (Ronneberger et al., 2015) +architecture with four downsample and upsample layers and 64 +convolutional filters were utilized. We designed the decoder +part of the network as the multi-class segmentation model, to +delineate the ROIs (brainstem, optic chiasm, mandible, optic +nerves, parotid glands, and submandibular glands) simultane- +ously. Following the previous work on MICCAI 2015 head and +neck segmentation challenge dataset, we used the combined fo- +cal loss (Lin et al., 2017) and Dice loss for the optimization. +The SGD optimizer was used for the segmentation model with +a learning rate of 0.01 and a batch size of 10. +For the downstream classification task, a simple linear layer +was added as the classification head. As we implemented the tu- +berculosis diagnosis task as the binary classification, the model +optimized with the BCE loss. Adam W optimizer was used to +fine-tune the model for the classification task, with a learning +rate of 0.0001 and a batch size of 16. +All experiments including data processing, foundation model +learning, fine-tuning, and evaluation were performed with +Python version 3.9 and Pytorch library 1.10 on NVIDIA +GeForce RTX 3090, and the experiments on distributed learning +were conducted with the FLOWER framework (Beutel et al., +2020). +5.2. Details of evaluation +To evaluate the segmentation performance, the Dice similar- +ity coefficient is used to quantify the overlap between the pre- +dicted segmentation mask and the ground truth label. To eval- +uate the classification performance, the area under the receiver +operating characteristics curve (AUC) was calculated to com- +pare the model performance, and the sensitivity, specificity, and +accuracy were reported separately to provide detailed perfor- +mances, after adjusting the thresholds to meet the pre-defined +sensitivity value ≥ 80%, if possible. +5.3. Simulation for Feature Inversion Attack +We evaluated whether the permutation model is effective to +prevent privacy attacks from feature hijacking, supposing the +optimal configuration for the malicious attacker. +Following the previous work on the encryption with random +patch permutation (Park and Ye, 2022), it is supposed that the +attacker hijacked all encrypted features transmitted from all +clients and knows the exact permutation ratio and patch size, +the architecture of the unknown patch embedder and the dimen- +sionality of position embedding. Specifically, the same archi- +tecture as the original patch embedder was used as the attacker- +side feature embedder, and the three-layered discriminator and +the four-layered generator from DCGAN (Radford et al., 2015) +were employed as the discriminator and decoder. As a jigsaw +solver to solve the random permutation in the feature space, the +transformer with 12 encoder layers and 12 attention heads was +used. The discriminator was optimized with the modified ver- +sion of GAN loss (Goodfellow et al., 2014) as formulated in +Eqs. (5) and (6), the combined L1 and L2 losses were used as +the learning objective for the decoder, and the L1 loss was used +as the objective for the jigsaw solver. +In addition, we supposed that the attacker possesses a suffi- +cient amount of data in the same domain, like the CT or CXR +available at the public repository, to train the attacker-side net- +works. As the attacker-side public data, 6,189 CT slices from +the TCIA HNSCC-3DCT-RT data (Clark et al., 2013; Bejarano +et al., 2018, 2019) and 9,577 normal CXRs data from the CheX- +pert data (Irvin et al., 2019) were utilized. +The model was trained for 5 epochs with a batch size of 1 +and a learning rate of 0.001. +6. Experimental results +6.1. Attention Changes with Foundation Model Training +Fig. 3 depict the changes in the last layer of multi-head at- +tentions within the ViT model. Before the foundation model + +8 +S. Park and JC Ye et al. +Fig. 3. Visualization of multi-head attention for (A) CT and (B) CXR. Compared with the attentions obtained with the pre-trained ImageNet weights, the +different attention heads got refined and tend to concentrate on different semantic components after MS-DINO learning. +Table 3. Segmentation results after fine-tuning in data-limited setting +Method +Overall +Brainsetm +Chiasm +Mandible +Optic n Lt. +Optic n Rt. +Parotid Lt. +Parotid Rt. +SMG Lt. +SMG Rt. +Unet +0.083 +0.182 +0.000 +0.359 +0.000 +0.000 +0.067 +0.134 +0.002 +0.000 +(0.036) +(0.162) +(0.000) +(0.090) +(0.000) +(0.000) +(0.065) +(0.168) +(0.004) +(0.000) +ViT +0.437 +0.628 +0.085 +0.602 +0.213 +0.060 +0.623 +0.607 +0.538 +0.577 +(0.002) +(0.006) +(0.010) +(0.006) +(0.018) +(0.007) +(0.002) +(0.004) +(0.005) +(0.006) +ViT (DC DINO) +0.557 +0.741 +0.250 +0.734 +0.338 +0.343 +0.721 +0.727 +0.595 +0.567 +(0.003) +(0.024) +(0.020) +(0.011) +(0.016) +(0.010) +(0.006) +(0.021) +(0.008) +(0.042) +ViT (FL DINO) +0.538 +0.732 +0.202 +0.716 +0.327 +0.286 +0.689 +0.716 +0.567 +0.605 +(0.013) +(0.018) +(0.020) +(0.025) +(0.014) +(0.036) +(0.030) +(0.025) +(0.030) +(0.035) +ViT (Proposed) +0.556 +0.737 +0.232 +0.740 +0.330 +0.320 +0.725 +0.726 +0.589 +0.606 +(0.010) +(0.025) +(0.006) +(0.006) +(0.004) +(0.030) +(0.001) +(0.012) +(0.043) +(0.028) +Values are presented in the mean (standard deviation) of three repeats with different seeds. +DC DINO, data-centralized DINO; FL DINO, federated learning DINO +Bold face and underline denote the best and the second best performances. +training, the attentions of the ViT model with the ImageNet +pre-trained with the original DINO disperse throughout the im- +age (upper row), and the differences in attention between the +individual heads were not prominent, implying that most at- +tention heads process the image less effectively. After learn- +ing with the MS-DINO method, the different heads tend to at- +tention to different semantic components (lower row), which +can offer improved model performances with diverse patterns +of self-attention. +6.2. Performance Comparison for Downstream CT Task +The comparison results for the CT task between the meth- +ods are provided in Table 3 and 4. The model fine-tuned from +the foundation models showed better performance than those +obtained with training from scratch. In particular, the model +trained with the proposed MS-DINO showed a performance +close to that of the DINO trained in a data-centralized man- +ner, while outperforming the DINO trained with FL. The per- +formance enhancement with the foundation model learning is + +Head 1 +Head 2 +Head 3 +Head 4 +Head 5 +Head 6 +(A) +After MS-DINO learning +(B) +After MS-DINO learningS. Park and JC Ye et al. +9 +Table 4. Segmentation results after fine-tuning in data-abundant setting +Method +Overall +Brainsetm +Chiasm +Mandible +Optic n Lt. +Optic n Rt. +Parotid Lt. +Parotid Rt. +SMG Lt. +SMG Rt. +Unet +0.625 +0.758 +0.273 +0.800 +0.431 +0.433 +0.739 +0.776 +0.717 +0.703 +(0.014) +(0.013) +(0.012) +(0.007) +(0.029) +(0.072) +(0.013) +(0.004) +(0.002) +(0.024) +ViT +0.584 +0.718 +0.243 +0.761 +0.370 +0.359 +0.737 +0.742 +0.697 +0.627 +(0.002) +(0.002) +(0.007) +(0.004) +(0.000) +(0.012) +(0.003) +(0.002) +(0.004) +(0.012) +ViT (DC DINO) +0.658 +0.764 +0.320 +0.814 +0.498 +0.500 +0.786 +0.819 +0.708 +0.714 +(0.006) +(0.015) +(0.011) +(0.013) +(0.003) +(0.004) +(0.004) +(0.002) +(0.031) +(0.005) +ViT (FL DINO) +0.653 +0.775 +0.278 +0.810 +0.485 +0.443 +0.778 +0.814 +0.730 +0.706 +(0.005) +(0.010) +(0.019) +(0.022) +(0.031) +(0.044) +(0.005) +(0.003) +(0.003) +(0.029) +ViT (Proposed) +0.656 +0.777 +0.313 +0.818 +0.494 +0.460 +0.781 +0.817 +0.740 +0.699 +(0.002) +(0.014) +(0.007) +(0.004) +(0.014) +(0.012) +(0.006) +(0.005) +(0.009) +(0.025) +Values are presented in the mean (standard deviation) of three repeats with different seeds. +DC DINO, data-centralized DINO; FL DINO, federated learning DINO +Bold face and underline denote the best and the second best performances. +Table 5. Classification results after fine-tuning in data-limited setting +Method +AUC +Sensitivity +Specificity +Accuracy +DenseNet-121 +0.514 +0.442 +0.588 +0.519 +(0.006) +(0.066) +(0.089) +(0.042) +ViT +0.590 +0.558 +0.598 +0.585 +(0.004) +(0.013) +(0.011) +(0.004) +ViT (DC DINO) +0.824 +0.739 +0.757 +0.751 +(0.016) +(0.022) +(0.045) +(0.025) +ViT (FL DINO) +0.695 +0.645 +0.609 +0.616 +(0.057) +(0.066) +(0.058) +(0.029) +ViT (Proposed) +0.844 +0.790 +0.772 +0.778 +(0.007) +(0.063) +(0.011) +(0.022) +Values are presented in the mean (standard deviation) of three repeats with different seeds. +DC DINO, data-centralized DINO; FL DINO, federated learning DINO +Bold face and underline denote the best and the second best performances. +Table 6. Classification results after fine-tuning in data-abundant setting +Method +AUC +Sensitivity +Specificity +Accuracy +DenseNet-121 +0.652 +0.688 +0.583 +0.618 +(0.033) +(0.090) +(0.094) +(0.041) +ViT +0.493 +0.478 +0.543 +0.522 +(0.023) +(0.210) +(0.171) +(0.044) +ViT (DC DINO) +0.837 +0.812 +0.728 +0.756 +(0.023) +(0.013) +(0.050) +(0.033) +ViT (FL DINO) +0.773 +0.681 +0.746 +0.725 +(0.022) +(0.126) +(0.071) +(0.022) +ViT (Proposed) +0.854 +0.783 +0.790 +0.787 +(0.008) +(0.058) +(0.006) +(0.022) +Values are presented in the mean (standard deviation) of three repeats with different seeds. +DC DINO, data-centralized DINO; FL DINO, federated learning DINO +Bold face and underline denote the best and the second best performances. +more prominent in the data-limited than the data-abundant set- +tings. +Fig. 4 depicts the qualitative comparison between methods. +The model fine-tuned from the foundation model with the pro- +posed MS-DINO offered overall more accurate predictions for +the areas of OARs, which is comparable to those with data- +centralized DINO and outperforms those of fine-tuning only +baseline. +6.3. Performance comparison for downstream CXR task +Table 5 and 6 show the comparison results for the CXR task +between the methods. Compared with the fine-tune-only base- +lines which were obtained by training the model from scratch, +those with the foundation model offered generally better perfor- +mance. Furthermore, among those with foundation models, the +model trained with the proposed MS-DINO method provided +the best performance, even outperforming the DINO trained in +a data-centralized manner as well as the DINO obtained in FL. +The benefit of the proposed method was remarkable in the data- +limited setting, providing a diagnostic performance close to that +of the data-abundant setting with only several hundred images +for each class. +6.4. Communication costs +When defining the number of data as D, total training rounds +as R, the round between aggregation and distribution as r, the +model parameter as P, and the size of encrypted feature for each +data as F, the total communication costs for foundation model +learning T for the FL and the MS-DINO can be formulated as +below: +TFL = 4R +r × P, +(8) +TMS-DINO = D × F +(9) +where the constant 4 is multiplied to account for the parameter +transmission of both the teacher and student models, and the +both-way transmissions between server and client for FL. As +the MS-DINO method does not require continuous communi- +cation between the server and the clients, the only communica- +tion between the server and clients occurs at the beginning of +the learning. +With this formulation, the numerical comparison results are +provided in Table 7. Compared to the training of DINO with +FL, the communication cost of the proposed MS-DINO is about +one-third, which can be further advantageous as the number of +total rounds increases. +6.5. Experiments on Privacy protection +The qualitative and quantitative analysis results of recon- +struction from the privacy attack are provided in Table 8 and +Fig. 5. + +10 +S. Park and JC Ye et al. +Fig. 4. Qualitative comparisons of the segmentation results. The model fine-tuned from the foundation model with the MS-DINO method shows results +comparable to those with the data-centralized DINO method and outperforms the fine-tune only baselines. The results are obtained from fine-tuning in a +data-abundant setting. +Table 7. Comparison of communication overheads between methods +Method +Continuous communication +Communication cost +Feature transmission +Model averaging +Federated learning + +- +12,229.7 M +Proposed + +4,866.9 M +- +Given the optimal configuration for the attacker and the set- +ting that the feature-space permutation module is not employed, +both the private CT and CXR data can be reconstructed to the +amount that the privacy like shape, gender, anatomic location, +and disease status of the subject can be inferred from the re- +construction results of hijacked feature to some degree. This +implies that the approximate solution of the unknown feature +embedder can be obtained by the attacker. However, when the +extracted features are randomly permutated in the feature space, +it was nearly impossible to reconstruct the data to the amount +that the privacy can be inferred (Fig. 5). +Combined, these results confirm our claim that simultane- +ously solving two optimization problems that need each other’s +solution can be considered to be an underdetermined problem, +and thus practically difficult to solve. +6.6. Ablation studies +Table 9 shows the ablation studies to verify the roles of the +components of the proposed MS-DINO. As the random masked +sampling and different augmentation for each image were uti- +lized in the proposed method, we ablated the components in or- +der. The ablation studies are performed in the data-limited set- +ting, as the benefit of the proposed method is more pronounced +in this setting. +6.6.1. Random masked sampling +Ablating the random masked sampling means that the model +is trained by the knowledge distillation between teacher and stu- +dent, with the two views of the image, the original and the aug- +mented versions of the same image, but without any random +sub-sampling. As shown in the Table 9, the performances were +suboptimal for both CT segmentation and CXR classification, +implying that this component is indispensable for the best per- +formance. +6.6.2. Augmentation +Next, we ablated the augmentation to offer two different +views of the image to evaluate whether this component is nec- +essary for performance. Compared with the proposed method +leveraging an augmented version of the given image, apply- +ing random sampling solely for the original image show lower +performance in both CT segmentation and CXR classification, + +ViT +ViT (DC DINO) +UNet +ViT (proposed) +Label +ViT (FL DINO)S. Park and JC Ye et al. +11 +Fig. 5. Reconstruction results of the privacy attack for (A) CT and (B) CXR. Without the feature-space permutation module, information (level of CT slice, +presence of disease, etc.) can be inferred from the reconstructed results, while it was infeasible when the feature-space permutation module was utilized. +Table 8. Quantitative comparison for privacy attack +Method +CT +CXR +MSE +SSIM +MSE +SSIM +No permutation +0.237 (0.005) +0.694 (0.008) +0.161 (0.051) +0.350 (0.014) +Permutation +0.272 (0.009) +0.479 (0.038) +0.267 (0.006) +0.018 (0.007) +Values are presented in the mean (standard deviation) of three repeats with different seeds. +MSE, mean squared error; SSIM, structural similarity index measure. +Table 9. Ablation studies +CT Segmentation +CXR classification +Method +Overall DSC +AUC +Proposed +0.556 (0.010) +0.844 (0.007) +No random masking +0.544 (0.003) +0.817 (0.045) +No augmentation +0.548 (0.004) +0.833 (0.019) +Values are presented in the mean (standard deviation) of three repeats with different seeds. +OAR, organ at risk; DSC, dice similairty coefficient; AUC, area under the curve. +which suggests that offering different views of an image en- +hances the performance regardless of the modalities and the +tasks. +7. Discussion and Conclusion +Despite surprising performance of the DL-based vision mod- +els, the data-driven learning paradigm of the DL has been pos- +ing practical hurdles in the development of the AI model for +healthcare research where the data utilized for training may +contain sensitive privacy of individuals (Perone and Cohen- +Adad, 2019). In addition, the label dependency throws another +challenge since the labels annotated by the medical experts are +usually expensive and hard to be obtained (Willemink et al., +2020). +Distributed learning methods have been introduced to tackle +these problems by enabling model training without directly +sharing private data, and self-supervised learning methods have +been investigated to alleviate the label dependency of the DL +model. +Several works have investigated the combination of +these two techniques, but there remain problems in those ap- +proaches reporting suboptimal performances compared with +data-centralized counterparts (Makhija et al., 2022; Zhuang +et al., 2022; Shi et al., 2021). Furthermore, the innate proper- +ties of model averaging during FL enforce continuous commu- +nication between server and clients for model averaging as well +as high communication overhead, and pose the privacy threats +that can be caused by the model inversion attack by malicious +attackers (Geiping et al., 2020; Lyu et al., 2020b; Hatamizadeh +et al., 2022). +To alleviate these issues, we have introduced a novel self- +supervised learning method, dubbed MS-DINO, that can be +used in a distributed manner without continuous communi- +cation but still attains a performance comparable to or even +outperforming the self-supervised learning method performed +in a data-centralized setting. Inspired by the previous works +that reported the intriguing properties of ViT (Naseer et al., +2021; Park and Kim, 2022), we utilized two key concepts, the +permutation-invariant property of the transformer layer and se- +mantic learning via local-to-global correspondence using the +teacher-student distillation, to devise our method. Specifically, +we replaced the multi-crop strategy of cropping multiple small +local crops and large global crops with the random masked +sampling strategy, which randomly samples large and multi- +ple small portions of the patch features. This random masking +strategy is also consistent with pioneering self-supervised learn- +ing approaches that conjugate masked image modeling with the +ViT models in a patch-wise manner (Bao et al., 2021; He et al., +2022; Xie et al., 2022). Furthermore, we amalgamated the pre- +viously proposed feature-space permutation module (Park and +Ye, 2022) to enhance privacy and to improve communication +efficiency without defacing the performances. By enabling to +save the encrypted patch features with the feature-space per- +mutation module, the foundation model learning based on the +masked random sampling of patch features can be performed +solely on the server-side device while at the same time protect- + +(A) +(B) +Original +Permutation +Original +No permutation +No permutation +Permutation12 +S. Park and JC Ye et al. +ing the privacy of the participating subjects. We performed ex- +tensive experiments on two imaging modalities and two tasks in +both data-abundant and data-limited settings to clarify the ben- +efit of the method in general application. In addition, we also +performed experiments on the privacy attack to verify the ef- +ficacy of the feature-space permutation module, demonstrating +that enhanced privacy can be offered with the proposed method +equipped with this module. +Our study is not free of limitations. First, as the proposed +MS-DINO method is built upon the intrinsic permutation- +invariant property of the Transformer, the model without this +property, for instance, CNN-based architecture, can not be uti- +lized. +Second, although we have demonstrated that the pri- +vacy attack through feature-space hijacking is infeasible for our +method, other types of malicious attacks including model poi- +soning and data poisoning (Lyu et al., 2020b; Tolpegin et al., +2020; Bagdasaryan et al., 2020) or privacy threats like the mem- +bership inference (Gupta et al., 2021; Shokri et al., 2017; Nasr +et al., 2019) were not investigated, which is beyond the scope of +this work. The existing methods for privacy protection can be +adopted in addition to the proposed method to enhance privacy +(Lyu et al., 2020a; Zhao et al., 2019; Li et al., 2020a). Third, +other considerations for the practical implementation like the +data skewness between clients were not investigated (Gao et al., +2020). However, as the permutated features from all clients are +saved and used by the server during the foundation model learn- +ing, the vulnerability to data skewness is expected to be lower +compared with the existing approaches. Finally, our learning +method is only applicable to the ViT-based models, since it +is based on the model-specific learning method, the random +masked sampling of the shuffled patch features. In addition, as +the ViT model processes the image by dividing it into multiple +patches and by calculating the attention scores, the fine-grained +segmentation of the thin and small organ (e.g. optic chiasm) is +difficult for the ViT-based model, as shown in the example in +the third row of Fig. 4. +Nevertheless, given the limited data availability and the +importance of privacy in health research, our method has +great promise to build a general foundation model in a +communication-efficient and privacy-protecting way. Since the +model obtained with the proposed MS-DINO has a general un- +derstanding of visual semantics, it can be used as the task- +agnostic foundation model to enhance the performances of the +downstream task, suggesting its widespread applicability in +healthcare research. +8. Acknowledgement +This research was supported in part by a grant of the MD- +PhD/Medical Scientist Training Program through the Korea +Health Industry Development Institute (KHIDI), funded by the +Ministry of Health & Welfare, Republic of Korea, by the Na- +tional Research Foundation of Korea (NRF) grant funded by +the Korean Government Ministry of Science and ICT (NRF- +2020R1A2C1102559), by a faculty research grant of Yon- +sei University College of Medicine (6-2019-0071), by the +National Research Foundation of Korea under Grant NRF- +2020R1A2B5B03001980, and by the KAIST Key Research In- +stitute (Interdisciplinary Research Group) Project. +Appendix A. Ethic Committee Approval +The CT and the ROI data collected for this study were eth- +ically approved by the Institutional Review Board (IRB) of +Gangnam Severance Hospital (IRB number: 3-2020-0289), and +the requirement for informed consent was waived due to the ret- +rospective nature of this study. +References +Bagdasaryan, E., Veit, A., Hua, Y., Estrin, D., Shmatikov, V., 2020. How to +backdoor federated learning, in: International Conference on Artificial In- +telligence and Statistics, PMLR. pp. 2938–2948. +Bao, H., Dong, L., Wei, F., 2021. Beit: Bert pre-training of image transformers. +arXiv preprint arXiv:2106.08254 . +Bejarano, T., De Ornelas Couto, M., Mihaylov, I.B., 2018. +Head-and-neck +squamous cell carcinoma patients with ct taken during pre-treatment, mid- +treatment, and post-treatment dataset. The Cancer Imaging Archive 10, K9. +Bejarano, T., De Ornelas-Couto, M., Mihaylov, I.B., 2019. 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Pdgan: A novel +poisoning defense method in federated learning using generative adversarial +network, in: International Conference on Algorithms and Architectures for +Parallel Processing, Springer. pp. 595–609. +Zhuang, W., Wen, Y., Zhang, S., 2022. +Divergence-aware federated self- +supervised learning. arXiv preprint arXiv:2204.04385 . + diff --git a/ftA0T4oBgHgl3EQfHv83/content/tmp_files/load_file.txt b/ftA0T4oBgHgl3EQfHv83/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6d41a7c3fceddf5041a26ead641d8885269f6a4e --- /dev/null +++ b/ftA0T4oBgHgl3EQfHv83/content/tmp_files/load_file.txt @@ -0,0 +1,1517 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf,len=1516 +page_content='MS-DINO: Efficient Distributed Training of Vision Transformer Foundation Model in Medical Domain through Masked Sampling Sangjoon Parka,' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Daejeon,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' South Korea bDepartment of Radiation Oncology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Gangnam Severance Hospital,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Seoul,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' South Korea cKim Jaechul Graduate School of AI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Korea Advanced Institute of Science and Technology (KAIST),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Daejeon,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' South Korea A R T I C L E I N F O 2000 MSC: 41A05,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' 41A10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' 65D05,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' 65D17 Keywords: Distributed learning,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Self-su- pervised learning,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Vision Transformer,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Random permutation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Computed To- mography,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Chest X-ray A B S T R A C T In spite of the recent success of deep learning in the medical domain,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' the problem of data scarcity in the medical domain gets aggravated due to privacy and data ownership issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Distributed learning approaches including federated learning have been studied to alleviate the problems, but they suffer from cumbersome communication overheads and weakness in privacy protection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' To address this, here we propose a self-supervised masked sampling distillation method for vision transformer that can be performed with- out continuous communication but still enhance privacy using a vision transformer- specific encryption method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' The effectiveness of our method is demonstrated with ex- tensive experiments on two medical domain data and two different downstream tasks, showing superior performances than those obtained with the existing distributed learn- ing strategy as well as the fine-tuning only baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' As the self-supervised model built with the proposed method is capable of having a general semantic understanding of the modality, we demonstrate its potential as a task-agnostic foundation model for various medical tasks, widening the applicability in the medical domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' © 2023 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Introduction The success of deep learning (DL) has made it de facto stan- dard in developing artificial intelligence (AI) powered medical tools (Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Ting et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Giger, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Pesapane et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=', 2018), but the data and label-driven nature of the deep learning requires imperative collaboration between multiple in- stitutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' However, strict legal or institutional regulations often forbid the free sharing of data derived from patients due to pri- vacy concerns (Edemekong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Hoofnagle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Usually, the de-identified data can be shared only between the collaborators under formal consents, which can be one of the most cumbersome obstacles in AI research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Distributed learn- ing methods like federated learning (FL) (Koneˇcn`y et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=', 2016) and split learning (SL) (Vepakomma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=', 2018) have been in- troduced to cope with this problem by alleviating the data gov- ernance and ownership issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' In the FL, the goal is to obtain a model on the server-side while training data remain unmoved over the edge devices of ∗Co-corresonding author: Tel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' : +82-2-2019-3152;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' fax: +82-2-2019-4855;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' ∗∗Co-corresonding author: Tel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' : +82-42-350-4320;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' fax: +82-42-350-4310;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' e-mail: junwon@yuhs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='ac, jong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='ye@kaist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='kr (Jong Chul Ye) multiple clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' In detail, the central server distributes the global model to each client, and the clients perform training iterations with their data in parallel, to return the results of lo- cal computations to the server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' The server then aggregates and averages local updates, and distributes again the updated global model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' This process is repeatedly performed until the model converges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' While the FL has resolved the issues of data shar- ing, it does not fully guarantee privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Specifically, data can leak by reconstructing the private data used in training with the inversion attack which uses the stolen gradients of local model updates from insecure aggregation (Geiping et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' In addition, it imposes heavy computational loads on the edge de- vices of the clients as the most of computations and updates of the model are performed in client-side devices (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=', 2020b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Mammen, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' In the SL, on the other hand, the entire model is split into several sub-networks trained separately on the server-side and client-side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Specifically, the first sub-network performs a for- ward pass on the client-side device and sends the smashed fea- tures to the second sub-network located on the server-side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' The server then performs forward propagation with these features to pass back the subsequent features to the third sub-network on the client side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' The third sub-network on the client side can arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='02064v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='CV] 5 Jan 2023 2 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Park and JC Ye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' yield the outcome of the model, and the loss can be calculated with the client-side label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Backpropagation through split sub- networks on clients and server-sides is performed in the exact opposite manner to the forward pass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Allocating small-sized sub-networks to train on the client-side device reduces the com- putational load of local clients that usually lack resources in practical implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' In addition, the SL offers model pri- vacy by inserting black-box to clients and server, preventing both from having access to the full network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' However, there remain privacy concerns like the hijacking of transmitted fea- tures to be inverted to the original data (Gawron and Stubbings, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Besides these limitations, these methods may impose sub- stantial communication overheads in practical implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' For instance, FL requires the entire model, which is usually large-sized, to be aggregated and distributed between the server and clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Meanwhile, the feature and gradients from the split subnetwork should be continuously interchanged in a relay- based manner in the SL, enforcing the clients to be connected during the entire training process (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=', 2020b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Recently, a pure attention-based DL model named Vision Transformer (ViT) (Dosovitskiy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=', 2020) has been intro- duced to the vision community and has become a core compo- nent of vision research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' The ViT has several desirable prop- erties thanks to its simple but powerful attention architecture, and recent efforts to understand the properties of the ViT have found that it has more shape-biased nature like human and is less susceptible to perturbation like occlusion or random patch permutation (Naseer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' In addition, several recent works on self-supervised learning have reported that the ViT- based model can benefit more from the various self-supervised learning schemes like learning semantic meaning with knowl- edge distillation (Caron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=', 2021) or masked patch prediction (Bao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Inspired by the properties of ViT, here we present a novel distributed self-supervised learning strategy to build a founda- tion model that does not require continuous communication be- tween server and clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' In detail, our insight comes from the permutation invariant properties of self-attention can be utilized to offer the encryption by feature-space random permutation (Park and Ye, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' On top of this, we exploit another im- portant property of ViT resulting from its patch-based image processing, which enables the random masked sampling-based self-supervised learning to train the foundation model solely on the server-side, eliminating the need for continuous communi- cation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Our contributions can be highlighted as follows: We tackle the cumbersome problems of distributed self- supervised learning by introducing a novel method that ef- fectively leverages the properties of ViT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' We conducte extensive experiments to show the superior- ity of the proposed method on two medical domain data and tasks, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=', organ-at-risk (OAR) segmentation in com- puted tomography (CT) and tuberculosis diagnosis in chest X-ray (CXR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' We experimentally prove the infeasibility of a privacy at- tack on the proposed methods, implying that the proposed method can offer better privacy compared with the existing approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Related Works 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Self-supervised Vision Transformer Self-supervised learning is gaining traction in the vision community due to its outstanding successes in recent years, re- ducing the gap with supervised learning (Jing and Tian, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' A line of works on self-supervised learn- ing leverages an approach to train the model by discriminating the differently augmented versions of the given image, com- monly called contrastive learning (Jaiswal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' In a pi- oneering work, Caron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' (2021) proposed a contrastive learn- ing method that allows the ViT to use contrastive learning via teacher-student knowledge distillation, called distillation with- out a label (DINO), eliminating the need for cumbersome neg- ative samples required in traditional contrastive learning meth- ods (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=', 2020b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Grill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Zbontar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' In this method, the model learns the task-agnostic semantic meaning of the image through local-to-global correspondence with a random multi-crop strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' When visualizing the self- attention of the ViT model with DINO, the instance-level visual semantics were well attended by the model, without any super- vision for instance segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' This property was only ob- servable in ViT, and better performances in both linear probes and fine-tuning were observed in ViT compared to the CNN- based model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Since the ViT model is designed to be a pure patch-based attention model, another strong self-supervised learning tech- nique, random masked patch prediction, can be utilized for ViT- based models, which resembles the masked language modeling for Bidirectional Encoder Representations from Transformers (BERT) pre-training in the field of natural language process- ing (NLP) (Devlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Bao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' (2021) proposed a Bidirectional Encoder representation from Image Transformers (BEiT) that learns to predict the masked patch with discrete vi- sual tokens obtained with the discrete tokenizer, demonstrating that the same strategy in BERT can also be leveraged in vision tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Instead of predicting discrete tokens, He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' (2022) proposed a rather simple strategy of learning directly from pre- dicting pixels within the masked patches called masked au- toencoder (MAE), by adopting computation efficient encoder- decoder design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' These random masking-based learning strate- gies are specially designed for the ViT that process the im- age in a patch-wise manner and are not suitable for a CNN- based model that leverages the shared convolution kernels stride across the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Federated Split Task-Agnostic Learning with Permutating Pure ViT (p-FeSTA) Inspired by the modular configuration of the ViT model that can be divided into the embedder head, transformer body, and task-specific tail (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=', 2020a), the Federated Split Task- agnostic (FeSTA) learning has been proposed to maximally ex- ploit the distinct strengths of the FL and SL methods (Park S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Park and JC Ye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' 3 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Overall framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' First, the participating clients transmit the patch features encrypted with the arbitrary patch embedder and the feature-space permutation module to the server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Then, the MS-DINO training is performed on a server-side device to make the foundation model to have a general semantic understanding of the modality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Finally, the trained foundation model can be accessed by authorized clients to be used for various downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=', 2021), and to improve the performances of the individual tasks with the collaboration between the participating clients with different tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' In more recent work, Federated Split Task- Agnostic Learning with Permutating Pure ViT (p-FeSTA) has been introduced to surmount the drawbacks of the FeSTA (Park and Ye, 2022) by leveraging the permutation-invariant property of the ViT by adopting random patch permutation to provide better privacy as well as fewer communication overheads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' The main motivation of the p-FeSTA method is to reduce the com- munication between the server and clients as well as enhance the privacy with the permutation module in the feature space, which is possible via the permutation invariant property of the transformer encoder layers, the main component of ViT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Specif- ically, the model can be trained with the patch features permu- tated in the feature space with the novel feature-space permuta- tion module, which provides privacy by keeping the malicious attackers from faithfully reconstructing the private data from the intermediate features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' These permutated features can be safely saved in the server-side memory and used throughout the en- tire learning process, easing the burden to approximately half of the FeSTA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' However, this method is also not free of limita- tions that restrict the generalized application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' First, continuous communication between the server and clients is mandatory for model training as the labels are required for the update of the shared transformer body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Second, multi-task learning was pos- sible only with the clients simultaneously participating in dis- tributed learning who want to perform relevant tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Proposed Framework The proposed self-supervised learning method, dubbed Masked Sampling Distillation with No Label (MS-DINO), is composed of three steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' First, the patch features from all images were extracted by an arbitrary patch embedder along with another arbitrary position embedding, and their sequence is randomly permutated with feature-space permutation mod- ule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Then, the encrypted patch features are transmitted to the server and stored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' The server-side computational device uses these encrypted features throughout the entire learning process of the foundation model with no further communication with the clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Finally, the trained foundation model can be ac- cessed by the authorized clients for the application of down- stream tasks, offering performance superior to the fine-tuning- only baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' The overall framework is proposed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Foundation model learning Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' 2 illustrates the foundation model learning process with the MS-DINO method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Similar to the approach proposed in the p-FeSTA, given the image data of client x, the intermediate patch features f are extracted from the arbitrary patch feature extractor F, and permutated with the feature-space permutation module permute, which can be defined as f = permute(F(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Then, the encrypted features are transmitted from each client c to the server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' This process is performed at the beginning of the learning process, and the remaining processes are performed solely on the server-side, eliminating the need for both further Server Local clients 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' C D Encrypted features preparation 2) MS-DINO training Permutation Saved module Masked Sampling Feature set (Large) Teacher : Encrypted Feature embed features Match Distillation : : Masked Sampling (Small) 口 Student : ③ Fine-tuning Client 1 Task- Foundation Segmentation specific Task-agnostic model module Foundation model Client C Task- Foundation Classification specific Upload model module : Download4 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Park and JC Ye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Comparison of (A) the Distillation with No Labels (DINO) trained with federated learning (FL) and (B) the Masked Sampling Distillation with No Labels (MS-DINO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Compared with the original DINO method where the model gets general semantic understanding via local-to-global correspon- dence between the student and the momentum teacher, the model learns smaller number-to-larger number correspondence between the student and the momentum teacher in the MS-DINO method, enabling the training with randomly permutated patch features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' communication and computation overheads in client-sides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' In the process of the feature extraction and encryption, we not only extracted features from the original image x but also from an- other randomly augmented version of the image ˆx to provide diversity in data, resulting in two different encrypted features data per image as below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' f = permutate(F(x)), ˆf = permutate(F(ˆx)) (1) As the features are encrypted with both the patch embedder unknown to the server along with the unknown position embed and the permutation module in the feature space, it is infeasible for the server to faithfully reconstruct the private data with the transmitted feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' The detailed formulation and experiments of the privacy protection will be discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='3 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' With the encrypted features from all participating clients, the server performs random masked sampling-based self- supervised learning, by substituting the local-to-global of the original DINO correspondence learning strategy with small- to-large patches correspondence (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' As illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' 2(A), the motivation of DINO is to teach the model the vi- sual semantics of the image with the teacher-student knowledge distillation, by providing the global views with larger crops to the teacher and the multiple local views with smaller crops to the student.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' We adopted and modified this key concept into random masked sampling, by substituting the global views as the larger number and the local views as the smaller number of patch features sampled as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' 2(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' In detail, for the permutated feature f obtained from an image, the majority of the permutated patch features are randomly sampled to make the feature fl, and a relatively smaller number of the patches are randomly sampled resulting in the feature fs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Like DINO, fl is ■ (A) Client-side: Random cropping Server-side: model averaging Global views (Large) Got Averaging embedder Projection teacher Momentum Patch Teacher C : Nc Get Go.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' N c=1 Distribute Random Match i :EMA Averaging cropping ■ student ！' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='！' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' c embedder 1 Projection Student Patch N Ps Transformer c=1 Gos ！' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Ges Local views (Small) Server-side: Random mask distillation (B) Random sampling (Large) Client-side: encrypted feature set preparation c Projection Permutation Momentum module N Pos →PT Embed Teacher 1 C=1 i× Nc= Got Patch : No further !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' embedder transfer EMA Match i ！' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='！' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' 1 ■ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' 1 Projection Student ■ Transformer Gos Saved Feature set ■ Random sampling (Small)S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Park and JC Ye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' 5 fed to the teacher, while both fl and fs are fed to the student.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Then, the student is optimized to match the prediction of the momentum teacher with relatively small information about the image, in line with the local-to-global correspondence strategy of the DINO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Supposing the original feature fo, large sampled feature fl, small sampled features fs, the set that contains fo, fl, and N differently sampled fs can be defined as V = {fo, fl, f 1 s , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='f N s }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Given the teacher and student models as Gθt and Gθs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' the cross- entropy loss as L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' the student is trained to mimic the teacher’s prediction with following optimization problem: min Gθs � f∈{fo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='fl} � f ′∈V f ′� f L(Gθs(f ′),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='Gθt( f)) (2) During the learning process,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' the momentum teacher model is updated with an exponential moving average (EMA) of the stu- dent’s update,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' where λ follows a cosine scheduling: Gθt = λGθt + (1 − λ)Gθs (3) The algorithm for the encrypted feature set preparation and the foundation model learning with MS-DINO are formally pre- sented in the algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Given the limited data availability for a single client, this foundation model may merit improved generalization perfor- mance, which is further investigated in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='2 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Fine-tuning for Tasks of Interest As depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' 1, the authorized clients can access the trained foundation model for their purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' For instance, if clients seek to train an OAR segmentation model for radiother- apy planning, they can leverage the foundation model to en- hance the generalization performance, since it holds the general ability to attend to visual semantics within the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Specifically, for client, c, supposing the pre-trained foun- dation model backbone as G and task-specific layer like de- coder as Hc, data and label for fine-tuning as xc and yc, task- specific loss function as Lc, the following optimization problem is solved for the fine-tuning: min G,H � i=1 Lc(Hc(G(xc)), yc) (4) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Protecting Privacy with Feature-space Permutation Mod- ule Protecting privacy by randomly permutating patch features in the feature space has been investigated in the previous work (Park and Ye, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' As the embedded features from the clients are saved on the server-side, privacy concerns may arise if the “honest-but-curious” server or the malicious attacker who has hijacked the features during the transmission may try to revert the features into the original data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Specifically, suppose that the encrypted features with permu- tation are hijacked during the communication and there are suf- ficient public data in the same domain available for the attacker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Then, the attacker needs to solve two problems simultaneously: (1) training the attacker-side feature extract that can embed the Algorithm 1 Proposed MS-DINO algorithm /* Run on Client c */ 1 Function ClientMain: 2 Client initialize with arbitrary feature embedder F 3 for data x ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' X} 4 {x, x′} ← Augment(x) 5 f = permute(F(x)) // original feature 6 f ′ = permute(F(x′)) // augmented feature 7 feature set f = {{ f1, f ′ 1}, {f2, f ′ 2} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' {fX, f ′ X}} ← {f, f ′} 8 return f /* Run on Main Server */ 9 Function ServerMain: 10 Server initializes student Gθs and teacher Gθt for clients c ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='C} do in parallel 11 fc ← ClientMain(c) Memory = {f1, f2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' fc} ← fc // Save all features fc in Memory 12 for epoch e ∈ Memory 13 for features f, f ′ ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' E} // Run MS-DINO learning in server-side device 14 LDINO = � f∈{fo,fl} � f ′∈V f ′� f L(Gθs( f ′),Gθt( f)) 15 θs ← θs − η N ∂LDINO ∂θs // update student model 16 θt = λθt+(1−λ)θs // EMA update of teacher model 17 for clients c ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='C} do in parallel 18 Distribute Gθs to authorized client // Distribute foundation model to clients image into the feature space the same as that of the unpermu- tated hijacked feature, and (2) training the jigsaw solver to un- permutate the encrypted features into unpermutated ones in the feature space of hijacked features, not in the image space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' More specifically, we denote the permutated and the origi- nal features embedded by the attacker-side model ˆF as ˜fpub and fpub, respectively, the number of attacker-side images (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' pub- licly available CT or CXR images) and the hijacked encrypted features as m, and the permutated features hijacked by the at- tacker during communication as ˜fpriv which is embedded by an arbitrary client-side feature embedder F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Then, the attacker-side model ˆF, discriminator D, and the de- coder G can be trained by optimizing the following two learning objectives: min ˆF max D m � i=1 n � j=1 [log(1 − D(J( ˜f (i) priv))) + log D(J( ˜f (j) pub))] (5) min G m � i=1 Ldecoder(G(J( ˜f (i) pub)), x(i) pub) (6) where Ldecoder denotes reconstruction loss for decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Mean- while, the second optimization problem for the jigsaw solve J 6 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Park and JC Ye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Data for foundation model learning Modality Client #1 Client #2 Client #3 Client #4 CT 7,993 8,063 8,182 7,973 CXR 8,422 8,424 8,422 8,422 can be formulated as follow: min J m � i=1 Ljigsaw(J( ˜f (i) pub), f (i) pub) (7) where Ljigsaw denotes similarity loss in the feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Note that to simultaneously solve the first two equations, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' (5) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' (6), the exact jigsaw solver should be unrav- eled, which can be obtained if the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' (7) is successfully solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' However, the jigsaw solver should be trained and utilized in the same feature space as the hijacked encrypted feature, which ne- cessitates knowing the correct solution for the attacker model ˆF to embed the same feature space, and this is conversely the tar- get of the optimization problem Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Combined, optimiza- tion of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' (5), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' (6), and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' (7) requires to already have each other’s solutions, indicating that the problems are under- determined and practically hard to solve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='5 provides experimental results to support our as- sertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Datasets We investigated the applicability of the proposed method in two different domain data and two different tasks, OAR seg- mentation with CT for radiotherapy planning and tuberculosis diagnosis with CXR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Datasets for foundation model learning For the foundation model learning, two open-sourced datasets, CT scans of the head and neck cancer patients from The Cancer Imaging Archive (TCIA) Head and Neck Squa- mous Cell Cancer (HNSCC) data (Grossberg A et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Grossberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Elhalawani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Clark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=', 2013) and the CXR images from the Radiological Society of North America (RSNA) pneumonia detection challenge data (of North America, 2018) were used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' To simulate the application in clinical collaboration, we sim- ulated the collaboration of four clients with different data di- visions, emulating the collaboration between four institutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' We divided the dataset into several subsets, defining each subset as the data of each client as shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Among the CT scans of a total of 619 patients from the HNSCC data, we used the CT scans of 200 patients, 7,993 CT slices from 50 patients (client #1), 8,063 from 50 patients (client #2), 8,182 from 50 patients (client #3) and 7,973 from 50 patients (client #4) were assigned for each client.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Similarly, among a total of 33,690 images from the RSNA pneumonia detection challenge dataset, 8,422 (client #1), 8,424 (client #2), 8,422 (client #3), and 8,422 (client #4) images were assigned for each client (Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Data for fine-tuning downstream task Task Description Fine-tune Test Full Limited CT segmentation OARs 2,911 608 521 CXR classification Normal 4,127 327 92 Tuberculosis 1,135 335 46 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Datasets for fine-tuning downstream task In practical implementation, the trained foundation model will be accessed and fine-tuned for the client’s task of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Therefore, we used the datasets containing both data and la- bels for the downstream tasks, namely OAR segmentation with CT and tuberculosis diagnosis with CXR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Given the problem of limited data and label availability frequently with a single client, we performed the experiments in two settings: data- abundant (full) and data-insufficient (limited) settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' For the downstream OAR segmentation task with CT, the Medical Image Computing and Computer Assisted Intervention (MICCAI) 2015 head and neck challenge dataset (Computing and Interventions, 2015) was used as the fine-tuning dataset for organ-at-risk segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' As the dataset contains a total of 2,911 CT slices and labels from 38 patients, all patients’ data were utilized for the data-abundant setting, while the 8 patients’ data containing 608 CT slices were used as the data-insufficient setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' For the downstream tuberculosis diagnosis task with CXR, 1,135 tuberculosis and 4,127 normal cases from two data sources, the Montgomery County (MC) dataset (Jaeger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=', 2014) and TBX 11K dataset (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=', 2020) were collected for data-abundant setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' For data-insufficient setting, only the MC dataset containing 335 tuberculosis and 327 normal cases was used (Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Datasets for evaluation To evaluate the benefit of the foundation model, we used two datasets for each task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' For evaluation of the segmentation per- formance, we used the CT and region-of-interest (ROI) data collected and delineated by the board-certified radiation oncol- ogists from the local institution (Gangnam Severance Hospital) to externally validate the performance of the developed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' From 2007 to 2021, data from 44 head and neck cancer patients were collected, and seven patient data containing all ROIs were used for the evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' For evaluation of the classification per- formance, we used a publicly accessible dataset (India tuber- culosis dataset) (radder), which can be regarded as the external validation that is collected from the data source different from the training and fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Implementation Details 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Details of Model development The CT images underwent preprocessing by cropping the center area of 224 × 224 from a total size of 512 × 512.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' The upper and lower window of the Hounsfield unit (HU) were set to -200 and 200, respectively, in consideration of the ranges S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Park and JC Ye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' 7 of HU of the OARs of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' As the data were from differ- ent sources, the pixel spacing was adjusted to match between datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' The CXR images were preprocessed with histogram equalization Gaussian blurring and normalization, and finally resized to 224 × 224.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' As the arbitrary feature embedder, the patch embedder of the DINO model pre-trained on ImageNet was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' The embed- ded features were permutated with the feature-space permuta- tion module proposed in Park and Ye (2022), yielding the en- crypted features for foundation model learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' For the body part of the Transformer, the transformer encoder of ViT small having 6 heads, 12 layers, and a patch size of 8 was used, which was initialized with the DINO self-supervised learning weights on the ImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' We used the same-sized teacher and student models, and the multi-masked sampling strategy was adopted instead of the multi-cropping strategy following the original implementation of Caron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' As a global view, the smaller number of patch features were masked, resulting in the ratio of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='9 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='0 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='8 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='0 being sampled for CT and CXR, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' For multiple local views, the larger number of patch features were masked, sampling with a ratio of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='3 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='5 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='3 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='4 for CT and CT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' The same configuration with the original work of DINO was adopted for comparison, with crop sizes of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='0 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='05 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='4 for global and local views, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Considering the relatively small dataset size and complexity of the medical image, we reduced the dimensionality of the DINO head output from 65,536 to 8,192.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' For the foundation model learning with the proposed MS- DINO, Adam W optimizer (Loshchilov and Hutter, 2017) was used along with a cosine decay scheduler with a maximum learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='00004 with a batch size of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' The model was trained in the server-side device for 5 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' For training the foundation model with the DINO method via FL, the same op- timizer, scheduler, and learning rate was used with a batch size of 4 per client, and the model was trained for 10,500 federated rounds in the client-side devices to match the total number of updates to MS-DINO learning in the server-side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' For FL, both the student and teacher model parameters were averaged every 100 rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' For the downstream OAR segmentation task, the foundation ViT model and the UperNet (Xiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=', 2018) decoder were used as the encoder and the decoder, respectively, following the implementation in Bao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' As the UNet for comparison, standard UNet (Ronneberger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=', 2015) architecture with four downsample and upsample layers and 64 convolutional filters were utilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' We designed the decoder part of the network as the multi-class segmentation model, to delineate the ROIs (brainstem, optic chiasm, mandible, optic nerves, parotid glands, and submandibular glands) simultane- ously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Following the previous work on MICCAI 2015 head and neck segmentation challenge dataset, we used the combined fo- cal loss (Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=', 2017) and Dice loss for the optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' The SGD optimizer was used for the segmentation model with a learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='01 and a batch size of 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' For the downstream classification task, a simple linear layer was added as the classification head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' As we implemented the tu- berculosis diagnosis task as the binary classification, the model optimized with the BCE loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Adam W optimizer was used to fine-tune the model for the classification task, with a learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='0001 and a batch size of 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' All experiments including data processing, foundation model learning, fine-tuning, and evaluation were performed with Python version 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='9 and Pytorch library 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='10 on NVIDIA GeForce RTX 3090, and the experiments on distributed learning were conducted with the FLOWER framework (Beutel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Details of evaluation To evaluate the segmentation performance, the Dice similar- ity coefficient is used to quantify the overlap between the pre- dicted segmentation mask and the ground truth label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' To eval- uate the classification performance, the area under the receiver operating characteristics curve (AUC) was calculated to com- pare the model performance, and the sensitivity, specificity, and accuracy were reported separately to provide detailed perfor- mances, after adjusting the thresholds to meet the pre-defined sensitivity value ≥ 80%, if possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Simulation for Feature Inversion Attack We evaluated whether the permutation model is effective to prevent privacy attacks from feature hijacking, supposing the optimal configuration for the malicious attacker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Following the previous work on the encryption with random patch permutation (Park and Ye, 2022), it is supposed that the attacker hijacked all encrypted features transmitted from all clients and knows the exact permutation ratio and patch size, the architecture of the unknown patch embedder and the dimen- sionality of position embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Specifically, the same archi- tecture as the original patch embedder was used as the attacker- side feature embedder, and the three-layered discriminator and the four-layered generator from DCGAN (Radford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=', 2015) were employed as the discriminator and decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' As a jigsaw solver to solve the random permutation in the feature space, the transformer with 12 encoder layers and 12 attention heads was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' The discriminator was optimized with the modified ver- sion of GAN loss (Goodfellow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=', 2014) as formulated in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' (5) and (6), the combined L1 and L2 losses were used as the learning objective for the decoder, and the L1 loss was used as the objective for the jigsaw solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' In addition, we supposed that the attacker possesses a suffi- cient amount of data in the same domain, like the CT or CXR available at the public repository, to train the attacker-side net- works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' As the attacker-side public data, 6,189 CT slices from the TCIA HNSCC-3DCT-RT data (Clark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Bejarano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=', 2018, 2019) and 9,577 normal CXRs data from the CheX- pert data (Irvin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=', 2019) were utilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' The model was trained for 5 epochs with a batch size of 1 and a learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Experimental results 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Attention Changes with Foundation Model Training Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' 3 depict the changes in the last layer of multi-head at- tentions within the ViT model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Before the foundation model 8 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Park and JC Ye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Visualization of multi-head attention for (A) CT and (B) CXR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Compared with the attentions obtained with the pre-trained ImageNet weights, the different attention heads got refined and tend to concentrate on different semantic components after MS-DINO learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Segmentation results after fine-tuning in data-limited setting Method Overall Brainsetm Chiasm Mandible Optic n Lt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Optic n Rt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Parotid Lt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Parotid Rt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' SMG Lt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' SMG Rt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Unet 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='083 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='182 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='359 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='000 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='012) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='043) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='028) Values are presented in the mean (standard deviation) of three repeats with different seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' DC DINO, data-centralized DINO;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' FL DINO, federated learning DINO Bold face and underline denote the best and the second best performances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' training, the attentions of the ViT model with the ImageNet pre-trained with the original DINO disperse throughout the im- age (upper row), and the differences in attention between the individual heads were not prominent, implying that most at- tention heads process the image less effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' After learn- ing with the MS-DINO method, the different heads tend to at- tention to different semantic components (lower row), which can offer improved model performances with diverse patterns of self-attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Performance Comparison for Downstream CT Task The comparison results for the CT task between the meth- ods are provided in Table 3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' The model fine-tuned from the foundation models showed better performance than those obtained with training from scratch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' In particular, the model trained with the proposed MS-DINO showed a performance close to that of the DINO trained in a data-centralized man- ner, while outperforming the DINO trained with FL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' The per- formance enhancement with the foundation model learning is Head 1 Head 2 Head 3 Head 4 Head 5 Head 6 (A) After MS-DINO learning (B) After MS-DINO learningS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Park and JC Ye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' 9 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Segmentation results after fine-tuning in data-abundant setting Method Overall Brainsetm Chiasm Mandible Optic n Lt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Optic n Rt.' metadata={'source': 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mean (standard deviation) of three repeats with different seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' DC DINO, data-centralized DINO;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' FL DINO, federated learning DINO Bold face and underline denote the best and the second best performances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Classification results after fine-tuning in data-limited setting Method AUC Sensitivity Specificity Accuracy DenseNet-121 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} 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+page_content='042) ViT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='590 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='558 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='598 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='585 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='004) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='013) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='011) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='004) ViT (DC DINO) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='824 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='739 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='757 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='751 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='016) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='022) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='045) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='025) ViT (FL DINO) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='695 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='645 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='609 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='616 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='057) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='066) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='058) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='029) ViT (Proposed) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='844 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='790 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='772 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='778 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='007) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='063) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='011) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='022) Values are presented in the mean (standard deviation) of three repeats with different seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' DC DINO, data-centralized DINO;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' FL DINO, federated learning DINO Bold face and underline denote the best and the second best performances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Classification results after fine-tuning in data-abundant setting Method AUC Sensitivity Specificity Accuracy DenseNet-121 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='652 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='688 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='583 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='618 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='033) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='090) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='094) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='041) ViT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='493 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='478 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='543 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='522 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='023) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='210) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='171) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='044) ViT (DC DINO) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='837 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='812 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='728 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='756 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='023) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='013) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='050) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='033) ViT (FL DINO) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='773 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='681 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='746 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='725 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='022) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='126) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='071) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='022) ViT (Proposed) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='854 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='783 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='790 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='787 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='008) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='058) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='006) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='022) Values are presented in the mean (standard deviation) of three repeats with different seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' DC DINO, data-centralized DINO;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' FL DINO, federated learning DINO Bold face and underline denote the best and the second best performances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' more prominent in the data-limited than the data-abundant set- tings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' 4 depicts the qualitative comparison between methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' The model fine-tuned from the foundation model with the pro- posed MS-DINO offered overall more accurate predictions for the areas of OARs, which is comparable to those with data- centralized DINO and outperforms those of fine-tuning only baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Performance comparison for downstream CXR task Table 5 and 6 show the comparison results for the CXR task between the methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Compared with the fine-tune-only base- lines which were obtained by training the model from scratch, those with the foundation model offered generally better perfor- mance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Furthermore, among those with foundation models, the model trained with the proposed MS-DINO method provided the best performance, even outperforming the DINO trained in a data-centralized manner as well as the DINO obtained in FL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' The benefit of the proposed method was remarkable in the data- limited setting, providing a diagnostic performance close to that of the data-abundant setting with only several hundred images for each class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Communication costs When defining the number of data as D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' total training rounds as R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' the round between aggregation and distribution as r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' the model parameter as P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' and the size of encrypted feature for each data as F,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' the total communication costs for foundation model learning T for the FL and the MS-DINO can be formulated as below: TFL = 4R r × P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' (8) TMS-DINO = D × F (9) where the constant 4 is multiplied to account for the parameter transmission of both the teacher and student models,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' and the both-way transmissions between server and client for FL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' As the MS-DINO method does not require continuous communi- cation between the server and the clients, the only communica- tion between the server and clients occurs at the beginning of the learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' With this formulation, the numerical comparison results are provided in Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Compared to the training of DINO with FL, the communication cost of the proposed MS-DINO is about one-third, which can be further advantageous as the number of total rounds increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Experiments on Privacy protection The qualitative and quantitative analysis results of recon- struction from the privacy attack are provided in Table 8 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' 10 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Park and JC Ye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Qualitative comparisons of the segmentation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' The model fine-tuned from the foundation model with the MS-DINO method shows results comparable to those with the data-centralized DINO method and outperforms the fine-tune only baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' The results are obtained from fine-tuning in a data-abundant setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Comparison of communication overheads between methods Method Continuous communication Communication cost Feature transmission Model averaging Federated learning \x13 12,229.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='7 M Proposed \x17 4,866.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='9 M Given the optimal configuration for the attacker and the set- ting that the feature-space permutation module is not employed, both the private CT and CXR data can be reconstructed to the amount that the privacy like shape, gender, anatomic location, and disease status of the subject can be inferred from the re- construction results of hijacked feature to some degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' This implies that the approximate solution of the unknown feature embedder can be obtained by the attacker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' However, when the extracted features are randomly permutated in the feature space, it was nearly impossible to reconstruct the data to the amount that the privacy can be inferred (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Combined, these results confirm our claim that simultane- ously solving two optimization problems that need each other’s solution can be considered to be an underdetermined problem, and thus practically difficult to solve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Ablation studies Table 9 shows the ablation studies to verify the roles of the components of the proposed MS-DINO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' As the random masked sampling and different augmentation for each image were uti- lized in the proposed method, we ablated the components in or- der.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' The ablation studies are performed in the data-limited set- ting, as the benefit of the proposed method is more pronounced in this setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Random masked sampling Ablating the random masked sampling means that the model is trained by the knowledge distillation between teacher and stu- dent, with the two views of the image, the original and the aug- mented versions of the same image, but without any random sub-sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' As shown in the Table 9, the performances were suboptimal for both CT segmentation and CXR classification, implying that this component is indispensable for the best per- formance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Augmentation Next, we ablated the augmentation to offer two different views of the image to evaluate whether this component is nec- essary for performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Compared with the proposed method leveraging an augmented version of the given image, apply- ing random sampling solely for the original image show lower performance in both CT segmentation and CXR classification, ViT ViT (DC DINO) UNet ViT (proposed) Label ViT (FL DINO)S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Park and JC Ye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' 11 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Reconstruction results of the privacy attack for (A) CT and (B) CXR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Without the feature-space permutation module, information (level of CT slice, presence of disease, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=') can be inferred from the reconstructed results, while it was infeasible when the feature-space permutation module was utilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Quantitative comparison for privacy attack Method CT CXR MSE SSIM MSE SSIM No permutation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='237 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='005) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='694 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='008) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='161 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='051) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='350 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='014) Permutation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='272 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='009) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='479 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='038) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='267 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='006) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='018 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='007) Values are presented in the mean (standard deviation) of three repeats with different seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' MSE, mean squared error;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' SSIM, structural similarity index measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Table 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Ablation studies CT Segmentation CXR classification Method Overall DSC AUC Proposed 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='556 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='010) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='844 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='007) No random masking 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='544 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='003) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='817 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='045) No augmentation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='548 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='004) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='833 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='019) Values are presented in the mean (standard deviation) of three repeats with different seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' OAR, organ at risk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' DSC, dice similairty coefficient;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' AUC, area under the curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' which suggests that offering different views of an image en- hances the performance regardless of the modalities and the tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Discussion and Conclusion Despite surprising performance of the DL-based vision mod- els, the data-driven learning paradigm of the DL has been pos- ing practical hurdles in the development of the AI model for healthcare research where the data utilized for training may contain sensitive privacy of individuals (Perone and Cohen- Adad, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' In addition, the label dependency throws another challenge since the labels annotated by the medical experts are usually expensive and hard to be obtained (Willemink et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Distributed learning methods have been introduced to tackle these problems by enabling model training without directly sharing private data, and self-supervised learning methods have been investigated to alleviate the label dependency of the DL model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Several works have investigated the combination of these two techniques, but there remain problems in those ap- proaches reporting suboptimal performances compared with data-centralized counterparts (Makhija et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Zhuang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Shi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Furthermore, the innate proper- ties of model averaging during FL enforce continuous commu- nication between server and clients for model averaging as well as high communication overhead, and pose the privacy threats that can be caused by the model inversion attack by malicious attackers (Geiping et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Lyu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=', 2020b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Hatamizadeh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' To alleviate these issues, we have introduced a novel self- supervised learning method, dubbed MS-DINO, that can be used in a distributed manner without continuous communi- cation but still attains a performance comparable to or even outperforming the self-supervised learning method performed in a data-centralized setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Inspired by the previous works that reported the intriguing properties of ViT (Naseer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Park and Kim, 2022), we utilized two key concepts, the permutation-invariant property of the transformer layer and se- mantic learning via local-to-global correspondence using the teacher-student distillation, to devise our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Specifically, we replaced the multi-crop strategy of cropping multiple small local crops and large global crops with the random masked sampling strategy, which randomly samples large and multi- ple small portions of the patch features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' This random masking strategy is also consistent with pioneering self-supervised learn- ing approaches that conjugate masked image modeling with the ViT models in a patch-wise manner (Bao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Furthermore, we amalgamated the pre- viously proposed feature-space permutation module (Park and Ye, 2022) to enhance privacy and to improve communication efficiency without defacing the performances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' By enabling to save the encrypted patch features with the feature-space per- mutation module, the foundation model learning based on the masked random sampling of patch features can be performed solely on the server-side device while at the same time protect- (A) (B) Original Permutation Original No permutation No permutation Permutation12 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Park and JC Ye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' ing the privacy of the participating subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' We performed ex- tensive experiments on two imaging modalities and two tasks in both data-abundant and data-limited settings to clarify the ben- efit of the method in general application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' In addition, we also performed experiments on the privacy attack to verify the ef- ficacy of the feature-space permutation module, demonstrating that enhanced privacy can be offered with the proposed method equipped with this module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Our study is not free of limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' First, as the proposed MS-DINO method is built upon the intrinsic permutation- invariant property of the Transformer, the model without this property, for instance, CNN-based architecture, can not be uti- lized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Second, although we have demonstrated that the pri- vacy attack through feature-space hijacking is infeasible for our method, other types of malicious attacks including model poi- soning and data poisoning (Lyu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=', 2020b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Tolpegin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Bagdasaryan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=', 2020) or privacy threats like the mem- bership inference (Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Shokri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Nasr et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=', 2019) were not investigated, which is beyond the scope of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' The existing methods for privacy protection can be adopted in addition to the proposed method to enhance privacy (Lyu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=', 2020a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=', 2020a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Third, other considerations for the practical implementation like the data skewness between clients were not investigated (Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' However, as the permutated features from all clients are saved and used by the server during the foundation model learn- ing, the vulnerability to data skewness is expected to be lower compared with the existing approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Finally, our learning method is only applicable to the ViT-based models, since it is based on the model-specific learning method, the random masked sampling of the shuffled patch features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' In addition, as the ViT model processes the image by dividing it into multiple patches and by calculating the attention scores, the fine-grained segmentation of the thin and small organ (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' optic chiasm) is difficult for the ViT-based model, as shown in the example in the third row of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Nevertheless, given the limited data availability and the importance of privacy in health research, our method has great promise to build a general foundation model in a communication-efficient and privacy-protecting way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Since the model obtained with the proposed MS-DINO has a general un- derstanding of visual semantics, it can be used as the task- agnostic foundation model to enhance the performances of the downstream task, suggesting its widespread applicability in healthcare research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Acknowledgement This research was supported in part by a grant of the MD- PhD/Medical Scientist Training Program through the Korea Health Industry Development Institute (KHIDI),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' funded by the Ministry of Health & Welfare,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Republic of Korea,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' by the Na- tional Research Foundation of Korea (NRF) grant funded by the Korean Government Ministry of Science and ICT (NRF- 2020R1A2C1102559),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' by a faculty research grant of Yon- sei University College of Medicine (6-2019-0071),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' by the National Research Foundation of Korea under Grant NRF- 2020R1A2B5B03001980,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' and by the KAIST Key Research In- stitute (Interdisciplinary Research Group) Project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' Ethic Committee Approval The CT and the ROI data collected for this study were eth- ically approved by the Institutional Review Board (IRB) of Gangnam Severance Hospital (IRB number: 3-2020-0289), and the requirement for informed consent was waived due to the ret- rospective nature of this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=' References Bagdasaryan, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=', Veit, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=', Hua, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=', Estrin, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=', Shmatikov, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftA0T4oBgHgl3EQfHv83/content/2301.02064v1.pdf'} +page_content=', 2020.' metadata={'source': 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b/hdE1T4oBgHgl3EQfMgPQ/content/tmp_files/2301.02991v1.pdf.txt @@ -0,0 +1,4020 @@ +arXiv:2301.02991v1 [stat.ME] 8 Jan 2023 +Test of Bivariate Independence Based on Angular +Probability Integral Transform with Emphasis on +Circular-Circular and Circular-Linear Data +Fernández-Durán, J.J. and Gregorio-Domínguez, M.M. +ITAM +E-mail: jfdez@itam.mx +Abstract +The probability integral transform (PIT) of a random variable X with distribution function +FX is a uniformly distributed random variable U = FX(X). We define the angular probabil- +ity integral transform (APIT) as θU = 2πU = 2πFX(X), which corresponds to a uniformly +distributed angle on the unit circle. For circular (angular) random variables, the sum of abso- +lutely continuous independent circular uniform random variables is a circular uniform random +variable, that is, the circular uniform distribution is closed under summation, and it is a stable +continuous distribution on the unit circle. If we consider the sum (difference) of the angular +probability integral transforms of two random variables, X1 and X2, and test for the circular +uniformity of their sum (difference), this is equivalent to test of independence of the original +variables. In this study, we used a flexible family of nonnegative trigonometric sums (NNTS) +circular distributions, which include the uniform circular distribution as a member of the fam- +ily, to evaluate the power of the proposed independence test by generating samples from NNTS +alternative distributions that could be at a closer proximity with respect to the circular uniform +null distribution. +Keywords: circular-circular dependence; circular-linear dependence; circular uniformity tests; +copula, dependence measures +1 + +2 +1 +Introduction +Testing the independence of a set of random variables is considered as one of the most important +task in many practical applications: while estimating the joint distribution of a set of random vari- +ables, constructing conditional models to explain one variable in terms of others as in regression +models, among many other problems in statistics. +According to Herwatz and Maxand (2020), one can consider the following tests of indepen- +dence: bivariate (pairwise), groupwise, and mutual independence tests. +Hereinafter, we only +consider absolutely continuous random variables. A random variable with a density function +with support on an interval of the real line is a linear random variable, and one with support +on the unit circle is a circular random variable. For two random variables, X1 and X2, bivari- +ate (pairwise) independence tests have null hypothesis, H0 : FX1,X2(x1, x2) = FX1(x1)FX2(x2), +where FX1,X2(x1, x2) = P{X1 ≤ x1, X2 ≤ x2} is the bivariate joint distribution function and +FX1(x1) = P{X1 ≤ x1} and FX2(x2) = P{X2 ≤ x2} are the corresponding marginal distri- +bution functions. For a set D of d (d > 2) random variables, X1, X2, . . . , Xd, in a groupwise +independence test one partitions the set D of d random variables into two nonempty disjoint sub- +sets, D1 and D2 such that D1 +� D2 = D and D1 +� D2 = ∅, and considers the null hypothesis, +H0 : FXD1,XD2(xD1, xD2) = FXD1(xD1)FXD2(xD2), where XD1 and XD2 are the vectors of ran- +dom variables in the sets D1 and D2, respectively. Groupwise independence tests can be extended +to the case of more than two nonempty disjoint subsets of random variables. Finally, a mutual +independence test for a set of random variables, X1, X2, . . . , Xd considers the null hypothesis, +H0 : FX1,X2,...,Xd(x1, x2, . . . , xd) = �d +k=1 FXk(xk), where FX1,X2,...,Xd(x1, x2, . . . , xd) is the joint +distribution and FXk(xk) for k = 1, 2, . . . , d are the marginal univariate distribution functions. +If the functional forms of FX1,X2,...,Xd and FXk for k = 1, 2, . . . , d are specified, a likelihood- +ratio test of independence from a sample of random vectors Xi = (Xi1, Xi2, . . . , Xid)⊤ of size n + +3 +(i = 1, 2, . . . , n), can be constructed by considering +Λ = −2 ln +� +maxΘjoint Ljoint(X1, X2, . . . , Xn) +maxΘindep Lindep(X1, X2, . . . , Xn) +� +(1) +where Ljoint(X1, X2, . . . , Xn) is the likelihood of the data under joint distribution function FX1,X2,...,Xd +with vector of parameters Θjoint and, Lindep(X1, X2, . . . , Xn) is the likelihood of the data under +the independence assumption with the distribution function being the product of the marginal uni- +variate distribution functions with vector of parameters Θindep. The statistic Λ asymptotically and +under regularity conditions has a chi-squared distribution with df degrees of freedom, where df is +equal to the difference between the dimensions of the parameter vectors Θjoint and Θindep. +The most commonly used test for independence is the chi-squared test of independence for +contingency tables, which is not adequate when dealing with absolutely continuous random vari- +ables. Alternatively to the likelihood ratio independence test, other tests for independence were +developed by considering nonparametric (distribution free) methods, rank tests (see Hoeffding, +1948 and Kendall and Stuart,1951), and measures of dependence (association) derived from the +empirical copula process (Deheuvels, 1981; Genest and Rémillard, 2004; Genest and Verret, 2005; +Genest et al., 2019; Roy, 2020 and Roy et al., 2020). The empirical copula, Cn, for a vector of d +absolutely continuous linear random variables, X⊤ = (X1, X2, . . . , Xd)⊤, and a sample of size n +is defined as follows: +Cn(u1, u2, . . . , ud) = 1 +n +n +� +j=1 +I( ˆF1(Xi1) ≤ u1, ˆF2(Xi2) ≤ u2, . . . , ˆFd(Xid) ≤ u2) +(2) +where I() is an indicator function, which is equal to one if the condition in its argument is satisfied +and zero otherwise, and ˆF1, ˆF2, . . . , ˆFd are the empirical distribution functions of the random vari- +ables X1, X2, . . . , Xd. A test of independence based on the distance between the empirical copula +and independence copula for absolutely continuous linear random variables is implemented in the +R package copula (Hofert et al., 2022 and Kojadinovic and Yan, 2010). +Among the nonparametric tests of independence, there are family of tests based on some func- +tional of the empirical independence process, which is defined as the distance between the empir- + +4 +ical joint distribution function and product of the empirical univariate distribution functions. His- +torically, the most used functionals have been the Cramér-von Mises and Kolmogorov-Smirnov +functionals (refer Blum, Keifer and Rosenblatt, 1961; DeWet, 1980 and Deheuvels, 1981). For +example, Hoeffding (1948) considered the Cramér-von Mises functional to generate a rank test +of independence between two random variables. Modern rank tests of independence have been +developed by Kallenberg and Ledwina (1999). Kernel-based methods have also been used to es- +timate the empirical independence process, as in Pfister et al. (2018). Mardia and Kent (1991) +used the general Rao score test to generate independence tests. Csörg˝o (1985) developed indepen- +dence tests based on the multivariate empirical characteristic function, and Einmahl and McKeague +(2003) developed the tests based on the empirical likelihood. The measures of dependence derived +from entropy were defined by Joe (1990) and from mutual information by Berrett and Samworth +(2019). +Tests of independence in specified multivariate distribution functions were simplified many +times with respect to the structure of the distribution. This is the case for the multivariate Gaussian +(normal) distribution where the null correlation implies independence, and for the test of mutual +independence under multivariate Gaussian distribution is equivalent to the one used to testing an +identity correlation matrix. Some of these pairwise tests are the Pearson (1920) product moment +correlation coefficient test, Kendall (1938) rank correlation coefficient test, and Spearman (1904) +rank correlation coefficient test. Of course, for a pair of Gaussian random variables, rejecting null +correlation implies rejecting pairwise independence, but applying pairwise independence (correla- +tion) tests is not adequate to test for mutual independence for a set with more than two Gaussian +random variables. The Wilks test (Wilks, 1935) is an optimal test of independence for multivariate +Gaussian populations and for the case of a bivariate groupwise independence test for the vectors +XD1 and XD2 with X = X⊤ +D1 +� D2 = (XD1, XD2)⊤ considers the following test statistic for a + +5 +sample of size n, +W = |ˆΣD1 +� D2| +|ˆΣD1||ˆΣD2| +(3) +where ˆΣD1 +� D2 = �n +j=1(xj−¯x)(xj−¯x)⊤ is the estimated covariance matrix of the complete vector +of observations x = xD1 +� D2 which is partitioned into ˆΣD1 and ˆΣD2 with ˆΣD1 = �n +j=1(xD1,j − +¯xD1)(xD1,j − ¯xD1)⊤ and ˆΣD2 = �n +j=1(xD2,j − ¯xD2)(xD2,j − ¯xD2)⊤. The statistic W then measures +the extent of the distance between the determinant of ˆΣD1 +� D2 and the product of the determinants +of ˆΣD2 and ˆΣD2. The equality relationship is satisfied in the multivariate Gaussian population under +the null hypothesis of independence between XD1 and XD2. Asymptotically and under regularity +conditions, −n ln(W) follows a chi-squared distribution with ♯D1♯D2 degrees of freedom, where +♯D1 and ♯D2 are the cardinalities of sets D1 and D2, respectively. +For the circular-circular (angular-angular) and circular-linear (angular-linear) cases, in which +the objective is to test for bivariate independence between two circular random variables and one +circular and one linear random variable, respectively, independence tests were developed by con- +sidering the specification of measures of dependence and studying their (asymptotic) distributions. +By applying Kendall’s tau and Spearman’s rho general measures of dependence based on the con- +cept of concordance, or the construction of distribution-free correlation coefficients based on ranks +to a pair of circular random variables or a circular and linear random variable, tests of indepen- +dence were developed by Fisher and Lee (1981, 1982 and 1983) and reviewed by Fisher (1993) +and Mardia and Jupp (2000). +The objective of this study is to develop a test of bivariate (pairwise) independence for two +random variables by considering the angular probability transform of each variables, which corre- +spond to circular uniform distributions on (0, 2π], and an additional result of the theory of circular +statistics (see Fisher, 1993; Upton and Fingleton, 1989; Mardia and Jupp, 2000 and Jammala- +madaka and SenGupta, 2001). The test was evaluated using flexible nonnegative trigonometric +sums (NNTS) distributions (Fernández-Durán, 2004b, 2007). Although the proposed test of in- +dependence is a general one, it is especially suitable to test for the independence in the circular- + +6 +circular and circular-linear cases. Thus, a measure of dependence was developed. Once the bivari- +ate test was developed, it was extended to a group-wise independence test for two disjoint subsets +of random variables. +This paper is divided into six sections, including the introduction. In the second section, John- +son and Wehrly (1977) model is presented as a motivation for performing the test of bivariate +independence for two random variables, and here, the theory of NNTS circular distributions is +included. The third section presents the proposed bivariate independence test, a measure of de- +pendence, and its application to simulated data to study the power of the test. In this section, we +explain how to extend the test of bivariate independence to a test of groupwise independence for +two disjoint subsets of random variables. The fourth section includes a simulation study to evaluate +the power of the proposed test in the linear-linear, circular-linear, and circular-circular cases. The +fifth section describes the application of the proposed independence test to real datasets. Finally, +conclusions are presented in the sixth section. +2 +Bivariate Johnson and Wehrly Model and NNTS Family of +Circular Densities +Johnson and Wehrly (1977), and Wehrly and Johnson (1980) developed joint density functions for +bivariate circular-circular, (Θ1, Θ2), and circular-linear (Θ, T) random vectors with Θ an angular +(circular) random variable with a 2π periodic density function with support on the unit circle, and +T is a linear random variable with a density function with support on an interval of the real line. +For the circular-circular case, +fΘ1,Θ2(θ1, θ2) = 2πg(2π(FΘ1(θ1) ± FΘ2(θ2))fΘ1(θ1)fΘ2(θ2). +(4) +For the circular-linear case, +fΘ,T(θ, t) = 2πg(2π(FΘ(θ) ± FT(t))fΘ(θ)fT(t). +(5) + +7 +Function g must be the density function of an angular (circular) random variable in the interval +(0, 2π]. +Fernández-Durán (2004) identified the structure of Johnson and Wehrly model in terms of the +theory of copula functions through the Sklar (1959) theorem (refer Nelsen, 1999) by satisfying +c(u, v) = 2πg(2π(u ± v)) +(6) +where c(u, v) is the density copula, and u and v are variables that take values in the interval (0, 1). +It should be noted that c(u, v) = ∂2C(u,v) +∂u∂v , where C(u, v) is the copula function that corresponds to +a bivariate joint distribution of two identically distributed uniform random variables in the interval +(0, 1]. A linear-linear copula function must be an increasing function that satisfies C(u, 1) = u, +C(1, v) = v, C(0, v) = C(u, 0) = 0. In the case of circular-circular and circular-linear bivariate +copulas, function C has to be periodic, and this is the reason why in the Johnson and Wehrly model, +the joining function g is the density of a circular random variable. +Johnson and Wehrly derived bivariate circular-circular and circular-linear models by consid- +ering conditional arguments. When function g corresponds to a uniform circular density on the +circle, g(θ) = +1 +2π, the joint density of the Johnson and Wehrly model corresponds to the indepen- +dence case in which the joint density of the circular-circular (circular-linear) model is the product +of the marginal univariate densities. +This property of the Johnson and Wehrly model motivated our independence test by considering +the g circular density function as a member of the flexible family of NNTS densities (Fernández- +Durán, 2004b and 2007), which includes uniform circular density as a particular case. In addition, +by using the NNTS family of circular densities, it is possible to generate joining densities g that +are in closer proximity to the circular uniform density as desired; this is explained below. +The circular density function based on nonnegative trigonometric sums (NNTS) for a circular +(angular) random variable Θ ∈ (0, 2π] (refer Fernández-Durán, 2004b) is defined as follows: +fΘ(θ; M, c) = 1 +2π +����� +����� +M +� +k=0 +ckeikθ +����� +����� +2 += 1 +2π +M +� +k=0 +M +� +l=0 +ck¯clei(k−l)θ +(7) + +8 +where i = √−1, ck are complex numbers ck = crk + icck for k = 0, . . . , M and ¯ck = crk − icck is +the conjugate of ck. To obtain a valid density function, fΘ(θ; M, c), which integrates to one, +M +� +k=0 +||ck||2 = 1 +(8) +where cc0 = 0 and cr0 ≥ 0, i.e., c0 is a nonnegative real number. The c parameter space is a +subset of the surface of a hypersphere, because c and −c give the same NNTS density, and the +conjugate of c written in reverse order also gives the same NNTS density as c. There are a total +of 2M free parameters c and M is an additional parameter that determines the total number of +terms in the sum defining the density, which is related to the maximum number of modes that +the density can have. It should be noted that the circular uniform density on (0, 2π] corresponds +to an NNTS density with M = 0, fΘ(θ; M = 0, c) = +1 +2π or equivalently, c = (1, 0, 0, . . . , 0)⊤, +that is, with c0 = 1 and the other elements of c equal to zero. As c0 approaches 1, the NNTS +density converges to a circular uniform density. This property is used to evaluate the power of the +proposed independence test by generating samples from NNTS alternative densities with values of +c0 as close to one as desired. It should be noted that an NNTS model with M = M1 is nested on +NNTS models with M = M2 such that M2 > M1. Fernández-Durán and Gregorio-Domínguez +(2010) developed an efficient algorithm based on optimization algorithms on manifolds to obtain +the maximum likelihood estimates of the c parameters. This algorithm was included with other +routines for the analysis of circular data based on NNTS models in the free R (R Core Team, +2021) package CircNNTSR (Fernández-Durán and Gregorio-Domínguez, 2016). +3 +Proposed Test for Bivariate Independence +For absolutely continuous independent and identically distributed (i.i.d.) circular uniform random +variables, U1, U2, . . . , Ud ∼ U(0, 2π], consider �d +k=1 Uk ∼ U(0, 2π], that is, the sum of i.i.d. +circular uniform random variables is also circular and uniformly distributed (refer Mardia and + +9 +Jupp, 2000 p. 35). The proposed test of independence is based on this result by considering the +angular probability integral transform of arbitrary (linear or circular) absolutely continuous random +variables. Let X1, X2, . . . , Xd be d arbitrary absolutely continuous random variables. The angular +probability transform of Xk is defined as the angular (circular) random variable APIT(Xk) = +2πFXk(Xk), which is uniformly distributed on the unit circle. By considering the null hypothesis +of mutual joint independence, APIT(X1), APIT(X2), . . . , APIT(Xd) are i.i.d. U(0, 2π], then +�d +k=1 ±APIT(Xk) ∼ U(0, 2π] is also circular and uniformly distributed. The proposed test for +bivariate independence is based on testing for the circular uniformity of APIT(X1) + APIT(X2) +(APIT(X1) − APIT(X2)) for absolutely continuous (circular or linear) random variables X1 +and X2. Testing for bivariate independence is equivalent to testing for uniformity of the sum +(difference) of the angular probability integral transforms. To test for circular uniformity of the +sum (difference) of the angular integral transforms we considered the tests of Rayleigh and Pycke +(Pycke, 2010). The Rayleigh test considers an alternative unimodal circular density and has a test +statistic for a sample θ1, θ2, . . . , θn, which is defined as (Mardia and Jupp, 2000) follows: +TRT = 2n ¯R2 +(9) +where ¯R is the sample mean resultant length. The statistic TRT asymptotically has a chi-squared +distribution with two degrees of freedom. The Pycke test considers an alternative multimodal +density, and its test statistic for a sample θ1, θ2, . . . , θn is defined as follows: +TP T = +� 1 +n +� +n +� +i=1 +n +� +j=1 +� +2(cos(θi − θj) − +√ +0.5) +1.5 − (2 +√ +0.5 cos(θi − θj)) +� +. +(10) +The critical values of the Pycke test are obtained via simulation. +The steps of the proposed independence test for two absolutely continuous random variables, +X1 and X2 are as follows. First, the values of the pseudo-observations (empirical distributions), +ˆFX1 and ˆFX2 were calculated from the observed values of each random variable. Second, the angu- +lar probability integral transforms (APITs) were calculated by multiplying the pseudo-observations + +10 +by 2π, (2π ˆFX1 and 2π ˆFX2). Third, the sum (difference) modulus 2π of the two angular probability +integral transforms was calculated. Finally, the Rayleigh or Pycke test was applied to the vector +of the observed values of the sum (difference) of the APITs. In the case of a positive association +between the random variables, the proposed independence test must be applied to the difference of +the APITs and in the case of a negative association, it must be applied to the sum of the APITs. The +case in which there is no prior indication regarding whether the association is positive or negative, +the maximum of the two test statistics values calculated on the sum and difference of the APITs is +used as a final value for the test statistic, and the p-value of the test is obtained. +The extension of a group-wise test of independence for two disjoint subsets of the original +set of random variables, X1, X2, . . . , Xd, divided into the disjoint subsets of random variables D1 +and D2 is equivalent to the bivariate test for independence. In the case of no prior information on +the type of association, either positive or negative, once the observed values of all APITs of all d +original variables are calculated, the Rayleigh and Pycke tests for circular uniformity are applied +to � +Xk∈D1 APIT(Xk)+� +Xm∈D1 APIT(Xm) and � +Xk∈D1 APIT(Xk)−� +Xm∈D1 APIT(Xm) +modulus 2π and the maximum of the test statistics of the two cases is used as the final value of the +test statistic to obtain the p-value of the uniformity test, and hence of the independence test. +Derived from the fitting of an NNTS model with M = 1 to the sum (difference) of APITs, the +following measure of dependence can be defined as follows: +λc0 = M + 1 +M +� +1 − ˆc2 +0 +� += 2 +� +1 − ˆc2 +0 +� +(11) +where the correction term M+1 +M += 2 comes from the fact that the NNTS density with the high- +est concentration around zero has a parameter vector c in which the squared norm of each of its +components is equal to +1 +M+1. This implies that +1 +M+1 ≤ c2 +0 ≤ 1 and λc0 takes values in the inter- +val [0, 1] with values close to zero, implying low dependence (independence) and values closer to +one, further implying high dependence between the considered random variables. The measure of +dependence λc0 is particularly useful in the circular-circular and circular-linear cases. + +11 +4 +Simulation Study +In this section, we present a simulation study to compare the power of the proposed test with the +Wilks test and a test of independence based on the empirical copula. We simulated the data from +different multivariate distributions using known parameters that define the dependence structure +and known marginal densities. We considered the sample sizes of 20, 50, 100, and 200. For a +given significance level α (10%, 5%, and 1%), the powers of the tests were obtained by generating +100 samples of the specified sample size from the bivariate density, by calculating the values of +the test statistics for each of the 100 samples, and determining the number of times that the test +statistics considered a value that rejected the null hypothesis of independence at the given value +of α. Thus, the reported powers of the tests considered values in the range of 0-100 and can be +interpreted in terms of percentages. The Rayleigh test of circular uniformity was performed using +the circular R package (Agostinelli and Lund, 2017). The Pycke test of circular uniformity was +performed using the CircMLE R package (Fitak and Johnsen, 2017 and Landler et al., 2019). The +R package copula was used to calculate the empirical copula test, and the measure of dependence +λc0 was obtained by fitting an NNTS model with M = 1 using the R package CircNNTSR +(Fernández-Durán and Gregorio-Domínguez, 2016). +4.1 +Linear-Linear Models +Tables 1 and 2 list the powers of different tests while simulating samples from a bivariate distri- +bution in which both variables are linear. In the first case, a Gaussian copula was used, and in the +second case a Frank copula was used. + +12 +4.1.1 +Bivariate Gaussian Copula +The bivariate Gaussian (normal) copula correspond to a multivariate distribution, which is defined +as follows: +C(u1, u2) = ΦΓ(Φ−1(u1), Φ−1(u2)) +(12) +where ΦΓ is the multivariate normal distribution with a zero mean vector and correlation matrix Γ +and Φ() is the univariate standard normal distribution function. By using an identity matrix as a +correlation matrix, Γ = I, the independence copula, C(u1, u2) = u1u2, is obtained. +Table 1 compares the powers of the proposed independence test when using a Rayleigh (ART) +and Pycke (APT) circular uniformity tests with respect to those of the Wilks (WT) and empiri- +cal copula (ECT) independence tests when using simulated samples from a bivariate linear-linear +distribution with a Gaussian copula and three different cases of marginal distributions following +Herwatz and Maxand (2020): exponential, Gaussian, and Cauchy. For the Gaussian copula, we +considered five different values of the correlation coefficient ρ (0, 0.25, 0.5, 0.75, and 0.99). When +the correlation coefficient is equal to zero, it corresponds to the null independence hypothesis, and +the reported powers correspond to the sizes of the tests expected to be similar to the significance +levels α (10%, 5%, and 1%). In general terms, both WT and ECT have higher powers than the +proposed ART and APT tests. However, for large sample sizes and high values of the correlation +coefficient (0.75 and 0.99),the ART and APT tests have powers similar to those of the WT and ECT +tests. As expected, for the exponential and Cauchy marginals, the power of the WT reduced when +compared to the Gaussian marginal case for which it was designed; that is, the power of the WT +deteriorates for marginals which are not Gaussian. In the case of Gaussian marginals, the WT for +large sample sizes has high power, and in some cases, its power is larger than the ECT power. In +terms of the sizes of all the tests, it appears that all tests have approximately the correct sizes, given +that a total of 100 samples were used. The fourth column of Table 1 includes the averages of the +values of the dependence measure λc0 which, as expected, increase as the value of the correlation + +13 +coefficient ρ increases. +4.1.2 +Bivariate Frank Copula +The Frank bivariate copula is defined as follows: +C(u, v) = − 1 +ϕ ln +� +1 + (e−ϕu − 1)(e−ϕv − 1) +e−ϕ − 1 +� +(13) +where u, v ∈ (0, 1]. Parameter ϕ assumes values in the interval (0, ∞) and in the limit ϕ = 0, +the Frank copula corresponds to the independence copula. In Table 2, the Gaussian copula of +the previous study is replaced by the Frank copula using five different values of the dependence +parameter ϕ (0, 5, 10, 15, and 50). For the case of independence obtained in the limit when +ϕ = 0, the powers are also obtained when the limit is considered as ϕ = 0. In general, in +Table 2, we obtained the same conclusions as those obtained in Table 1, further indicating that the +characteristics of the proposed tests: ART and APT are more suitable for the circular-circular and +circular-linear cases than to the linear-linear case. +4.2 +Circular-Linear Models +Table 3 includes the powers of the proposed ART and APT tests, and the WT and ECT tests +when simulating samples from the circular-linear model of Johnson and Wehrly with the circular +marginal density being an NNTS density with M = 3, which is plotted in the first plot of Figure 1; +three different linear models (exponential, Gaussian, and Cauchy), and a joining circular density +that corresponds to an NNTS density with M = 3 with five different values of the parameter c0 +(0.7, 0.8, 0.9, 0.99, and 0.9999) to account for different degrees of association between the circular +and linear random variables, as depicted in the last plot of Figure 1, which includes the plots of +the angular joining functions for the five different values of parameter c0. The case c0 = 0.9999 +corresponds to an almost circular uniform density and to the null hypothesis of independence (refer +the last plot of Figure 1). In general, the proposed ART and APT tests demonstrated significantly + +14 +larger powers when compared to the ECT test, which was only similar for large sample sizes +and low values of c0 (0.7, 0.8, and 0.9), further representing highly dependent circular and linear +random variables. The power of the WT was considerably lower when compared to that of the +ART, APT and ECT tests. +4.3 +Circular-Circular Models +For Johnson and Wehrly’s circular-circular model, we used the same angular joining density and +one of the marginal circular densities as that used in the circular-linear model. Figure 1 depicts the +plots of the marginal circular densities that correspond to NNTS densities with M = 3 and M = 2, +and angular joining densities that correspond to NNTS densities with M = 3 for five different val- +ues of the parameter c0 (0.7, 0.8, 0.9, 0.99, and 0.9999) for the circular-circular model of Johnson +and Wehrly. Similar to the results in the circular-linear model, the ART and APT demonstrated +larger power values when compared to the WT and ECT tests. Given the multimodality of all the +circular densities involved, of the two proposed independence tests, APT exhibited a larger power +when compared to ART, which was similar only for the largest sample size of 200. Moreover, for +highly dependent circular random variables (c0 = 0.7, 0.8, or 0.9) and large sample sizes of 100 +or 200, the power of the ECT test was similar to those of the ART and APT tests. The average +values of the dependence measure λc0 listed in the fourth column of Tables 3 and 4 assumed similar +values when c0 =0.7, 0.8, and 0.9 but these values were smaller when c0=0.99 and 0.9999, further +reflecting the fact that values of c0 near one are associated with random variables with a very weak +association. + +15 +5 +Application to Real Circular-Circular and Circular-Linear +Data +5.1 +Test of Bivariate Independence +5.1.1 +Circular-Linear Real Examples +Figure 2 depicts the scatterplots of the considered real examples. We applied the proposed indepen- +dence test to the circular-linear data on wind direction (circular variable) and ozone concentration +(linear variable) originally analyzed by Johnson and Wehrly (1977), and later included them as +dataset B.18 in Fisher (1993). A total of 19 measurements were taken at a weather station in Mil- +waukee at 6 o’clock in the morning every fourth day starting on April 18th and ending on June +29, 1975. The scatterplot of this data is included in the top left plot of Figure 2, which presents +the values of the wind direction and ozone concentration, further indicating a possible positive as- +sociation between the circular and linear variables, and considering the periodicity of the circular +random variable. By applying the Pycke and Rayleigh circular uniformity tests to the difference +of the angular probability transforms of the circular and linear variables, we obtained p-values of +0.0133 and 0.0077, respectively, thus rejecting the null hypothesis of independence (uniformity) +at a 5% significance level for the wind direction and ozone concentration. The value of the de- +pendence measure λc0 was calculated to be 0.4383. Fisher (1993) reached the same conclusion +by considering an expected sine-wave functional form for the conditional expected value of ozone +concentration given the wind direction. +A second example analyzed by Fisher (1993) is a dataset on the directions and distances trav- +elled by 31 small blue periwinkles after undergoing transplantation from their normal place of +living. The top-right plot depicted in Figure 2 includes the scatterplot for this dataset, further indi- +cating a possible negative association between the direction and travelled distance. When applying +the uniformity test to the sum of the angular probability transforms of the direction and distance, + +16 +a p-value of 0.0087 for the Pycke test and a p-value of 0.0096 for the Rayleigh test were obtained, +which rejected independence in accordance with the results obtained by Fisher (1993) when fit- +ting a circular-linear regression model with von Mises errors with non-constant dispersion. The +dependence measure λc0 was calculated to be 0.2668. +5.1.2 +Circular-Circular Real Examples +The first example in the circular-circular test of independence corresponds to pairs of wind di- +rections measured at a weather monitoring station at Milwaukee. The measurements were taken +at 6:00 and 12:00 o’clock for 21 consecutive days and were originally included in Johnson and +Wehrly (1977). The bottom-left plot depicted in Figure 2 includes a scatterplot of the pairs of wind +directions, which indicates a possible positive association between the two angles. Fisher (1993) +listed this dataset as the B.21 dataset, and the main conclusion of Fisher (1993) was that there +exists a strong positive association between the wind directions when applying a hypothesis test +based on a circular-circular correlation coefficient. When applying the proposed methodology to +the difference of the angular probability transforms, a p-value of 0.0148 for the Pycke test and a +p-value of 0.0075 for the Rayleigh test were obtained, which rejected the null hypothesis of in- +dependence between the two angles at a 5% significance level. The dependence measure λc0 was +calculated to be one. +The second example corresponds to 233 pairs (φ,ψ) of dihedral angles in segments alanine- +alanine-alanine of proteins that were originally analyzed by Fernández-Durán (2007) using bivari- +ate NNTS models demonstrating the dependence between the two angles. The scatterplot of these +two angles is included in the bottom-right plot of Figure 2, and the type of association between the +two angles, whether it is positive or negative is not evident. When applying the independence test +to the difference of the angular probability transforms of the two angles, we obtained p-values of +0.0014 and 0.0064 for the Rayleigh and Pycke circular uniformity tests, respectively, thus clearly +rejecting the null hypothesis of independence in favor of a positive association. + +17 +5.2 +Test for Groupwise Independence +As a final example, we applied the proposed groupwise independence test to a dataset on meteo- +rological variables and concentration of pollutants in Mexico City. The measurements are average +values that were measured from 14:30-15:30 h of the wind direction (radians), temperature (de- +grees Celsius), carbon monoxide CO (ppm) and inhalable particles of diameter 10 microns or less +PM10 (ppm) during the spring days from 1993-1999 inclusive at the monitoring station "Pedregal" +in the southwest of Mexico City. A total of 578 observations were analyzed in Fernández-Durán +(2007). Figure 3 depicts the matrix of bivariate scatterplots of four variables. We applied the pro- +posed independence test to test for independence between the set of meteorological variables (wind +direction and temperature) and, the set of concentration of pollutant variables (CO and PM10). We +calculated the angular probability integral transforms of each of the four variables, sum of the an- +gular probability transforms for the meteorological variables wind direction and temperature, and +the corresponding sum of the probability integral transforms for the pollutant concentrations of CO +and PM10. Thereafter, we applied the Rayleigh circular uniformity test for the sum (difference) of +the two previously calculated sums and obtained a p-value of 0.7128 (0.7165), which did not re- +ject the null hypothesis of groupwise independence between the sets of variables {Wind Direction, +Temperature} and {CO, PM10}. The corresponding p-value for the Pycke test was calculated to be +0.5330 (0.8377), further confirming the results of the Rayleigh test. The value of the dependence +measure λc0 was calculated to be 0.0013 (0.0011). +6 +Conclusions +By using the result that the sum of independent circular uniform random variables is circular and +uniformly distributed, a general test of independence based on the angular integral probability +transform was developed. We demonstrated its use, particularly when at least one of the variables +is an angle, that is a circular random variable, further implying that testing for independence in + +18 +the circular case could be equivalent to testing for circular uniformity. From the simulation study +presented in this paper, it is clear that the proposed test is particularly useful for bivariate cases +of circular-circular and circular-linear pairs of random variables with more power than the Wilks +and empirical copula independence tests. We reached this conclusion by simulating samples from +NNTS densities, in which the degree of closeness to the circular uniform distribution was under +control. Although the proposed independence test can be applied to the linear-linear case or for +testing the groupwise independence among two sets of linear random variables, its power is smaller +than that of the commonly used independence tests converging to the power of the common inde- +pendence tests when the sample size increases. The proposed independence test can be put in +practice and it demonstrated superior performance when at least one circular random variable is +among the random variables used to test for independence. 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On the Independence of k Sets of Normally Distributed Statistical Variables. +Econometrica, 3, pp. 309-326. + +24 +0 +1 +2 +3 +4 +5 +6 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +Marginal 1 + M=3 +Angle θ1 +Density Function +0 +1 +2 +3 +4 +5 +6 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +Marginal 2 + M=2 +Angle θ2 +Density Function +0 +1 +2 +3 +4 +5 +6 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +Joining Function + M=3 +Angle θ +Density Function +Figure 1: Circular-circular copula model: The first two plots indicate the marginal NNTS circular +densities (M1 = 3 and M2 = 2) and the last third plot indicates the angular (circular) joining +density for different values of the parameter c0 (0.7, 0.8, 0.9, 0.99, and 0.9999). The case c0 = 1 +corresponds to the circular uniform density (null independence model). + +25 +λc0 = +α = 10% +α = 5% +α = 1% +Marginals +SS +ρ +2(1 − ˆc2 +0) +ART +APT +WT +ECT +ART +APT +WT +ECT +ART +APT +WT +ECT +X1 ∼ Exp(1) +20 +0.99 +1 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +X2 ∼ Exp(2) +20 +0.75 +0.57 +75 +69 +92 +97 +68 +60 +87 +92 +50 +45 +71 +75 +20 +0.5 +0.25 +38 +36 +45 +60 +24 +25 +39 +47 +9 +7 +31 +24 +20 +0.25 +0.15 +17 +20 +17 +29 +10 +8 +13 +15 +3 +4 +7 +3 +20 +0 +0.14 +10 +14 +7 +10 +7 +9 +3 +5 +0 +0 +1 +3 +50 +0.99 +1 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +50 +0.75 +0.48 +99 +98 +100 +100 +99 +96 +100 +100 +91 +89 +100 +100 +50 +0.5 +0.14 +57 +51 +83 +94 +46 +38 +78 +90 +23 +21 +57 +85 +50 +0.25 +0.06 +18 +17 +28 +42 +10 +7 +21 +28 +2 +1 +9 +15 +50 +0 +0.05 +7 +6 +3 +8 +1 +3 +2 +4 +1 +0 +0 +0 +100 +0.99 +1 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +0.75 +0.43 +100 +100 +100 +100 +100 +100 +100 +100 +99 +99 +100 +100 +100 +0.5 +0.11 +79 +76 +99 +99 +72 +61 +99 +99 +50 +43 +91 +98 +100 +0.25 +0.03 +32 +30 +57 +69 +20 +15 +45 +55 +4 +5 +24 +36 +100 +0 +0.02 +8 +12 +4 +10 +4 +3 +1 +5 +1 +1 +0 +1 +200 +0.99 +1 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +200 +0.75 +0.4 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +200 +0.5 +0.08 +99 +98 +100 +100 +97 +97 +100 +100 +88 +82 +100 +100 +200 +0.25 +0.02 +33 +27 +72 +94 +21 +24 +63 +90 +10 +7 +45 +74 +200 +0 +0.01 +10 +13 +2 +6 +6 +7 +0 +1 +1 +1 +0 +0 +X1 ∼ N(0, 1) +20 +0.99 +1 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +X2 ∼ N(0, 1) +20 +0.75 +0.6 +81 +75 +98 +97 +70 +64 +95 +93 +50 +37 +88 +72 +20 +0.5 +0.26 +27 +21 +63 +66 +19 +13 +51 +54 +7 +4 +25 +25 +20 +0.25 +0.14 +13 +10 +9 +21 +4 +7 +4 +12 +1 +2 +1 +2 +20 +0 +0.15 +13 +8 +4 +7 +4 +6 +3 +5 +1 +1 +1 +2 +50 +0.99 +1 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +50 +0.75 +0.49 +99 +99 +100 +100 +98 +97 +100 +100 +96 +92 +100 +100 +50 +0.5 +0.12 +59 +51 +96 +95 +40 +40 +94 +92 +26 +22 +82 +75 +50 +0.25 +0.05 +19 +15 +41 +45 +8 +11 +29 +32 +1 +1 +10 +13 +50 +0 +0.04 +5 +8 +4 +10 +1 +5 +3 +5 +1 +1 +0 +3 +100 +0.99 +1 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +0.75 +0.44 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +0.5 +0.1 +81 +75 +100 +100 +69 +66 +100 +100 +47 +44 +99 +100 +100 +0.25 +0.03 +23 +18 +69 +71 +12 +11 +55 +56 +3 +4 +30 +33 +100 +0 +0.02 +4 +6 +3 +5 +1 +5 +2 +1 +0 +0 +0 +0 +200 +0.99 +1 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +200 +0.75 +0.4 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +200 +0.5 +0.09 +98 +97 +100 +100 +96 +92 +100 +100 +88 +83 +100 +100 +200 +0.25 +0.02 +41 +28 +94 +93 +22 +19 +88 +90 +9 +8 +75 +79 +200 +0 +0.01 +11 +11 +6 +11 +8 +5 +0 +5 +0 +0 +0 +2 +X1 ∼ Cauchy(0, 1) +20 +0.99 +1 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +X2 ∼ Cauchy(0, 1) +20 +0.75 +0.57 +75 +69 +69 +97 +68 +60 +62 +92 +50 +45 +50 +80 +20 +0.5 +0.25 +38 +36 +37 +60 +24 +25 +28 +47 +9 +7 +19 +26 +20 +0.25 +0.15 +17 +20 +13 +30 +10 +8 +8 +14 +3 +4 +6 +6 +20 +0 +0.14 +10 +14 +8 +10 +7 +9 +5 +5 +0 +0 +2 +4 +50 +0.99 +1 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +50 +0.75 +0.48 +99 +98 +79 +100 +99 +96 +76 +100 +91 +89 +61 +100 +50 +0.5 +0.14 +57 +51 +38 +94 +46 +38 +32 +90 +23 +21 +22 +85 +50 +0.25 +0.06 +18 +17 +11 +42 +10 +7 +7 +28 +2 +1 +7 +15 +50 +0 +0.05 +7 +6 +4 +8 +1 +3 +3 +4 +1 +0 +1 +0 +100 +0.99 +1 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +0.75 +0.43 +100 +100 +86 +100 +100 +100 +82 +100 +99 +99 +76 +100 +100 +0.5 +0.11 +79 +76 +45 +99 +72 +61 +41 +99 +50 +43 +32 +98 +100 +0.25 +0.03 +32 +30 +10 +69 +20 +15 +8 +55 +4 +5 +5 +36 +100 +0 +0.02 +8 +12 +4 +10 +4 +3 +3 +5 +1 +1 +2 +1 +200 +0.99 +1 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +200 +0.75 +0.4 +100 +100 +88 +100 +100 +100 +87 +100 +100 +100 +81 +100 +200 +0.5 +0.08 +99 +98 +45 +100 +97 +97 +35 +100 +88 +82 +27 +100 +200 +0.25 +0.02 +33 +27 +11 +94 +21 +24 +7 +90 +10 +7 +4 +74 +200 +0 +0.01 +10 +13 +4 +6 +6 +7 +4 +1 +1 +1 +3 +0 +Table 1: Gaussian copula linear-linear power study: The powers of the proposed test implemented +using the Rayleigh (ART) and Pycke (APT) circular uniformity tests, Wilks (WT) and empirical +copula (ECT) tests are compared when simulating 100 times samples of sizes 20, 50, 100, and +200 from a linear-linear density function constructed from a Gaussian copula and three different +marginals (exponential, Gaussian and Cauchy). The Gaussian copula is defined with an equicor- +related correlation matrix with five different common correlation values of 0, 0.25, 0.5, 0.75, and +0.99. The case with common correlation equal to zero corresponds to the null independence model. + +26 +λc0 = +α = 10% +α = 5% +α = 1% +Marginals +SS +ϕ +2(1 − ˆc2 +0) +ART +APT +WT +ECT +ART +APT +WT +ECT +ART +APT +WT +ECT +X1 ∼ Exp(1) +20 +50 +1 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +X2 ∼ Exp(2) +20 +15 +0.95 +100 +100 +100 +100 +100 +99 +100 +100 +99 +97 +98 +100 +20 +10 +0.83 +97 +95 +98 +100 +96 +90 +96 +99 +88 +78 +88 +98 +20 +5 +0.4 +55 +48 +67 +89 +46 +41 +58 +77 +31 +19 +37 +59 +20 +0 +0.14 +14 +10 +4 +6 +4 +2 +3 +3 +1 +0 +1 +0 +50 +50 +1 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +50 +15 +0.99 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +50 +10 +0.9 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +50 +5 +0.34 +97 +89 +96 +100 +94 +84 +95 +99 +81 +67 +88 +99 +50 +0 +0.05 +14 +12 +7 +13 +6 +7 +5 +7 +1 +2 +1 +1 +100 +50 +1 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +15 +1 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +10 +0.89 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +5 +0.31 +100 +100 +100 +100 +100 +100 +100 +100 +100 +97 +100 +100 +100 +0 +0.02 +9 +9 +4 +9 +4 +4 +3 +4 +1 +0 +1 +0 +200 +50 +1 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +200 +15 +1 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +200 +10 +0.88 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +200 +5 +0.29 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +200 +0 +0.01 +8 +10 +3 +8 +4 +5 +2 +6 +1 +1 +1 +2 +X1 ∼ N(0, 1) +20 +50 +1 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +X2 ∼ N(0, 1) +20 +15 +0.95 +100 +100 +100 +100 +100 +99 +100 +100 +99 +97 +100 +100 +20 +10 +0.83 +97 +95 +100 +100 +96 +90 +99 +99 +88 +78 +99 +97 +20 +5 +0.4 +55 +48 +79 +90 +46 +41 +64 +72 +31 +19 +44 +54 +20 +0 +0.14 +14 +10 +4 +6 +4 +2 +0 +3 +1 +0 +0 +0 +50 +50 +1 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +50 +15 +0.99 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +50 +10 +0.9 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +50 +5 +0.34 +97 +89 +99 +100 +94 +84 +99 +99 +81 +67 +98 +99 +50 +0 +0.05 +14 +12 +7 +13 +6 +7 +2 +7 +1 +2 +0 +1 +100 +50 +1 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +15 +1 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +10 +0.89 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +5 +0.31 +100 +100 +100 +100 +100 +100 +100 +100 +100 +97 +100 +100 +100 +0 +0.02 +9 +9 +3 +9 +4 +4 +2 +4 +1 +0 +0 +0 +200 +50 +1 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +200 +15 +1 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +200 +10 +0.88 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +200 +5 +0.29 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +200 +0 +0.01 +8 +10 +3 +8 +4 +5 +2 +6 +1 +1 +0 +2 +X1 ∼ Cauchy(0, 1) +20 +50 +1 +100 +100 +95 +100 +100 +100 +93 +100 +100 +100 +86 +100 +X2 ∼ Cauchy(0, 1) +20 +15 +0.95 +100 +100 +65 +100 +100 +99 +56 +100 +99 +97 +39 +100 +20 +10 +0.83 +97 +95 +48 +100 +96 +90 +36 +99 +88 +78 +28 +97 +20 +5 +0.4 +55 +48 +25 +90 +46 +41 +21 +72 +31 +19 +14 +54 +20 +0 +0.14 +14 +10 +8 +6 +4 +2 +5 +3 +1 +0 +3 +0 +50 +50 +1 +100 +100 +92 +100 +100 +100 +91 +100 +100 +100 +89 +100 +50 +15 +0.99 +100 +100 +70 +100 +100 +100 +65 +100 +100 +100 +54 +100 +50 +10 +0.9 +100 +100 +58 +100 +100 +100 +51 +100 +100 +100 +38 +100 +50 +5 +0.34 +97 +89 +34 +100 +94 +84 +26 +99 +81 +67 +18 +99 +50 +0 +0.05 +14 +12 +6 +13 +6 +7 +5 +7 +1 +2 +5 +1 +100 +50 +1 +100 +100 +87 +100 +100 +100 +85 +100 +100 +100 +80 +100 +100 +15 +1 +100 +100 +62 +100 +100 +100 +59 +100 +100 +100 +42 +100 +100 +10 +0.89 +100 +100 +47 +100 +100 +100 +41 +100 +100 +100 +31 +100 +100 +5 +0.31 +100 +100 +27 +100 +100 +100 +24 +100 +100 +97 +14 +100 +100 +0 +0.02 +9 +9 +7 +9 +4 +4 +5 +4 +1 +0 +4 +0 +200 +50 +1 +100 +100 +81 +100 +100 +100 +77 +100 +100 +100 +72 +100 +200 +15 +1 +100 +100 +63 +100 +100 +100 +56 +100 +100 +100 +44 +100 +200 +10 +0.88 +100 +100 +52 +100 +100 +100 +42 +100 +100 +100 +30 +100 +200 +5 +0.29 +100 +100 +17 +100 +100 +100 +14 +100 +100 +100 +9 +100 +200 +0 +0.01 +8 +10 +1 +9 +4 +5 +1 +7 +1 +1 +1 +2 +Table 2: Frank copula linear-linear power study: The powers of the proposed test implemented +using the Rayleigh (ART) and Pycke (APT) circular uniformity tests, Wilks (WT) and empirical +copula (ECT) tests are compared when simulating 100 times samples of sizes 20, 50, 100,and 200 +from a linear-linear density function constructed from a Frank copula and three different marginals +(exponential, Gaussian and Cauchy). The Frank copula is defined with five different values of the +dependence parameter ϕ (0, 5, 10, 15, and 50). The limit case with ϕ = 0 corresponds to the null +independence model. + +27 +λc0 = +α = 10% +α = 5% +α = 1% +Marginals +SS +c0 +2(1 − ˆc2 +0) +ART +APT +WT +ECT +ART +APT +WT +ECT +ART +APT +WT +ECT +Θ ∼ NNTS(M = 3) +20 +0.7 +0.61 +91 +89 +15 +43 +89 +88 +8 +32 +68 +64 +1 +3 +X ∼ Exp(1) +20 +0.8 +0.66 +88 +85 +11 +48 +83 +79 +7 +39 +67 +64 +4 +5 +20 +0.9 +0.62 +86 +85 +14 +47 +83 +80 +8 +35 +58 +61 +2 +11 +20 +0.99 +0.22 +34 +35 +6 +11 +28 +26 +2 +9 +10 +7 +2 +2 +20 +0.9999 +0.15 +13 +10 +7 +10 +3 +5 +4 +7 +1 +1 +3 +2 +50 +0.7 +0.46 +100 +100 +13 +79 +100 +99 +6 +57 +97 +97 +1 +28 +50 +0.8 +0.51 +100 +100 +17 +86 +100 +100 +10 +61 +99 +98 +2 +29 +50 +0.9 +0.5 +100 +100 +34 +89 +100 +100 +24 +72 +99 +98 +7 +48 +50 +0.99 +0.15 +71 +81 +10 +40 +64 +70 +3 +20 +40 +41 +0 +3 +50 +0.9999 +0.04 +12 +10 +4 +10 +6 +6 +1 +4 +1 +2 +1 +2 +100 +0.7 +0.45 +100 +100 +24 +100 +100 +100 +17 +100 +100 +100 +8 +89 +100 +0.8 +0.49 +100 +100 +19 +100 +100 +100 +15 +99 +100 +100 +4 +94 +100 +0.9 +0.47 +100 +100 +51 +100 +100 +100 +37 +100 +100 +100 +13 +90 +100 +0.99 +0.12 +90 +95 +13 +55 +82 +92 +9 +41 +66 +80 +0 +18 +100 +0.9999 +0.02 +10 +7 +1 +9 +5 +4 +1 +4 +1 +0 +0 +2 +200 +0.7 +0.44 +100 +100 +53 +100 +100 +100 +41 +100 +100 +100 +15 +100 +200 +0.8 +0.49 +100 +100 +42 +100 +100 +100 +29 +100 +100 +100 +14 +100 +200 +0.9 +0.45 +100 +100 +85 +100 +100 +100 +73 +100 +100 +100 +46 +100 +200 +0.99 +0.11 +100 +100 +17 +97 +100 +100 +11 +79 +99 +100 +4 +55 +200 +0.9999 +0.01 +11 +14 +5 +12 +9 +7 +2 +8 +3 +1 +0 +0 +Θ ∼ NNTS(M = 3) +20 +0.7 +0.62 +89 +86 +8 +36 +79 +80 +2 +25 +55 +52 +1 +4 +X ∼ N(0, 1) +20 +0.8 +0.63 +89 +85 +9 +39 +78 +76 +5 +23 +53 +52 +0 +6 +20 +0.9 +0.64 +94 +88 +14 +48 +86 +78 +6 +37 +62 +57 +5 +9 +20 +0.99 +0.32 +38 +30 +16 +23 +28 +19 +10 +19 +11 +10 +2 +8 +20 +0.9999 +0.16 +13 +18 +6 +14 +7 +8 +5 +12 +1 +2 +1 +3 +50 +0.7 +0.56 +100 +100 +10 +92 +100 +100 +7 +77 +99 +99 +1 +34 +50 +0.8 +0.59 +100 +100 +13 +89 +100 +100 +5 +78 +99 +99 +2 +33 +50 +0.9 +0.49 +100 +100 +16 +85 +100 +100 +10 +64 +98 +99 +4 +30 +50 +0.99 +0.13 +59 +63 +8 +27 +52 +55 +5 +11 +30 +33 +2 +5 +50 +0.9999 +0.04 +14 +12 +7 +10 +5 +6 +1 +5 +2 +2 +0 +1 +100 +0.7 +0.43 +100 +100 +10 +100 +100 +100 +4 +100 +100 +100 +0 +95 +100 +0.8 +0.5 +100 +100 +11 +100 +100 +100 +8 +100 +100 +100 +2 +97 +100 +0.9 +0.48 +100 +100 +31 +100 +100 +100 +21 +98 +100 +100 +8 +93 +100 +0.99 +0.13 +91 +97 +9 +64 +86 +91 +5 +36 +66 +76 +2 +23 +100 +0.9999 +0.02 +10 +14 +0 +13 +5 +7 +0 +3 +1 +1 +0 +2 +200 +0.7 +0.44 +100 +100 +12 +100 +100 +100 +5 +100 +100 +100 +0 +100 +200 +0.8 +0.48 +100 +100 +11 +100 +100 +100 +6 +100 +100 +100 +0 +100 +200 +0.9 +0.46 +100 +100 +53 +100 +100 +100 +41 +100 +100 +100 +21 +100 +200 +0.99 +0.11 +100 +100 +23 +92 +100 +100 +13 +81 +97 +100 +2 +50 +200 +0.9999 +0.01 +11 +7 +1 +4 +6 +3 +0 +1 +0 +0 +0 +0 +Θ ∼ NNTS(M = 3) +20 +0.7 +0.52 +81 +75 +8 +28 +73 +66 +6 +14 +46 +46 +1 +6 +X ∼ Cauchy(0, 1) +20 +0.8 +0.58 +82 +84 +11 +40 +78 +72 +7 +22 +53 +50 +3 +5 +20 +0.9 +0.6 +81 +79 +7 +34 +70 +71 +3 +22 +56 +52 +1 +9 +20 +0.99 +0.26 +34 +37 +8 +21 +22 +25 +5 +9 +7 +10 +2 +4 +20 +0.9999 +0.15 +12 +16 +8 +15 +6 +10 +7 +6 +2 +6 +1 +0 +50 +0.7 +0.49 +100 +100 +5 +84 +100 +100 +2 +68 +99 +99 +1 +35 +50 +0.8 +0.5 +100 +100 +10 +89 +100 +100 +7 +80 +99 +98 +0 +35 +50 +0.9 +0.48 +100 +100 +4 +88 +100 +100 +2 +67 +99 +99 +1 +32 +50 +0.99 +0.13 +62 +69 +5 +28 +52 +56 +2 +19 +26 +26 +1 +5 +50 +0.9999 +0.05 +12 +12 +7 +9 +4 +5 +5 +3 +1 +2 +1 +1 +100 +0.7 +0.45 +100 +100 +7 +100 +100 +100 +4 +100 +100 +100 +1 +89 +100 +0.8 +0.49 +100 +100 +7 +100 +100 +100 +5 +99 +100 +100 +0 +89 +100 +0.9 +0.47 +100 +100 +2 +100 +100 +100 +2 +97 +100 +100 +0 +83 +100 +0.99 +0.12 +97 +94 +6 +52 +88 +94 +5 +34 +74 +84 +1 +15 +100 +0.9999 +0.02 +9 +14 +5 +7 +4 +5 +4 +5 +0 +1 +1 +0 +200 +0.7 +0.43 +100 +100 +3 +100 +100 +100 +1 +100 +100 +100 +0 +100 +200 +0.8 +0.48 +100 +100 +2 +100 +100 +100 +1 +100 +100 +100 +0 +100 +200 +0.9 +0.46 +100 +100 +3 +100 +100 +100 +1 +100 +100 +100 +0 +100 +200 +0.99 +0.1 +100 +100 +2 +92 +99 +100 +1 +84 +97 +100 +1 +56 +200 +0.9999 +0.01 +14 +13 +5 +7 +7 +5 +4 +2 +2 +1 +1 +1 +Table 3: NNTS angular joining density circular-linear power study: The powers of the proposed +test implemented using the Rayleigh (ART) and Pycke (APT) circular uniformity tests, Wilks (WT) +and empirical copula (ECT) tests are compared when simulating 100 times samples of sizes 20, +50, 100, and 200 from a Johnson and Wehrly circular-linear density function constructed from an +NNTS angular joining density with M = 3, an NNTS marginal density function with M = 3 (see +Figure 1) and, three different linear marginals (exponential, Gaussian and Cauchy). The NNTS +angular joining density is defined with five different values of the parameter c0 (0.7, 0.8, 0.9, 0.99, +and 0.9999). The case with c0 = 1 corresponds to the null independence model. + +28 +λc0 = +α = 10% +α = 5% +α = 1% +Marginals +SS +c0 +2(1 − ˆc2 +0) +ART +APT +WT +ECT +ART +APT +WT +ECT +ART +APT +WT +ECT +Θ1 ∼ NNTS(M1 = 3) +20 +0.7 +0.63 +87 +80 +11 +38 +78 +73 +7 +29 +59 +61 +4 +11 +Θ2 ∼ NNTS(M2 = 2) +20 +0.8 +0.6 +84 +81 +15 +41 +72 +74 +13 +28 +57 +57 +5 +10 +20 +0.9 +0.59 +81 +79 +9 +36 +76 +67 +4 +28 +46 +49 +1 +7 +20 +0.99 +0.25 +34 +33 +9 +15 +26 +23 +4 +12 +10 +8 +0 +4 +20 +0.9999 +0.13 +11 +13 +3 +10 +7 +7 +2 +10 +1 +0 +2 +1 +50 +0.7 +0.45 +100 +100 +11 +88 +99 +99 +7 +63 +99 +98 +2 +41 +50 +0.8 +0.5 +100 +100 +10 +91 +100 +100 +8 +70 +98 +98 +1 +38 +50 +0.9 +0.49 +100 +100 +17 +92 +100 +100 +11 +70 +100 +100 +3 +34 +50 +0.99 +0.14 +70 +73 +11 +36 +55 +63 +4 +17 +36 +36 +1 +6 +50 +0.9999 +0.05 +17 +14 +2 +12 +6 +6 +1 +3 +0 +1 +0 +1 +100 +0.7 +0.42 +100 +100 +14 +100 +100 +100 +10 +100 +100 +100 +5 +92 +100 +0.8 +0.47 +100 +100 +13 +100 +100 +100 +9 +100 +100 +100 +3 +93 +100 +0.9 +0.46 +100 +100 +22 +100 +100 +100 +14 +100 +100 +100 +6 +96 +100 +0.99 +0.12 +90 +95 +10 +60 +83 +92 +4 +41 +64 +85 +1 +15 +100 +0.9999 +0.02 +9 +14 +3 +8 +4 +7 +0 +1 +3 +1 +0 +0 +200 +0.7 +0.43 +100 +100 +23 +100 +100 +100 +19 +100 +100 +100 +6 +100 +200 +0.8 +0.48 +100 +100 +18 +100 +100 +100 +13 +100 +100 +100 +4 +100 +200 +0.9 +0.45 +100 +100 +34 +100 +100 +100 +21 +100 +100 +100 +10 +100 +200 +0.99 +0.1 +100 +100 +18 +86 +100 +100 +10 +80 +99 +99 +3 +40 +200 +0.9999 +0.01 +8 +10 +5 +9 +4 +6 +2 +4 +0 +0 +1 +1 +Table 4: NNTS angular joining density circular-circular power study: The powers of the proposed +test implemented using the Rayleigh (ART) and Pycke (APT) circular uniformity tests, Wilks +(WT) and empirical copula (ECT) tests are compared when simulating 100 times samples of sizes +20, 50, 100, and 200 from a Johnson and Wehrly circular-circular density function constructed +from an NNTS angular joining density with M = 3 and NNTS marginal density functions with +M1 = 3 and M2 = 2. The NNTS angular joining density is defined with five different values of +the dependence parameter c0 (0.7, 0.8, 0.9, 0.99, and 0.9999). The case with c0 = 1 corresponds to +the null independence model. The plots of the NNTS angular joining density and NNTS circular +marginal densities are shown in Figure 1. + +29 +0 +1 +2 +3 +4 +5 +6 +20 +40 +60 +80 +100 +Wind and Ozone +Wind Direction +Ozone Concentration +0 +1 +2 +3 +4 +5 +6 +0 +20 +40 +60 +80 +120 +Travelled Direction and Distance +Direction +Travelled Distance +0 +1 +2 +3 +4 +5 +6 +0 +1 +2 +3 +4 +5 +6 +Pairs of Wind Directions +Direction 6:00am +Direction 12:00 noon +0 +1 +2 +3 +4 +5 +6 +0 +1 +2 +3 +4 +5 +6 +Pairs of Dihedral Angles +Angle φ +Angle ψ +Figure 2: Scatterplots of the circular-linear (upper plots) and circular-circular (lower plots) real +datasets. The upper-left scatterplot indicates the wind direction and ozone concentration datapoints +of the dataset of Johnson and Wehrly (1977). The upper-right plot corresponds to the small blue +periwinkles dataset on travelled distance and direction analized by Fisher (1993). The bottom-left +plot corresponds to the Johnson and Wehrly (1977) dataset on pairs of wind directions at 6:00am +and 12:00 noon in a weather monitoring station. Finally, the bottom right includes the pairs of +dihedral angles in segments alanine-alanine-alanine of proteins originally analyzed by Fernández- +Durán (2007). + +30 +Wind Direction +10 +15 +20 +25 +30 +0 +2 +4 +6 +0 +50 +150 +250 +10 +15 +20 +25 +30 +Temperature +CO +1 +2 +3 +4 +5 +6 +0 +50 +150 +250 +0 +2 +4 +6 +1 +2 +3 +4 +5 +6 +PM10 +Figure 3: Scatterplot matrix of the meteorological variables (wind direction and temperature) and +pollutants concentration variables (CO and PM10) registered at a monitoring station at the south- +west of Mexico City. The values of the variables are averages from 14:30-15:30 h for the spring +days from 1993-1999 inclusive. + diff --git a/hdE1T4oBgHgl3EQfMgPQ/content/tmp_files/load_file.txt b/hdE1T4oBgHgl3EQfMgPQ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..59fa26c5fdb8019cd4f02a635234dd2ef5f33f23 --- /dev/null +++ b/hdE1T4oBgHgl3EQfMgPQ/content/tmp_files/load_file.txt @@ -0,0 +1,1098 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf,len=1097 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='02991v1 [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='ME] 8 Jan 2023 Test of Bivariate Independence Based on Angular Probability Integral Transform with Emphasis on Circular-Circular and Circular-Linear Data Fernández-Durán, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' and Gregorio-Domínguez, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' ITAM E-mail: jfdez@itam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='mx Abstract The probability integral transform (PIT) of a random variable X with distribution function FX is a uniformly distributed random variable U = FX(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' We define the angular probabil- ity integral transform (APIT) as θU = 2πU = 2πFX(X), which corresponds to a uniformly distributed angle on the unit circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' For circular (angular) random variables, the sum of abso- lutely continuous independent circular uniform random variables is a circular uniform random variable, that is, the circular uniform distribution is closed under summation, and it is a stable continuous distribution on the unit circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' If we consider the sum (difference) of the angular probability integral transforms of two random variables, X1 and X2, and test for the circular uniformity of their sum (difference), this is equivalent to test of independence of the original variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' In this study, we used a flexible family of nonnegative trigonometric sums (NNTS) circular distributions, which include the uniform circular distribution as a member of the fam- ily, to evaluate the power of the proposed independence test by generating samples from NNTS alternative distributions that could be at a closer proximity with respect to the circular uniform null distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' Keywords: circular-circular dependence;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' circular-linear dependence;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' circular uniformity tests;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' copula, dependence measures 1 2 1 Introduction Testing the independence of a set of random variables is considered as one of the most important task in many practical applications: while estimating the joint distribution of a set of random vari- ables, constructing conditional models to explain one variable in terms of others as in regression models, among many other problems in statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' According to Herwatz and Maxand (2020), one can consider the following tests of indepen- dence: bivariate (pairwise), groupwise, and mutual independence tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' Hereinafter, we only consider absolutely continuous random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' A random variable with a density function with support on an interval of the real line is a linear random variable, and one with support on the unit circle is a circular random variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' For two random variables, X1 and X2, bivari- ate (pairwise) independence tests have null hypothesis, H0 : FX1,X2(x1, x2) = FX1(x1)FX2(x2), where FX1,X2(x1, x2) = P{X1 ≤ x1, X2 ≤ x2} is the bivariate joint distribution function and FX1(x1) = P{X1 ≤ x1} and FX2(x2) = P{X2 ≤ x2} are the corresponding marginal distri- bution functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' For a set D of d (d > 2) random variables, X1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' , Xd, in a groupwise independence test one partitions the set D of d random variables into two nonempty disjoint sub- sets, D1 and D2 such that D1 � D2 = D and D1 � D2 = ∅, and considers the null hypothesis, H0 : FXD1,XD2(xD1, xD2) = FXD1(xD1)FXD2(xD2), where XD1 and XD2 are the vectors of ran- dom variables in the sets D1 and D2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' Groupwise independence tests can be extended to the case of more than two nonempty disjoint subsets of random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' Finally, a mutual independence test for a set of random variables, X1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' , Xd considers the null hypothesis, H0 : FX1,X2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=',Xd(x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' , xd) = �d k=1 FXk(xk), where FX1,X2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=',Xd(x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' , xd) is the joint distribution and FXk(xk) for k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' , d are the marginal univariate distribution functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' If the functional forms of FX1,X2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=',Xd and FXk for k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' , d are specified, a likelihood- ratio test of independence from a sample of random vectors Xi = (Xi1, Xi2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' , Xid)⊤ of size n 3 (i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' , n), can be constructed by considering Λ = −2 ln � maxΘjoint Ljoint(X1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' , Xn) maxΘindep Lindep(X1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' , Xn) � (1) where Ljoint(X1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' , Xn) is the likelihood of the data under joint distribution function FX1,X2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=',Xd with vector of parameters Θjoint and, Lindep(X1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' , Xn) is the likelihood of the data under the independence assumption with the distribution function being the product of the marginal uni- variate distribution functions with vector of parameters Θindep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' The statistic Λ asymptotically and under regularity conditions has a chi-squared distribution with df degrees of freedom, where df is equal to the difference between the dimensions of the parameter vectors Θjoint and Θindep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' The most commonly used test for independence is the chi-squared test of independence for contingency tables, which is not adequate when dealing with absolutely continuous random vari- ables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' Alternatively to the likelihood ratio independence test, other tests for independence were developed by considering nonparametric (distribution free) methods, rank tests (see Hoeffding, 1948 and Kendall and Stuart,1951), and measures of dependence (association) derived from the empirical copula process (Deheuvels, 1981;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' Genest and Rémillard, 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' Genest and Verret, 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' Genest et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' Roy, 2020 and Roy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' The empirical copula, Cn, for a vector of d absolutely continuous linear random variables, X⊤ = (X1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' , Xd)⊤, and a sample of size n is defined as follows: Cn(u1, u2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' , ud) = 1 n n � j=1 I( ˆF1(Xi1) ≤ u1, ˆF2(Xi2) ≤ u2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' , ˆFd(Xid) ≤ u2) (2) where I() is an indicator function, which is equal to one if the condition in its argument is satisfied and zero otherwise, and ˆF1, ˆF2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' , ˆFd are the empirical distribution functions of the random vari- ables X1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' , Xd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' A test of independence based on the distance between the empirical copula and independence copula for absolutely continuous linear random variables is implemented in the R package copula (Hofert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=', 2022 and Kojadinovic and Yan, 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' Among the nonparametric tests of independence, there are family of tests based on some func- tional of the empirical independence process, which is defined as the distance between the empir- 4 ical joint distribution function and product of the empirical univariate distribution functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' His- torically, the most used functionals have been the Cramér-von Mises and Kolmogorov-Smirnov functionals (refer Blum, Keifer and Rosenblatt, 1961;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' DeWet, 1980 and Deheuvels, 1981).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' For example, Hoeffding (1948) considered the Cramér-von Mises functional to generate a rank test of independence between two random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' Modern rank tests of independence have been developed by Kallenberg and Ledwina (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' Kernel-based methods have also been used to es- timate the empirical independence process, as in Pfister et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' Mardia and Kent (1991) used the general Rao score test to generate independence tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' Csörg˝o (1985) developed indepen- dence tests based on the multivariate empirical characteristic function, and Einmahl and McKeague (2003) developed the tests based on the empirical likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' The measures of dependence derived from entropy were defined by Joe (1990) and from mutual information by Berrett and Samworth (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' Tests of independence in specified multivariate distribution functions were simplified many times with respect to the structure of the distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' This is the case for the multivariate Gaussian (normal) distribution where the null correlation implies independence, and for the test of mutual independence under multivariate Gaussian distribution is equivalent to the one used to testing an identity correlation matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' Some of these pairwise tests are the Pearson (1920) product moment correlation coefficient test, Kendall (1938) rank correlation coefficient test, and Spearman (1904) rank correlation coefficient test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' Of course, for a pair of Gaussian random variables, rejecting null correlation implies rejecting pairwise independence, but applying pairwise independence (correla- tion) tests is not adequate to test for mutual independence for a set with more than two Gaussian random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' The Wilks test (Wilks,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' 1935) is an optimal test of independence for multivariate Gaussian populations and for the case of a bivariate groupwise independence test for the vectors XD1 and XD2 with X = X⊤ D1 � D2 = (XD1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' XD2)⊤ considers the following test statistic for a 5 sample of size n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' W = |ˆΣD1 � D2| |ˆΣD1||ˆΣD2| (3) where ˆΣD1 � D2 = �n j=1(xj−¯x)(xj−¯x)⊤ is the estimated covariance matrix of the complete vector of observations x = xD1 � D2 which is partitioned into ˆΣD1 and ˆΣD2 with ˆΣD1 = �n j=1(xD1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='j − ¯xD1)(xD1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='j − ¯xD1)⊤ and ˆΣD2 = �n j=1(xD2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='j − ¯xD2)(xD2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='j − ¯xD2)⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' The statistic W then measures the extent of the distance between the determinant of ˆΣD1 � D2 and the product of the determinants of ˆΣD2 and ˆΣD2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' The equality relationship is satisfied in the multivariate Gaussian population under the null hypothesis of independence between XD1 and XD2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' Asymptotically and under regularity conditions, −n ln(W) follows a chi-squared distribution with ♯D1♯D2 degrees of freedom, where ♯D1 and ♯D2 are the cardinalities of sets D1 and D2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' For the circular-circular (angular-angular) and circular-linear (angular-linear) cases, in which the objective is to test for bivariate independence between two circular random variables and one circular and one linear random variable, respectively, independence tests were developed by con- sidering the specification of measures of dependence and studying their (asymptotic) distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' By applying Kendall’s tau and Spearman’s rho general measures of dependence based on the con- cept of concordance, or the construction of distribution-free correlation coefficients based on ranks to a pair of circular random variables or a circular and linear random variable, tests of indepen- dence were developed by Fisher and Lee (1981, 1982 and 1983) and reviewed by Fisher (1993) and Mardia and Jupp (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' The objective of this study is to develop a test of bivariate (pairwise) independence for two random variables by considering the angular probability transform of each variables, which corre- spond to circular uniform distributions on (0, 2π], and an additional result of the theory of circular statistics (see Fisher, 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' Upton and Fingleton, 1989;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' Mardia and Jupp, 2000 and Jammala- madaka and SenGupta, 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' The test was evaluated using flexible nonnegative trigonometric sums (NNTS) distributions (Fernández-Durán, 2004b, 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' Although the proposed test of in- dependence is a general one, it is especially suitable to test for the independence in the circular- 6 circular and circular-linear cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' Thus, a measure of dependence was developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' Once the bivari- ate test was developed, it was extended to a group-wise independence test for two disjoint subsets of random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' This paper is divided into six sections, including the introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' In the second section, John- son and Wehrly (1977) model is presented as a motivation for performing the test of bivariate independence for two random variables, and here, the theory of NNTS circular distributions is included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' The third section presents the proposed bivariate independence test, a measure of de- pendence, and its application to simulated data to study the power of the test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' In this section, we explain how to extend the test of bivariate independence to a test of groupwise independence for two disjoint subsets of random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' The fourth section includes a simulation study to evaluate the power of the proposed test in the linear-linear, circular-linear, and circular-circular cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' The fifth section describes the application of the proposed independence test to real datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' Finally, conclusions are presented in the sixth section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' 2 Bivariate Johnson and Wehrly Model and NNTS Family of Circular Densities Johnson and Wehrly (1977), and Wehrly and Johnson (1980) developed joint density functions for bivariate circular-circular, (Θ1, Θ2), and circular-linear (Θ, T) random vectors with Θ an angular (circular) random variable with a 2π periodic density function with support on the unit circle, and T is a linear random variable with a density function with support on an interval of the real line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' For the circular-circular case, fΘ1,Θ2(θ1, θ2) = 2πg(2π(FΘ1(θ1) ± FΘ2(θ2))fΘ1(θ1)fΘ2(θ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' (4) For the circular-linear case, fΘ,T(θ, t) = 2πg(2π(FΘ(θ) ± FT(t))fΘ(θ)fT(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' (5) 7 Function g must be the density function of an angular (circular) random variable in the interval (0, 2π].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' Fernández-Durán (2004) identified the structure of Johnson and Wehrly model in terms of the theory of copula functions through the Sklar (1959) theorem (refer Nelsen, 1999) by satisfying c(u, v) = 2πg(2π(u ± v)) (6) where c(u, v) is the density copula, and u and v are variables that take values in the interval (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' It should be noted that c(u, v) = ∂2C(u,v) ∂u∂v , where C(u, v) is the copula function that corresponds to a bivariate joint distribution of two identically distributed uniform random variables in the interval (0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' A linear-linear copula function must be an increasing function that satisfies C(u, 1) = u, C(1, v) = v, C(0, v) = C(u, 0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' In the case of circular-circular and circular-linear bivariate copulas, function C has to be periodic, and this is the reason why in the Johnson and Wehrly model, the joining function g is the density of a circular random variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' Johnson and Wehrly derived bivariate circular-circular and circular-linear models by consid- ering conditional arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' When function g corresponds to a uniform circular density on the circle, g(θ) = 1 2π, the joint density of the Johnson and Wehrly model corresponds to the indepen- dence case in which the joint density of the circular-circular (circular-linear) model is the product of the marginal univariate densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' This property of the Johnson and Wehrly model motivated our independence test by considering the g circular density function as a member of the flexible family of NNTS densities (Fernández- Durán, 2004b and 2007), which includes uniform circular density as a particular case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' In addition, by using the NNTS family of circular densities, it is possible to generate joining densities g that are in closer proximity to the circular uniform density as desired;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' this is explained below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' The circular density function based on nonnegative trigonometric sums (NNTS) for a circular (angular) random variable Θ ∈ (0, 2π] (refer Fernández-Durán, 2004b) is defined as follows: fΘ(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' M, c) = 1 2π ����� ����� M � k=0 ckeikθ ����� ����� 2 = 1 2π M � k=0 M � l=0 ck¯clei(k−l)θ (7) 8 where i = √−1, ck are complex numbers ck = crk + icck for k = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' , M and ¯ck = crk − icck is the conjugate of ck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' To obtain a valid density function, fΘ(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' M, c), which integrates to one, M � k=0 ||ck||2 = 1 (8) where cc0 = 0 and cr0 ≥ 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=', c0 is a nonnegative real number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' The c parameter space is a subset of the surface of a hypersphere, because c and −c give the same NNTS density, and the conjugate of c written in reverse order also gives the same NNTS density as c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' There are a total of 2M free parameters c and M is an additional parameter that determines the total number of terms in the sum defining the density, which is related to the maximum number of modes that the density can have.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' It should be noted that the circular uniform density on (0, 2π] corresponds to an NNTS density with M = 0, fΘ(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' M = 0, c) = 1 2π or equivalently, c = (1, 0, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' , 0)⊤, that is, with c0 = 1 and the other elements of c equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' As c0 approaches 1, the NNTS density converges to a circular uniform density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' This property is used to evaluate the power of the proposed independence test by generating samples from NNTS alternative densities with values of c0 as close to one as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' It should be noted that an NNTS model with M = M1 is nested on NNTS models with M = M2 such that M2 > M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' Fernández-Durán and Gregorio-Domínguez (2010) developed an efficient algorithm based on optimization algorithms on manifolds to obtain the maximum likelihood estimates of the c parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' This algorithm was included with other routines for the analysis of circular data based on NNTS models in the free R (R Core Team, 2021) package CircNNTSR (Fernández-Durán and Gregorio-Domínguez, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' 3 Proposed Test for Bivariate Independence For absolutely continuous independent and identically distributed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=') circular uniform random variables, U1, U2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' , Ud ∼ U(0, 2π], consider �d k=1 Uk ∼ U(0, 2π], that is, the sum of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' circular uniform random variables is also circular and uniformly distributed (refer Mardia and 9 Jupp, 2000 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' 35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' The proposed test of independence is based on this result by considering the angular probability integral transform of arbitrary (linear or circular) absolutely continuous random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' Let X1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' , Xd be d arbitrary absolutely continuous random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' The angular probability transform of Xk is defined as the angular (circular) random variable APIT(Xk) = 2πFXk(Xk), which is uniformly distributed on the unit circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' By considering the null hypothesis of mutual joint independence, APIT(X1), APIT(X2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' , APIT(Xd) are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' U(0, 2π], then �d k=1 ±APIT(Xk) ∼ U(0, 2π] is also circular and uniformly distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' The proposed test for bivariate independence is based on testing for the circular uniformity of APIT(X1) + APIT(X2) (APIT(X1) − APIT(X2)) for absolutely continuous (circular or linear) random variables X1 and X2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' Testing for bivariate independence is equivalent to testing for uniformity of the sum (difference) of the angular probability integral transforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' To test for circular uniformity of the sum (difference) of the angular integral transforms we considered the tests of Rayleigh and Pycke (Pycke, 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' The Rayleigh test considers an alternative unimodal circular density and has a test statistic for a sample θ1, θ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' , θn, which is defined as (Mardia and Jupp, 2000) follows: TRT = 2n ¯R2 (9) where ¯R is the sample mean resultant length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' The statistic TRT asymptotically has a chi-squared distribution with two degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' The Pycke test considers an alternative multimodal density, and its test statistic for a sample θ1, θ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' , θn is defined as follows: TP T = � 1 n � n � i=1 n � j=1 � 2(cos(θi − θj) − √ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='5) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='5 − (2 √ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='5 cos(θi − θj)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' (10) The critical values of the Pycke test are obtained via simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' The steps of the proposed independence test for two absolutely continuous random variables, X1 and X2 are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' First, the values of the pseudo-observations (empirical distributions), ˆFX1 and ˆFX2 were calculated from the observed values of each random variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' Second, the angu- lar probability integral transforms (APITs) were calculated by multiplying the pseudo-observations 10 by 2π, (2π ˆFX1 and 2π ˆFX2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' Third, the sum (difference) modulus 2π of the two angular probability integral transforms was calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' Finally, the Rayleigh or Pycke test was applied to the vector of the observed values of the sum (difference) of the APITs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' In the case of a positive association between the random variables, the proposed independence test must be applied to the difference of the APITs and in the case of a negative association, it must be applied to the sum of the APITs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' The case in which there is no prior indication regarding whether the association is positive or negative, the maximum of the two test statistics values calculated on the sum and difference of the APITs is used as a final value for the test statistic, and the p-value of the test is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' The extension of a group-wise test of independence for two disjoint subsets of the original set of random variables, X1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' , Xd, divided into the disjoint subsets of random variables D1 and D2 is equivalent to the bivariate test for independence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' In the case of no prior information on the type of association,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' either positive or negative,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' once the observed values of all APITs of all d original variables are calculated,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' the Rayleigh and Pycke tests for circular uniformity are applied to � Xk∈D1 APIT(Xk)+� Xm∈D1 APIT(Xm) and � Xk∈D1 APIT(Xk)−� Xm∈D1 APIT(Xm) modulus 2π and the maximum of the test statistics of the two cases is used as the final value of the test statistic to obtain the p-value of the uniformity test,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' and hence of the independence test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' Derived from the fitting of an NNTS model with M = 1 to the sum (difference) of APITs, the following measure of dependence can be defined as follows: λc0 = M + 1 M � 1 − ˆc2 0 � = 2 � 1 − ˆc2 0 � (11) where the correction term M+1 M = 2 comes from the fact that the NNTS density with the high- est concentration around zero has a parameter vector c in which the squared norm of each of its components is equal to 1 M+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' This implies that 1 M+1 ≤ c2 0 ≤ 1 and λc0 takes values in the inter- val [0, 1] with values close to zero, implying low dependence (independence) and values closer to one, further implying high dependence between the considered random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' The measure of dependence λc0 is particularly useful in the circular-circular and circular-linear cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' 11 4 Simulation Study In this section, we present a simulation study to compare the power of the proposed test with the Wilks test and a test of independence based on the empirical copula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' We simulated the data from different multivariate distributions using known parameters that define the dependence structure and known marginal densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' We considered the sample sizes of 20, 50, 100, and 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' For a given significance level α (10%, 5%, and 1%), the powers of the tests were obtained by generating 100 samples of the specified sample size from the bivariate density, by calculating the values of the test statistics for each of the 100 samples, and determining the number of times that the test statistics considered a value that rejected the null hypothesis of independence at the given value of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' Thus, the reported powers of the tests considered values in the range of 0-100 and can be interpreted in terms of percentages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' The Rayleigh test of circular uniformity was performed using the circular R package (Agostinelli and Lund, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' The Pycke test of circular uniformity was performed using the CircMLE R package (Fitak and Johnsen, 2017 and Landler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' The R package copula was used to calculate the empirical copula test, and the measure of dependence λc0 was obtained by fitting an NNTS model with M = 1 using the R package CircNNTSR (Fernández-Durán and Gregorio-Domínguez, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='1 Linear-Linear Models Tables 1 and 2 list the powers of different tests while simulating samples from a bivariate distri- bution in which both variables are linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' In the first case, a Gaussian copula was used, and in the second case a Frank copula was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' 12 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='1 Bivariate Gaussian Copula The bivariate Gaussian (normal) copula correspond to a multivariate distribution, which is defined as follows: C(u1, u2) = ΦΓ(Φ−1(u1), Φ−1(u2)) (12) where ΦΓ is the multivariate normal distribution with a zero mean vector and correlation matrix Γ and Φ() is the univariate standard normal distribution function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' By using an identity matrix as a correlation matrix, Γ = I, the independence copula, C(u1, u2) = u1u2, is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' Table 1 compares the powers of the proposed independence test when using a Rayleigh (ART) and Pycke (APT) circular uniformity tests with respect to those of the Wilks (WT) and empiri- cal copula (ECT) independence tests when using simulated samples from a bivariate linear-linear distribution with a Gaussian copula and three different cases of marginal distributions following Herwatz and Maxand (2020): exponential, Gaussian, and Cauchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' For the Gaussian copula, we considered five different values of the correlation coefficient ρ (0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='75, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='99).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' When the correlation coefficient is equal to zero, it corresponds to the null independence hypothesis, and the reported powers correspond to the sizes of the tests expected to be similar to the significance levels α (10%, 5%, and 1%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' In general terms, both WT and ECT have higher powers than the proposed ART and APT tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' However, for large sample sizes and high values of the correlation coefficient (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='75 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='99),the ART and APT tests have powers similar to those of the WT and ECT tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' As expected, for the exponential and Cauchy marginals, the power of the WT reduced when compared to the Gaussian marginal case for which it was designed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' that is, the power of the WT deteriorates for marginals which are not Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' In the case of Gaussian marginals, the WT for large sample sizes has high power, and in some cases, its power is larger than the ECT power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' In terms of the sizes of all the tests, it appears that all tests have approximately the correct sizes, given that a total of 100 samples were used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' The fourth column of Table 1 includes the averages of the values of the dependence measure λc0 which, as expected, increase as the value of the correlation 13 coefficient ρ increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='2 Bivariate Frank Copula The Frank bivariate copula is defined as follows: C(u, v) = − 1 ϕ ln � 1 + (e−ϕu − 1)(e−ϕv − 1) e−ϕ − 1 � (13) where u, v ∈ (0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' Parameter ϕ assumes values in the interval (0, ∞) and in the limit ϕ = 0, the Frank copula corresponds to the independence copula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' In Table 2, the Gaussian copula of the previous study is replaced by the Frank copula using five different values of the dependence parameter ϕ (0, 5, 10, 15, and 50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' For the case of independence obtained in the limit when ϕ = 0, the powers are also obtained when the limit is considered as ϕ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' In general, in Table 2, we obtained the same conclusions as those obtained in Table 1, further indicating that the characteristics of the proposed tests: ART and APT are more suitable for the circular-circular and circular-linear cases than to the linear-linear case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='2 Circular-Linear Models Table 3 includes the powers of the proposed ART and APT tests, and the WT and ECT tests when simulating samples from the circular-linear model of Johnson and Wehrly with the circular marginal density being an NNTS density with M = 3, which is plotted in the first plot of Figure 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' three different linear models (exponential, Gaussian, and Cauchy), and a joining circular density that corresponds to an NNTS density with M = 3 with five different values of the parameter c0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='7, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='8, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='9, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='99, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='9999) to account for different degrees of association between the circular and linear random variables, as depicted in the last plot of Figure 1, which includes the plots of the angular joining functions for the five different values of parameter c0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' The case c0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='9999 corresponds to an almost circular uniform density and to the null hypothesis of independence (refer the last plot of Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' In general, the proposed ART and APT tests demonstrated significantly 14 larger powers when compared to the ECT test, which was only similar for large sample sizes and low values of c0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='7, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='8, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='9), further representing highly dependent circular and linear random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' The power of the WT was considerably lower when compared to that of the ART, APT and ECT tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='3 Circular-Circular Models For Johnson and Wehrly’s circular-circular model, we used the same angular joining density and one of the marginal circular densities as that used in the circular-linear model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' Figure 1 depicts the plots of the marginal circular densities that correspond to NNTS densities with M = 3 and M = 2, and angular joining densities that correspond to NNTS densities with M = 3 for five different val- ues of the parameter c0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='7, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='8, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='9, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='99, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='9999) for the circular-circular model of Johnson and Wehrly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' Similar to the results in the circular-linear model, the ART and APT demonstrated larger power values when compared to the WT and ECT tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' Given the multimodality of all the circular densities involved, of the two proposed independence tests, APT exhibited a larger power when compared to ART, which was similar only for the largest sample size of 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' Moreover, for highly dependent circular random variables (c0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='7, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='8, or 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='9) and large sample sizes of 100 or 200, the power of the ECT test was similar to those of the ART and APT tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' The average values of the dependence measure λc0 listed in the fourth column of Tables 3 and 4 assumed similar values when c0 =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='7, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='8, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='9 but these values were smaller when c0=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='99 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='9999, further reflecting the fact that values of c0 near one are associated with random variables with a very weak association.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' 15 5 Application to Real Circular-Circular and Circular-Linear Data 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='1 Test of Bivariate Independence 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='1 Circular-Linear Real Examples Figure 2 depicts the scatterplots of the considered real examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' We applied the proposed indepen- dence test to the circular-linear data on wind direction (circular variable) and ozone concentration (linear variable) originally analyzed by Johnson and Wehrly (1977), and later included them as dataset B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='18 in Fisher (1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' A total of 19 measurements were taken at a weather station in Mil- waukee at 6 o’clock in the morning every fourth day starting on April 18th and ending on June 29, 1975.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' The scatterplot of this data is included in the top left plot of Figure 2, which presents the values of the wind direction and ozone concentration, further indicating a possible positive as- sociation between the circular and linear variables, and considering the periodicity of the circular random variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' By applying the Pycke and Rayleigh circular uniformity tests to the difference of the angular probability transforms of the circular and linear variables, we obtained p-values of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='0133 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='0077, respectively, thus rejecting the null hypothesis of independence (uniformity) at a 5% significance level for the wind direction and ozone concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' The value of the de- pendence measure λc0 was calculated to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='4383.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' Fisher (1993) reached the same conclusion by considering an expected sine-wave functional form for the conditional expected value of ozone concentration given the wind direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' A second example analyzed by Fisher (1993) is a dataset on the directions and distances trav- elled by 31 small blue periwinkles after undergoing transplantation from their normal place of living.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' The top-right plot depicted in Figure 2 includes the scatterplot for this dataset, further indi- cating a possible negative association between the direction and travelled distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' When applying the uniformity test to the sum of the angular probability transforms of the direction and distance, 16 a p-value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='0087 for the Pycke test and a p-value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='0096 for the Rayleigh test were obtained, which rejected independence in accordance with the results obtained by Fisher (1993) when fit- ting a circular-linear regression model with von Mises errors with non-constant dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' The dependence measure λc0 was calculated to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='2668.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='2 Circular-Circular Real Examples The first example in the circular-circular test of independence corresponds to pairs of wind di- rections measured at a weather monitoring station at Milwaukee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' The measurements were taken at 6:00 and 12:00 o’clock for 21 consecutive days and were originally included in Johnson and Wehrly (1977).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' The bottom-left plot depicted in Figure 2 includes a scatterplot of the pairs of wind directions, which indicates a possible positive association between the two angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' Fisher (1993) listed this dataset as the B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='21 dataset, and the main conclusion of Fisher (1993) was that there exists a strong positive association between the wind directions when applying a hypothesis test based on a circular-circular correlation coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' When applying the proposed methodology to the difference of the angular probability transforms, a p-value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='0148 for the Pycke test and a p-value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='0075 for the Rayleigh test were obtained, which rejected the null hypothesis of in- dependence between the two angles at a 5% significance level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' The dependence measure λc0 was calculated to be one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' The second example corresponds to 233 pairs (φ,ψ) of dihedral angles in segments alanine- alanine-alanine of proteins that were originally analyzed by Fernández-Durán (2007) using bivari- ate NNTS models demonstrating the dependence between the two angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' The scatterplot of these two angles is included in the bottom-right plot of Figure 2, and the type of association between the two angles, whether it is positive or negative is not evident.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' When applying the independence test to the difference of the angular probability transforms of the two angles, we obtained p-values of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='0014 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='0064 for the Rayleigh and Pycke circular uniformity tests, respectively, thus clearly rejecting the null hypothesis of independence in favor of a positive association.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' 17 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='2 Test for Groupwise Independence As a final example, we applied the proposed groupwise independence test to a dataset on meteo- rological variables and concentration of pollutants in Mexico City.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' The measurements are average values that were measured from 14:30-15:30 h of the wind direction (radians), temperature (de- grees Celsius), carbon monoxide CO (ppm) and inhalable particles of diameter 10 microns or less PM10 (ppm) during the spring days from 1993-1999 inclusive at the monitoring station "Pedregal" in the southwest of Mexico City.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' A total of 578 observations were analyzed in Fernández-Durán (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' Figure 3 depicts the matrix of bivariate scatterplots of four variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' We applied the pro- posed independence test to test for independence between the set of meteorological variables (wind direction and temperature) and, the set of concentration of pollutant variables (CO and PM10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' We calculated the angular probability integral transforms of each of the four variables, sum of the an- gular probability transforms for the meteorological variables wind direction and temperature, and the corresponding sum of the probability integral transforms for the pollutant concentrations of CO and PM10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' Thereafter, we applied the Rayleigh circular uniformity test for the sum (difference) of the two previously calculated sums and obtained a p-value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='7128 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='7165), which did not re- ject the null hypothesis of groupwise independence between the sets of variables {Wind Direction, Temperature} and {CO, PM10}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' The corresponding p-value for the Pycke test was calculated to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='5330 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='8377), further confirming the results of the Rayleigh test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' The value of the dependence measure λc0 was calculated to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='0013 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='0011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' 6 Conclusions By using the result that the sum of independent circular uniform random variables is circular and uniformly distributed, a general test of independence based on the angular integral probability transform was developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' We demonstrated its use, particularly when at least one of the variables is an angle, that is a circular random variable, further implying that testing for independence in 18 the circular case could be equivalent to testing for circular uniformity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' From the simulation study presented in this paper, it is clear that the proposed test is particularly useful for bivariate cases of circular-circular and circular-linear pairs of random variables with more power than the Wilks and empirical copula independence tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' We reached this conclusion by simulating samples from NNTS densities, in which the degree of closeness to the circular uniform distribution was under control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' Although the proposed independence test can be applied to the linear-linear case or for testing the groupwise independence among two sets of linear random variables, its power is smaller than that of the commonly used independence tests converging to the power of the common inde- pendence tests when the sample size increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' The proposed independence test can be put in practice and it demonstrated superior performance when at least one circular random variable is among the random variables used to test for independence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' Additionally, a new measure of depen- dence was introduced based on the fitting of an NNTS density by considering M = 1 to the sum (difference) of the angular probability integral transforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' Acknowledgements We express our sincere gratitude to the Asociación Mexicana de Cultura, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' for their support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' References [1] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' Agostinelli, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' and Lund, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' R 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' 556-596.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' [41] Sklar, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' (1959).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' Fonctions de Répartition à n Dimensions et Leurs Marges, Publications de l’Institut de Statistique de l’Université de Paris, 8, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' 229–231.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' 23 [42] Spearman, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' (1904).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' The Proof and Measurement of Association between Two Things.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' The American Journal of Psychology, 15 (1), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' 72–101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' [43] Upton, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' and Fingleton, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' (1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' Spatial Data Analysis by Example Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' 2 (Categorical and Directional Data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' Chichester, New York: John Wiley and Sons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' [44] Wehrly, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' and Johnson, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' (1980).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' Bivariate Models for Dependence of Angular Observa- tions and a Related Markov Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' Biometrika, 67 (1), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' 255-256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' [45] Wilks, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' (1935).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' On the Independence of k Sets of Normally Distributed Statistical Variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' Econometrica, 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' 309-326.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' 24 0 1 2 3 4 5 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='5 Marginal 1 M=3 Angle θ1 Density Function 0 1 2 3 4 5 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='5 Marginal 2 M=2 Angle θ2 Density Function 0 1 2 3 4 5 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='5 Joining Function M=3 Angle θ Density Function Figure 1: Circular-circular copula model: The first two plots indicate the marginal NNTS circular densities (M1 = 3 and M2 = 2) and the last third plot indicates the angular (circular) joining density for different values of the parameter c0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='7, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='8, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='9, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='99, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='9999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' The case c0 = 1 corresponds to the circular uniform density (null independence model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' 25 λc0 = α = 10% α = 5% α = 1% Marginals SS ρ 2(1 − ˆc2 0) ART APT WT ECT ART APT WT ECT ART APT WT ECT X1 ∼ Exp(1) 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='99 1 100 100 100 100 100 100 100 100 100 100 100 100 X2 ∼ Exp(2) 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='57 75 69 92 97 68 60 87 92 50 45 71 75 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='5 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='08 99 98 100 100 97 97 100 100 88 82 100 100 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='02 33 27 72 94 21 24 63 90 10 7 45 74 200 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='01 10 13 2 6 6 7 0 1 1 1 0 0 X1 ∼ N(0, 1) 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='99 1 100 100 100 100 100 100 100 100 100 100 100 100 X2 ∼ N(0, 1) 20 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='1 81 75 100 100 69 66 100 100 47 44 99 100 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='03 23 18 69 71 12 11 55 56 3 4 30 33 100 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='02 4 6 3 5 1 5 2 1 0 0 0 0 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='99 1 100 100 100 100 100 100 100 100 100 100 100 100 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='4 100 100 100 100 100 100 100 100 100 100 100 100 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='09 98 97 100 100 96 92 100 100 88 83 100 100 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='02 41 28 94 93 22 19 88 90 9 8 75 79 200 0 0.' metadata={'source': 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20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='15 17 20 13 30 10 8 8 14 3 4 6 6 20 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='14 10 14 8 10 7 9 5 5 0 0 2 4 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='99 1 100 100 100 100 100 100 100 100 100 100 100 100 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='48 99 98 79 100 99 96 76 100 91 89 61 100 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='14 57 51 38 94 46 38 32 90 23 21 22 85 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='06 18 17 11 42 10 7 7 28 2 1 7 15 50 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='05 7 6 4 8 1 3 3 4 1 0 1 0 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='99 1 100 100 100 100 100 100 100 100 100 100 100 100 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='43 100 100 86 100 100 100 82 100 99 99 76 100 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='11 79 76 45 99 72 61 41 99 50 43 32 98 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='03 32 30 10 69 20 15 8 55 4 5 5 36 100 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='02 8 12 4 10 4 3 3 5 1 1 2 1 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='99 1 100 100 100 100 100 100 100 100 100 100 100 100 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='4 100 100 88 100 100 100 87 100 100 100 81 100 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='08 99 98 45 100 97 97 35 100 88 82 27 100 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='02 33 27 11 94 21 24 7 90 10 7 4 74 200 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='01 10 13 4 6 6 7 4 1 1 1 3 0 Table 1: Gaussian copula linear-linear power study: The powers of the proposed test implemented using the Rayleigh (ART) and Pycke (APT) circular uniformity tests, Wilks (WT) and empirical copula (ECT) tests are compared when simulating 100 times samples of sizes 20, 50, 100, and 200 from a linear-linear density function constructed from a Gaussian copula and three different marginals (exponential, Gaussian and Cauchy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' The Gaussian copula is defined with an equicor- related correlation matrix with five different common correlation values of 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='75, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' The case with common correlation equal to zero corresponds to the null independence model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' 26 λc0 = α = 10% α = 5% α = 1% Marginals SS ϕ 2(1 − ˆc2 0) ART APT WT ECT ART APT WT ECT ART APT WT ECT X1 ∼ Exp(1) 20 50 1 100 100 100 100 100 100 100 100 100 100 100 100 X2 ∼ Exp(2) 20 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='95 100 100 100 100 100 99 100 100 99 97 98 100 20 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='83 97 95 98 100 96 90 96 99 88 78 88 98 20 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='4 55 48 67 89 46 41 58 77 31 19 37 59 20 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='14 14 10 4 6 4 2 3 3 1 0 1 0 50 50 1 100 100 100 100 100 100 100 100 100 100 100 100 50 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='99 100 100 100 100 100 100 100 100 100 100 100 100 50 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='9 100 100 100 100 100 100 100 100 100 100 100 100 50 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='34 97 89 96 100 94 84 95 99 81 67 88 99 50 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='05 14 12 7 13 6 7 5 7 1 2 1 1 100 50 1 100 100 100 100 100 100 100 100 100 100 100 100 100 15 1 100 100 100 100 100 100 100 100 100 100 100 100 100 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='89 100 100 100 100 100 100 100 100 100 100 100 100 100 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='31 100 100 100 100 100 100 100 100 100 97 100 100 100 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='02 9 9 4 9 4 4 3 4 1 0 1 0 200 50 1 100 100 100 100 100 100 100 100 100 100 100 100 200 15 1 100 100 100 100 100 100 100 100 100 100 100 100 200 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='88 100 100 100 100 100 100 100 100 100 100 100 100 200 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='29 100 100 100 100 100 100 100 100 100 100 100 100 200 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='01 8 10 3 8 4 5 2 6 1 1 1 2 X1 ∼ N(0, 1) 20 50 1 100 100 100 100 100 100 100 100 100 100 100 100 X2 ∼ N(0, 1) 20 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='95 100 100 100 100 100 99 100 100 99 97 100 100 20 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='83 97 95 100 100 96 90 99 99 88 78 99 97 20 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='4 55 48 79 90 46 41 64 72 31 19 44 54 20 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='14 14 10 4 6 4 2 0 3 1 0 0 0 50 50 1 100 100 100 100 100 100 100 100 100 100 100 100 50 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='99 100 100 100 100 100 100 100 100 100 100 100 100 50 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='9 100 100 100 100 100 100 100 100 100 100 100 100 50 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='34 97 89 99 100 94 84 99 99 81 67 98 99 50 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='05 14 12 7 13 6 7 2 7 1 2 0 1 100 50 1 100 100 100 100 100 100 100 100 100 100 100 100 100 15 1 100 100 100 100 100 100 100 100 100 100 100 100 100 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='89 100 100 100 100 100 100 100 100 100 100 100 100 100 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='31 100 100 100 100 100 100 100 100 100 97 100 100 100 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='02 9 9 3 9 4 4 2 4 1 0 0 0 200 50 1 100 100 100 100 100 100 100 100 100 100 100 100 200 15 1 100 100 100 100 100 100 100 100 100 100 100 100 200 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='88 100 100 100 100 100 100 100 100 100 100 100 100 200 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='29 100 100 100 100 100 100 100 100 100 100 100 100 200 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='01 8 10 3 8 4 5 2 6 1 1 0 2 X1 ∼ Cauchy(0, 1) 20 50 1 100 100 95 100 100 100 93 100 100 100 86 100 X2 ∼ Cauchy(0, 1) 20 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='95 100 100 65 100 100 99 56 100 99 97 39 100 20 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='83 97 95 48 100 96 90 36 99 88 78 28 97 20 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='4 55 48 25 90 46 41 21 72 31 19 14 54 20 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='14 14 10 8 6 4 2 5 3 1 0 3 0 50 50 1 100 100 92 100 100 100 91 100 100 100 89 100 50 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='99 100 100 70 100 100 100 65 100 100 100 54 100 50 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='9 100 100 58 100 100 100 51 100 100 100 38 100 50 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='34 97 89 34 100 94 84 26 99 81 67 18 99 50 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='05 14 12 6 13 6 7 5 7 1 2 5 1 100 50 1 100 100 87 100 100 100 85 100 100 100 80 100 100 15 1 100 100 62 100 100 100 59 100 100 100 42 100 100 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='89 100 100 47 100 100 100 41 100 100 100 31 100 100 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='31 100 100 27 100 100 100 24 100 100 97 14 100 100 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='02 9 9 7 9 4 4 5 4 1 0 4 0 200 50 1 100 100 81 100 100 100 77 100 100 100 72 100 200 15 1 100 100 63 100 100 100 56 100 100 100 44 100 200 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='88 100 100 52 100 100 100 42 100 100 100 30 100 200 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='29 100 100 17 100 100 100 14 100 100 100 9 100 200 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='01 8 10 1 9 4 5 1 7 1 1 1 2 Table 2: Frank copula linear-linear power study: The powers of the proposed test implemented using the Rayleigh (ART) and Pycke (APT) circular uniformity tests, Wilks (WT) and empirical copula (ECT) tests are compared when simulating 100 times samples of sizes 20, 50, 100,and 200 from a linear-linear density function constructed from a Frank copula and three different marginals (exponential, Gaussian and Cauchy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' The Frank copula is defined with five different values of the dependence parameter ϕ (0, 5, 10, 15, and 50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' The limit case with ϕ = 0 corresponds to the null independence model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' 27 λc0 = α = 10% α = 5% α = 1% Marginals SS c0 2(1 − ˆc2 0) ART APT WT ECT ART APT WT ECT ART APT WT ECT Θ ∼ NNTS(M = 3) 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='61 91 89 15 43 89 88 8 32 68 64 1 3 X ∼ Exp(1) 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='8 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='47 100 100 2 100 100 100 2 97 100 100 0 83 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='12 97 94 6 52 88 94 5 34 74 84 1 15 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='9999 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='02 9 14 5 7 4 5 4 5 0 1 1 0 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='43 100 100 3 100 100 100 1 100 100 100 0 100 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='48 100 100 2 100 100 100 1 100 100 100 0 100 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='46 100 100 3 100 100 100 1 100 100 100 0 100 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='1 100 100 2 92 99 100 1 84 97 100 1 56 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='9999 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='01 14 13 5 7 7 5 4 2 2 1 1 1 Table 3: NNTS angular joining density circular-linear power study: The powers of the proposed test implemented using the Rayleigh (ART) and Pycke (APT) circular uniformity tests,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' Wilks (WT) and empirical copula (ECT) tests are compared when simulating 100 times samples of sizes 20,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' 50,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' 100,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' and 200 from a Johnson and Wehrly circular-linear density function constructed from an NNTS angular joining density with M = 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' an NNTS marginal density function with M = 3 (see Figure 1) and,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' three different linear marginals (exponential,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' Gaussian and Cauchy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' The NNTS angular joining density is defined with five different values of the parameter c0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='7, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='8, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='9, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='99, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='9999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' The case with c0 = 1 corresponds to the null independence model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' 28 λc0 = α = 10% α = 5% α = 1% Marginals SS c0 2(1 − ˆc2 0) ART APT WT ECT ART APT WT ECT ART APT WT ECT Θ1 ∼ NNTS(M1 = 3) 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='63 87 80 11 38 78 73 7 29 59 61 4 11 Θ2 ∼ NNTS(M2 = 2) 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='8 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='01 8 10 5 9 4 6 2 4 0 0 1 1 Table 4: NNTS angular joining density circular-circular power study: The powers of the proposed test implemented using the Rayleigh (ART) and Pycke (APT) circular uniformity tests,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' Wilks (WT) and empirical copula (ECT) tests are compared when simulating 100 times samples of sizes 20,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' 50,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' 100,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' and 200 from a Johnson and Wehrly circular-circular density function constructed from an NNTS angular joining density with M = 3 and NNTS marginal density functions with M1 = 3 and M2 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' The NNTS angular joining density is defined with five different values of the dependence parameter c0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='7, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='8, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='9, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='99, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='9999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' The case with c0 = 1 corresponds to the null independence model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' The plots of the NNTS angular joining density and NNTS circular marginal densities are shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='29 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='0 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='Direction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='Travelled Distance ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='4 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='Pairs of Wind Directions ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='Direction 6:00am ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='Direction 12:00 noon ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='1 ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='Pairs of Dihedral Angles ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='Angle φ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='Angle ψ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='Figure 2: Scatterplots of the circular-linear (upper plots) and circular-circular (lower plots) real ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content='datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' The upper-left scatterplot indicates the wind direction and ozone concentration datapoints of the dataset of Johnson and Wehrly (1977).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' The upper-right plot corresponds to the small blue periwinkles dataset on travelled distance and direction analized by Fisher (1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' The bottom-left plot corresponds to the Johnson and Wehrly (1977) dataset on pairs of wind directions at 6:00am and 12:00 noon in a weather monitoring station.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' Finally, the bottom right includes the pairs of dihedral angles in segments alanine-alanine-alanine of proteins originally analyzed by Fernández- Durán (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' 30 Wind Direction 10 15 20 25 30 0 2 4 6 0 50 150 250 10 15 20 25 30 Temperature CO 1 2 3 4 5 6 0 50 150 250 0 2 4 6 1 2 3 4 5 6 PM10 Figure 3: Scatterplot matrix of the meteorological variables (wind direction and temperature) and pollutants concentration variables (CO and PM10) registered at a monitoring station at the south- west of Mexico City.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} +page_content=' The values of the variables are averages from 14:30-15:30 h for the spring days from 1993-1999 inclusive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE1T4oBgHgl3EQfMgPQ/content/2301.02991v1.pdf'} diff --git a/idAyT4oBgHgl3EQfXvfg/content/tmp_files/2301.00191v1.pdf.txt b/idAyT4oBgHgl3EQfXvfg/content/tmp_files/2301.00191v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..e4b222d9a84574434285f11ec23c2b157f855b04 --- /dev/null +++ b/idAyT4oBgHgl3EQfXvfg/content/tmp_files/2301.00191v1.pdf.txt @@ -0,0 +1,2058 @@ +Using Affine Policies to Reformulate Two-Stage Wasserstein +Distributionally Robust Linear Programs to be Independent of +Sample Size∗ +Youngchae Cho +Insoon Yang† +Abstract +Intensively studied in theory as a promising data-driven tool for decision-making under +ambiguity, two-stage distributionally robust optimization (DRO) problems over Wasserstein +balls are not necessarily easy to solve in practice. +This is partly due to large sample size. +In this article, we study a generic two-stage distributionally robust linear program (2-DRLP) +over a 1-Wasserstein ball using an affine policy. The 2-DRLP has right-hand-side uncertainty +with a rectangular support. Our main contribution is to show that the 2-DRLP problem has +a tractable reformulation with a scale independent of sample size. The reformulated problem +can be solved within a pre-defined optimality tolerance using robust optimization techniques. +To reduce the inevitable conservativeness of the affine policy while preserving independence of +sample size, we further develop a method for constructing an uncertainty set with a probabilistic +guarantee over which the Wasserstein ball is re-defined. As an application, we present a novel +unit commitment model for power systems under uncertainty of renewable energy generation to +examine the effectiveness of the proposed 2-DRLP technique. Extensive numerical experiments +demonstrate that our model leads to better out-of-sample performance on average than other +state-of-the-art distributionally robust unit commitment models while staying computationally +competent. +1 +Introduction +Two-stage optimization is a popular tool for sequential decision-making under uncertainty, where +the decision maker makes two kinds of decisions, i.e., here-and-now and wait-and-see decisions, be- +fore and after observing the realization of uncertainty, respectively. Due to its generality, two-stage +optimization has seen many applications in various research fields such as inventory management [2], +workforce management [3], location planning [4], and power system operations [5,6]. In the present +article, we consider a class of two-stage optimization problems based on distributionally robust +optimization (DRO) with the Wasserstein metric. +1.1 +Backgrounds +Two-stage optimization approaches can be conveniently classified by the stochastic optimization +method. Among the most-studied stochastic optimization methods for two-stage optimization are +∗This +work +was +supported +in +part +by +the +National +Research +Foundation +of +Korea +funded +by +MSIT(2020R1C1C1009766, 2021R1A4A2001824), the Information and Communications Technology Planning and +Evaluation grant funded by MSIT(2022-0-00480), and Samsung Electronics. A preliminary version of this work was +presented at the 61st IEEE Conference on Decision and Control [1]. +†A. Hakobyan, and I. Yang are with the Department of Electrical and Computer Engineering and ASRI, Seoul +National University, Seoul, 08826, South Korea {youngchaecho, insoonyang}@snu.ac.kr +1 +arXiv:2301.00191v1 [math.OC] 31 Dec 2022 + +2 +stochastic programming (SP), robust optimization (RO) and DRO. A usual objective of SP is +to minimize the expected total cost, i.e., a sum of the deterministic cost associated with here- +and-now decisions and the expected cost associated with wait-and-see decisions, with respect to +a probability distribution of uncertainty [7]. As the true distribution of uncertainty is difficult to +obtain, an empirical distribution is used instead in most cases. For this reason, SP works well only +with large sample datasets. Without struggling to acquire the true distribution, RO uses worst-case +analyses over an uncertainty set (a set of possible scenarios of uncertainty) with the common aim +of minimizing the worst-case total cost, i.e., a sum of the deterministic cost associated with here- +and-now decisions and the worst-case cost associated with wait-and-see decisions [8]. However, RO +is often overly conservative as it ignores probabilistic features of uncertainty, which can be partially +obtained through samples. +To mitigate the disadvantages of SP and RO simultaneously, DRO uses worst-case analyses for +an ambiguity set, i.e., a family of probability distributions of uncertainty. A typical goal of DRO +is to minimize the expected total cost with respect to worst-case distributions in an ambiguity set. +Incorporating probabilistic features while hedging against the potential inappropriateness of any +single pre-specified distribution, DRO better balances efficiency and robustness compared to SP +and RO. For details of general DRO problems, see, for example, [9] and the references therein. +Performances of DRO greatly depend on how the ambiguity set is chosen. For example, ambigu- +ity sets can be defined using f-divergences [10], e.g., the Kullback–Leibler (KL) divergence [11] and +the total variation distance [12], as well as moment conditions [13,14]. However, these ambiguity +sets have a few limitations. First, ambiguity sets based on f-divergences may not be rich enough as +they include only distributions that are absolutely continuous with respect to a nominal distribu- +tion. Moreover, the underlying assumption of moment information known a priori for DRO based +on moment conditions hardly seems justifiable [13]. Reportedly, moment-based DRO solutions may +also be overly conservative [15]. +Ambiguity sets can be constructed using the Wasserstein metric as well [16,17]. A Wasserstein +ball is defined as a statistical ball in the space of probability distributions, the radius of which is +measured using the Wasserstein metric. Intuitively, the Wasserstein distance of two distributions +is interpreted as the minimum cost of redistributing the probability mass from one distribution to +the other. The center of a Wasserstein ball is mostly an empirical distribution constructed with a +finite number of samples. As the elements of a Wasserstein ball are perturbations of the nominal +distribution that are obtained considering the distance of uncertain scenarios, Wasserstein DRO +does not suffer from the aforementioned drawbacks of DRO based on f-divergences or moment +conditions. Moreover, Wasserstein DRO offers a strong finite-sample performance guarantee [18]. +For these reasons, we focus on two-stage Wasserstein DRO in this article. +1.2 +Related Work +Research works providing solution methods for two-stage Wasserstein DRO in general forms includes +[18–29] all of which, except for [18], consider linear costs of here-and-now and wait-and-see decisions. +Specifically, [19–26] deal with two-stage distributionally robust linear programs (2-DRLPs) over +Wasserstein balls, where the second-stage problem to optimize wait-and-see decisions is a linear +program (LP) while here-and-now decision variables can be integer or continuous. In [19], it is +briefly mentioned that 2-DRLPs over 1-Wasserstein balls can be reformulated as tractable semi- +infinite or finite-dimensional optimization problems if the 1-, 2- or ∞-norm is used as the metric +on the support. In [20], decomposition algorithms are developed for solving exact reformulations of +2-DRLPs over 1-Wasserstein balls with the 1- and ∞-norm, assuming right-hand-side uncertainty +and a rectangular uncertainty set. The algorithms build on Benders decomposition [30] and the + +3 +column-and-constraint generation (C&CG) method [31]. In [21], a second-order conic programming +approach is employed to derive tractable reformulations of 2-DRLPs over 1-Wasserstein balls with +the 2-norm, assuming that uncertainty appears in either the objective function or the constraints. In +[22], cutting-plane algorithms are used to exactly solve 2-DRLPs over 1-Wasserstein balls with either +the generic p-norm for p ≥ 1 or a class of quadratic functions. In [23], 2-DRLPs with the Wasserstein +metric of order 2 are exactly solved using conic programming approaches. In [24], 2-DRLPs over ∞- +Wasserstein balls with the p-norm are approximately solved by applying multiple decision policies, +one for an uncertainty set associated with each sample data point. This scheme achieves optimality +asymptotically, i.e., as the number of samples goes to infinity. In [25], tractable reformulations +of 2-DRLPs over ∞-Wasserstein balls with uncertainty in either the objective function or the +constraints are presented for different continuity conditions on the uncertainty. In [26], a sequential +algorithm is developed for general two-stage DRO problems and applied to 2-DRLPs over 1- and +∞-Wasserstein balls for demonstration. This algorithm creates at each iteration a Wasserstein ball +using only a finite subset of the support as an approximation to the original ambiguity set. With a +new observation added at each iteration, the algorithm is proved to achieve asymptotic optimality. +References [18, 27–29] address more general classes of two-stage Wasserstein DRO problems +than 2-DRLPs. In [27], two-stage distributionally robust conic LPs over 1-Wasserstein balls are +considered, for which a cutting-plane algorithm based on Benders decomposition is suggested. +In [28] and [29], decomposition methods are developed assuming that both here-and-now and wait- +and-see decisions are at least partially binary. The authors of [18] study a class of two-stage DRO +problems over 1-Wasserstein balls where the costs of wait-and-see decisions are written as point- +wise maximums of finitely many concave functions of uncertainty. Using main results, tractable +reformulations of 2-DRLPs over 1-Wasserstein balls with uncertainty in either the objective function +or the right-hand-side of constraints are presented in [18]. +Notably, most of the existing solution methods for two-stage Wasserstein DRO problems, in- +cluding those suggested in [18–29], have a scalability issue regarding sample size, i.e., the number +of historical sample data. In other words, the existing solution methods require more computa- +tional resources for more samples. This implies that two-stage Wasserstein DRO problems may not +yield desired solutions that fully exploit historical data at hand when computational resources are +limited. +1.3 +Contributions +In this article, we study a generic 2-DRLP over a 1-Wasserstein ball, which has right-hand-side +uncertainty with a rectangular support, using an affine policy. +Affine policies are a frequently +used solution method for two-stage optimization problems which impose the linear dependence +of wait-and-see decisions on uncertain parameters. First developed in the context of SP [32–34], +affine policies had been disregarded by the operations research community due to their intrinsic +conservativeness that is hard to meaningfully quantify [35]. A few decades later, however, affine +policies have gained wide attention in the fields of not only SP [36] but also RO [8,37,38] as well +as control theory for dynamical systems [39–44] due to their superior tractability and desirable +properties related to cost performances such as robust invariance [45]. Not only studied in theory, +affine policies have seen many applications thereafter as well, e.g., in portfolio management [46,47] +and power system operations [48–51]. Furthermore, researchers have successfully extended these +approaches by using piecewise affine [52,53], segregated affine [54,55] and polynomial [56] policies. +The main contributions of this study are three-fold. First, we show that the 2-DRLP of our +interest has a tractable reformulation with a scale independent of sample size. For this, we first +recast the worst-case expectation problem nested in it, which is infinite-dimensional, as a finite + +4 +convex program with a scale that grows with sample size. We then aggregate optimization vari- +ables associated with different sample indices, which intuitively represent perturbation of samples, +exploiting the fact that they have the same cost coefficient due to the affine policy. This yields +an LP equivalent to the nested infinite-dimensional program, the scale of which is invariant with +sample size. Finally, using duality in LPs, we obtain a finite-dimensional mixed-integer LP (MILP) +as an exact reformulation of the 2-DRLP. The reformulated problem can be solved up to a pre- +defined precision by RO techniques. We also present a cutting-plane algorithm for the reformulated +problem. As a result, many samples can be efficiently exploited without relying on computation- +ally expensive decomposition algorithms. To the best of our knowledge, our study is the first to +reveal that affine policies can resolve the scalability issue regarding sample size in a general class +of two-stage Wasserstein DRO problems.1 +Meanwhile, the optimality gap incurred by the affine policy can be arbitrarily large when the +size of the Wasserstein ambiguity set is big enough. We assert that it is also true for any value of +the radius, because the optimality gap as a function of the radius is a difference of two concave +functions, which can be neither increasing nor decreasing in general. +To reduce the inevitable +conservativeness of the affine policy, we re-define the Wasserstein ball on an uncertainty set smaller +than the support. Our second main contribution is to design a data-driven method for constructing +an uncertainty set with a bounded worst-case confidence level, over which the Wasserstein ball is +rebuilt. Since the feasibility of the affine policy is guaranteed on a smaller uncertainty set, more +efficient solutions can be obtained by using our method. Unlike existing data-driven methods for +building an uncertainty set with a similar probabilistic guarantee, our method ensures that the +2-DRLP does not depend on sample size. +Finally, to illustrate the applicability and effectiveness of the 2-DRLP approach using an affine +policy for practical decision-making problems, we develop a novel UC model for power systems +under the uncertainty of renewable generation. Extensive numerical experiments demonstrate that +the proposed UC model outperforms not only classical models based on SP and RO but four +state-of-the-art models based on DRO using ambiguity sets with the moment conditions [58], KL +divergence [59], 1-norm distance [60] and cumulative density function (CDF) [61] in terms of out- +of-sample performance, while staying computationally competent. +The rest of this article is organized as follows. In Section 2, we formulate the 2-DRLP of our +interest. In Section 3, we show that the 2-DRLP has a tractable reformulation with a scale indepen- +dent of sample size. Furthermore, we provide a cutting-plane algorithm for solving the reformulated +problem. In Section 4, we explain how to construct an uncertainty set with a probabilistic guaran- +tee, over which we rebuild the Wasserstein ball to reduce conservativeness. In Section 5, we present +the novel UC model based on the 2-DRLP approach using an affine policy and discuss simulation +results. In Section 6, we give concluding remarks. +Notation. We denote by R, R+, and R− the sets of all real numbers, non-negative real numbers, +and non-positive real numbers, respectively. For a natural number n, we denote by 1n, 0n, In, and +On the vector of ones, vector of zeros, identical matrix, and square zero matrix, respectively, all +of dimension n. Furthermore, [·]n represents the nth entry of a vector. We use | · | to denote the +1-norm of a vector or the cardinality of a finite set. We also denote by (·)⊤, δ(·), E, ◦, (·)◦, and V (·) +the transpose of a vector or matrix, Dirac delta distribution centered at a given point, expectation +operator, entrywise product operator for two vectors, interior of a subset of a Euclidean space, and +vertex set of a convex polytope, respectively. +1Although the scalability issue is addressed by [57] for the unit commitment (UC) problem, the method in [57] is +applicable only when the cost of wait-and-see decisions calculated using an affine policy is univariate. In contrast, we +do not impose any special assumption on the affine policy. + +5 +2 +Problem Formulation +In this section, we formulate a two-stage Wasserstein DRO problem of our interest using an affine +policy. To this end, we first consider the 2-DRLP +min +x1∈X1 c⊤ +1 x1 + max +P∈Pε(Ξ) EP [f (x1, ξ)] +(1) +where +f (x1, ξ) := +min +x2∈X2(x1,ξ) c⊤ +2 x2 +(2) +denotes the optimal cost of wait-and-see decisions. Here, ξ ∈ Rm and Ξ ⊂ Rm denote a random +vector and its support, respectively. The support Ξ is a bounded box that is known, i.e., Ξ = [ξ, ξ] +where ξ, ξ ∈ Rm can be obtained using a priori knowledge. We assume that N historical samples +ξ1, . . . , ξN of ξ are available and denote the index set of samples by I := {1, . . . , N}. +In (1), x1 ∈ {0, 1}n11 ×Rn12 and c1 ∈ Rn1 with n1 := n11+n12 represent a here-and-now decision +vector and its cost coefficient vector, respectively. The feasible set X1 of x1 is defined with finitely +many linear inequalities. The symbol Pε (·) denotes a 1-Wasserstein ball on a given uncertainty +set, which is a ball of radius ε > 0 centered at an empirical distribution Pe := +1 +N +� +i∈I δξi in the +space of probability distributions supported on the given uncertainty set. Specifically, we let +Pε (·) := {P ∈ P (·) : d (P, Pe) ≤ ε} +where P (·) represents the family of all probability distributions supported on a given uncertainty +set. Furthermore, d denotes the Wasserstein metric of order 1 defined with the 1-norm, i.e., +d +� +P, P′� +:= +inf +π∈Π(P,P′) +� +Ξ×Ξ +��ξ − ξ′�� π +� +dξ, dξ′� +where Π (·, ·) denotes the set of all joint distributions supported on Ξ × Ξ with marginals equal to +two given distributions. In (2), x2 ∈ Rn2 and c2 ∈ Rn2 represent a wait-and-see decision vector and +its cost coefficient vector, respectively. The feasible set of x2 is defined as +X2 (x1, ξ) := +� +x2 ∈ Rn2 : Ain +1 x1 + Ain +2 x2 + Ain +3 ξ ≤ bin� +where Ain +1 ∈ RL×n1, Ain +2 ∈ RL×n2, Ain +3 ∈ RL×m, and bin ∈ RL. In the above formulations, m, n11, +n2, and L are natural numbers, while n12 is a non-negative integer. Throughout the study, we +assume that (1) is feasible, as is standard in the literature [45]. However, we do not impose the +(relative) complete recourse condition, which is also usual (see, e.g., [27,28]) but might be restrictive +for some real-world problems [24]. +Problem (1) is general enough to model diverse decision-making problems in the real world. +For example, the biomass network design [62], unmanned aerial vehicle network design [63], and +railway scheduling [64] problems have been addressed in the form of (1). +However, it is often +computationally demanding to exactly solve a two-stage optimization problem such as (1) [65]. +In this article, we focus on affine policies that approximately solve (1). Affine policies are a +popular solution method for two-stage optimization problems, where wait-and-see decision variables +are restricted to be affine functions of uncertainty. Due to their computational efficiency, affine +policies have been studied extensively for practical two-stage RO [48, 66, 67] and DRO [68–70] +problems. Specifically, we use the affine function +xa +2 (ξ) := Aξ + a + +6 +as our decision rule for x2, where A ∈ Rn2×m and a ∈ Rn2 are determined simultaneously with x1 +at the first stage. Thus, the 2-DRLP of our interest is formulated as +min +x1∈X1,(A,a)∈A(x1,Ξ) c⊤ +1 x1 + hΞ (A, a) +(3) +where +hΞ (A, a) := +max +P∈Pε(Ξ) EP +� +c⊤ +2 (Aξ + a) +� +(4) +denotes the worst-case expected cost of wait-and-see decisions using the affine policy over Pε (Ξ). +To guarantee that xa +2 is feasible over Ξ, we define +A (x1, Ξ) := +� +(A, a) ∈ Rn2×m × Rn2 : Ain +1 x1 + Ain +2 (Aξ + a) + Ain +3 ξ ≤ bin +∀ξ ∈ Ξ +� +. +In this study, we assume that (3) is feasible.2 +One reason (1) is hard to solve in practice is its scalability issue regarding sample size. In- +tractable in the current form due to the nested infinite-dimensional optimization problem, (1) can +be rewritten in a tractable form using well-studied Wasserstein DRO techniques. However, the +scale of any tractable reformulation of (1) grows with sample size. We present such a tractable +reformulation in the following proposition, which can be proven by duality theory; see, e.g., [18,19]. +Proposition 1. Problem (1) can be rewritten as the two-stage RO problem +min +x1∈X1,λ≥0,η∈RN +c⊤ +1 x1 + λε + 1 +N +� +i∈I +[η]i +s.t. +f (x1, ξ) − λ |ξ − ξi| ≤ [η]i +∀ξ ∈ Ξ, i ∈ I. +(5) +Problem (5) can be solved using decomposition algorithms such as Benders decomposition, the +C&CG algorithm and variants of these methods [20]. In these algorithms, (5) is decomposed into +a master problem and two types of subproblems that are iteratively solved. Each of the master +problem and subproblems is written as an MILP. The scalability issue regarding sample size is +problematic specifically for the following two reasons. First, one of the two subproblems, which +has a size independent of sample size, has to be solved for each sample at each iteration. Second, +a set of decision variables and/or constraints, the number of which is proportional to sample size, +can be added to the master problem at each iteration. As empirically shown in [71], this may well +cause the actual computation time of the master problem to increase superlinearly with sample +size. Moreover, undoubtedly, the master problem with a scale increasing with sample size makes a +decomposition algorithm for (5) susceptible to memory-outage errors when many samples are used. +Considering the superior tractability of affine policies, one natural question arises: Does (3) +suffer from the same scalability issue regarding sample size as (1)? In the following section, we +show that the answer is no, i.e., (3) has a tractable reformulation with a scale independent of +sample size. +Remark 1. The feasibility of (1) implies that any feasible point x1 should be such that X2 (x1, ξ) +is non-empty for any ξ ∈ Ξ, i.e., +ff (x1, ξ) = 0, +∀ξ ∈ Ξ +(6) +2Unless m = 1, however, (3) might be infeasible even when (1) is feasible [45, 54]. In this case, the following +discussions throughout the article do not apply. + +7 +where ff (x1, ξ) is equal to the optimal value of the LP +min +x2∈Rn2,y∈R+ +y +s.t. +Ain +1 x1 + Ain +2 x2 + Ain +3 ξ ≤ bin + Iky. +(7) +In words, ff (x1, ξ) denotes the maximum violation of constraints in (2). By taking the dual for- +mulation of (7), we observe that ff (x1, ξ) is convex in ξ for a fixed x1. Thus, (6) is rewritten as +ff (x1, ξ) = 0, +∀ξ ∈ V (Ξ) . +(8) +We make explicit use of (8) in Section 4, where we construct a Wasserstein ball different from +Pε (Ξ) and (6) may not be implied. +Remark 2. Problem (3) can also express a “multi-stage” DRLP over 1-Wasserstein balls using an +affine policy. Specifically, we consider the multi-stage DRLP +min +x1∈X1 c⊤ +1 x1 + +max +P2∈P2ε (Ξ2) EP2 +� +min +z2∈Z2(x1,ξ2) e⊤ +2 z2 + +max +P3∈P3ε (Ξ3) EP3 +� +min +z3∈Z3(x1,z2,ξ2,ξ3) e⊤ +3 z3 + · · · ++ +max +PT ∈PT +ε (ΞT ) EPT +� +min +zT ∈ZT (x1,z2,...,zT −1,ξ2,...,ξT ) e⊤ +T zT +��� +(9) +where ξt, Ξt and Pt +ε +� +Ξt� +denote a random vector, its rectangular support, and a 1-Wasserstein ball +for each stage t = 2, . . . , T, respectively. Furthermore, zt, Zt, and et denote a real decision vector, +its feasible set defined using a finite number of linear inequalities with right-hand-side uncertainty, +and its cost coefficient vector for each stage t, respectively. Using the affine function +za +t +� +ξ2, . . . , ξt� +:= +t +� +τ=2 +Aτξτ + aτ +as a decision rule for zt in (9), which depends on the realization of uncertainty only up to stage +t, we can formulate a multi-stage problem in the form of (3) for ξ = +� +ξ2, . . . , ξT � +. +Here, the +matrices Aτ and vectors aτ to be determined at the first stage are of appropriate dimensions. +However, it is unclear if affine policies for multi-stage DRLPs over Wasserstein balls with different +assumptions and problem structures, possibly of greater practical importance, lead to (3) in a similar +way. Thus, we focus on the two-stage formulation (1) in this article. For details on general multi- +stage DRO or distributionally robust dynamic programming problems, the reader is referred to, for +example, [72–75]. +3 +Independence of Sample Size +Similar to (1), (3) is intractable in the current form as (4) is infinite-dimensional. In this section, we +prove that (3) has a tractable reformulation with a scale independent of sample size. In particular, +we derive a finite-dimensional MILP equivalent to (3), the scale of which is invariant with sample +size. Subsequently, we present a cutting-plane algorithm for solving the reformulated problem. To +this end, we first prove the following theorem. + +8 +Theorem 1. Problem (4) is rewritten as an LP with a scale independent of N. Specifically, we +have +hΞ (A, a) = +max +˜q+,˜q−∈Rm ++ +c⊤ +2 +� +A +� +˜ξ + ˜q+ − ˜q−� ++ a +� +s.t. +1⊤ +m +� +˜q+ + ˜q−� +≤ ε +(10) +˜q+ ≤ ξ − ˜ξ +(11) +˜q− ≤ ˜ξ − ξ +(12) +where ˜ξ := 1 +N +� +i∈I ξi. +Proof. Let +hΞ (A) := +max +P∈Pε(Ξ) EP +� +c⊤ +2 Aξ +� +. +From Theorem 4.4 in [18], it follows that +hΞ (A) = max +q∈RNm +1 +N +� +i∈I +c⊤ +2 A (ξi + qi) +s.t. +1 +N +� +i∈I +|qi| ≤ ε +(13) +ξ ≤ ξi + qi ≤ ξ +∀i ∈ I +where q := (q1, . . . , qN) is a vector concatenating qi ∈ Rm ++ for all i ∈ I. Introducing auxiliary +decision vectors q+ +i , q− +i ∈ Rm ++ such that qi = q+ +i − q− +i and q+ +i ◦ q− +i = 0m for each i ∈ I to linearize +the norm constraint (13), we observe that +hΞ (A) = +max +q+,q−∈RNm ++ +1 +N +� +i∈I +c⊤ +2 A +� +ξi + q+ +i − q− +i +� +s.t. +1 +N +� +i∈I +1⊤ +m +� +q+ +i + q− +i +� +≤ ε +(14) +q+ +i ≤ ξ − ξi +∀i ∈ I +(15) +q− +i ≤ ξi − ξ +∀i ∈ I +(16) +q+ +i ◦ q− +i = 0m +∀i ∈ I +(17) +where q+ := +� +q+ +1 , . . . , q+ +N +� +and q− := +� +q− +1 , . . . , q− +N +� +. Note that the mutual exclusivity constraint (17) +is redundant and thus can be omitted without affecting optimality. Adding up the N inequalities +in (15) and those in (16) respectively, we further have +hΞ (A) ≤ +max +q+,q−∈RNm ++ +1 +N +� +i∈I +c⊤ +2 A +� +ξi + q+ +i − q− +i +� +s.t. +(14) +� +i∈I +q+ +i ≤ Nξ − +� +i∈I +ξi +(18) +� +i∈I +q− +i ≤ +� +i∈I +ξi − Nξ. +(19) + +9 +In what follows, we show that this holds as equality. For any q+′ = +� +q+′ +1 , . . . , q+′ +N +� +∈ +� +q+ ∈ RNm ++ +: (18) +� +, +there exists q+′′ = +� +q+′′ +1 , . . . , q+′′ +N +� +∈ +� +q+ ∈ RNm ++ +: (15) +� +such that � +i∈I q+′′ +i += � +i∈I q+′ +i . For exam- +ple, one such q+′′ can be obtained by letting +q+′′ +i +:= +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +min +� +� +�ξ − ξi, +� +j∈I +q+′ +j +� +� +� +i = 1 +min +� +� +�ξ − ξi, +� +i∈I +q+′ +i +− +� +j 0. Thus, LP monotonically converges to the optimal value of (3). +Moreover, as |V (Ξ)| < ∞, the algorithm yields a solution optimal within the optimality tolerance of +the off-the-shelf MILP solver in finitely many iterations. We provide a pseudocode of the algorithm +in Algorithm 1. +Note that we do not actually use the optimized affine policy xa +2 (ξ) = A∗ξ + a∗ in any decision- +making stage. Rather, we enjoy only the computational tractability of affine policies when deter- +mining x1 in the first stage. In the second stage, we do not rely on the affine policy to determine x2 +as it may be overly conservative. Instead, we solve (2) for x1 = x∗ +1 to determine x2, the feasibility +of which for any ξ ∈ Ξ is implied by the feasibility of (3). As (2) is a standard LP, we can always +make more efficient wait-and-see decisions compared to using the affine policy. +According to [45], the optimality gap of (1) and (3) incurred by the affine policy can be arbi- +trarily large, when ε is big enough so that (1) is identical to its RO counterpart. This can also be +true for any ε > 0 as discussed in what follows. We first present the following theorem. +Theorem 2. The optimal values of (1) and (3) as a function of ε > 0 are piecewise affine and +concave. +Proof. It is enough to address only the optimal value of (1). For any x1 ∈ X1 such that (8) holds, +λ ≥ 0, and i ∈ I, we consider the problem +max +ξ∈Ξ f (x1, ξ) − λ |ξ − ξi| . +(24) +Introducing decision vectors r+ +i , r− +i ∈ Rm ++ such that ξ = ξi + r+ +i − r− +i , (24) is rewritten as +max +(r+ +i ,r− +i )∈R(σ),σ∈{0,1}m f +� +x1, ξi + r+ +i − r− +i +� +− λ +� +r+ +i + r− +i +� +(25) + +11 +Algorithm 1 Algorithm for (3) +Input: Feasibility tolerance ρ ≥ 0, any ξl1 ∈ V (Ξ) for each l ∈ L +Output: Solution (x∗ +1, A∗, a∗) to (3) +for l ← 1 to L do +Ξv +l1 ← {ξl1} +end for +F1 ← ∞, P ← 1 +while FP > ρ do +Solve (22), (x∗ +1, A∗, a∗) ← (x1P , AP , aP ) +for l ← 1 to L do +Solve (23) +if FlP > ρ then +Ξv +l(P+1) ← Ξv +lP ∪ {ξlP } +else +Ξv +l(P+1) ← Ξv +lP +end if +end for +FP+1 ← maxl FlP , P ← P + 1 +end while +where +R (σ) := +�� +r+ +i , r− +i +� +∈ Rm ++ × Rm ++ : r+ +i ≤ +� +ξ − ξi +� +◦ σ, r− +i ≤ +� +ξi − ξ +� +◦ (1m − σ) +� +. +As the dual of (2) for ξ = ξi +r+ +i −r− +i is a maximization problem with an objective function that is +linear in (r+ +i , r− +i ) for any fixed dual vector, the feasible set R (σ) of (r+ +i , r− +i ) in (25) can be replaced +with its vertex set. This implies that a solution ξ to (24) can be assumed to be equal to ξi, ξ, or ξ +in each entry independently. Thus, (24) is rewritten as the integer program +max +σ+ +i ,σ− +i ∈{0,1}m +f (x1, ξ) − λ |ξ − ξi| +s.t. +ξ = ξi + +� +ξ − ξi +� +◦ σ+ +i − +� +ξi − ξ +� +◦ σ− +i +σ+ +i + σ− +i ≤ 1m. +Based on this equivalence, (5) can be rewritten as a finite-dimensional MILP by introducing wait- +and-see decision vectors associated with uncertain scenarios corresponding to each pair of σ+ +i , σ− +i ∈ +{0, 1}m such that σ+ +i + σ− +i ≤ 1m for each i ∈ I. Thus, the optimal value of (1) can be considered +as a point-wise minimum of finitely many affine functions of ε. Hence the statement holds. +Theorem 2 suggests that the optimality gap between (1) and (3) as a function of ε > 0 is a +difference of two concave functions, which can be neither increasing nor decreasing in general. As +there exists some ε such that the optimality gap can be arbitrarily large, we assert that this holds +for any ε > 0. Although conditions under which affine policies for two-stage Wasserstein DRO +problems can be optimal are studied in [35], we do not have such special assumptions on (1) as +most real-world problems do not satisfy them. To reduce the inevitable conservativeness of the +affine policy, in the following section, we build an uncertainty set smaller than Ξ and re-define the +Wasserstein ball on it. + +12 +4 +Conservativeness Reduction via Wasserstein Ball Refinement +To reduce the conservativeness of the affine policy, we propose to use a Wasserstein ball Pε (Ω) +instead of Pε (Ξ), which is defined on a data-driven uncertainty set Ω := Ξ ∩ Ξa. Here, we define +Ξa := [ξl − ε∆1m, ξu + ε∆1m] +where ξl ∈ Rm and ξu ∈ Rm are the entry-wise minimum and maximum vectors of the samples, +respectively, i.e., [ξl, ξu] is the box hull of the samples. We let ∆ := max {N, β} where β > 0 is a +user-defined parameter. +Built this way, the uncertainty set Ω is endowed with a probabilistic property stated in the +following theorem. +Theorem 3. The worst-case probability of the realization of ξ being outside Ω over Pε (Ξ) is bounded +by ∆−1, i.e., +sup +P∈Pε(Ξ) +P [ξ /∈ Ω] ≤ ∆−1. +Proof. Assume for the proof that ξ has a compact convex support Ξu whose interior contains Ξ∪Ξa. +According to Theorem 4.4 and Corollary 5.3 in [18], +V := +sup +P∈Pε(Ξu) +P [ξ /∈ (Ξa)◦] +is equal to the optimal value of the problem +sup +α∈RN(2N+1) ++ +, +p∈RmN(2N+1) +1 +N +� +i∈I +� +k∈K +αiklk +� +ξi + pik +αik +� +s.t. +1 +N +� +i∈I +� +k∈K +|pik| ≤ ε, +� +k∈K +αik = 1 +∀i ∈ I, +ξi + pik +αik +∈ Ξu +∀k ∈ K, i ∈ I +(26) +where α and p are a vector of αik ∈ R+ and a vector concatenating pik ∈ Rm for all (i, k) ∈ I × K +with K := {1, 2, . . . , 2N + 1}, respectively. Moreover, we define +lk (ξ) := +�1 +if +[ξ]k ≥ [ξu + ε∆]k +− ∞ +otherwise +∀k ∈ I, +lN+k (ξ) := +�1 +if +[ξ]k ≤ [ξl − ε∆]k +− ∞ +otherwise +∀k ∈ I, +and l2N+1 (ξ) := 0. In (26), the conventional extended arithmetics apply. For example, we have +1/0 = ∞, 0/0 = 0, and 0·∞ = 0. The optimal value of (26) is obtained if either [pik]k = εN for any +(i, k) ∈ I×I such that i = arg maxi′∈I [ξi′]k or [pik]k−N = −εN for any (i, k) ∈ I×{N + 1, . . . , 2N} +such that i = arg mini′∈I [ξi′]k, with αik = max {1, N/β} in either case. These cases are when a + +13 +sample originally closest to the boundary of Ξa moves along the shortest path to reach it by ε∆. +Thus, we have V = ∆−1. Further, we observe that +V ≥ +sup +P∈Pε(Ξu) +P [ξ /∈ Ξa] +≥ +sup +P∈Pε(Ξu)∩{P′∈P(Ξu):P′(ξ∈Ξ)=1} +P [ξ /∈ Ξa] += +sup +P∈Pε(Ξu)∩{P′∈P(Ξu):P′(ξ∈Ξ)=1} +P [ξ /∈ Ω] += +sup +P∈Pε(Ξ) +P [ξ /∈ Ω] . +Hence the statement holds. +Replacing the Wasserstein ball Pε (Ξ) in (3) with Pε (Ω), we obtain the problem +min +x1∈X1,(A,a)∈A(x1,Ω) c⊤ +1 x1 + hΩ (A, a) +s.t. +(8) +(27) +where we impose (8) because the second-stage problem should always be feasible, as stated in +Remark 1. Since the feasibility of the affine policy is ensured over Ω ⊆ Ξ, (27) can yield a less +conservative solution than (3). +One of the biggest advantages of constructing Ω in the above-described way is that we can +preserve the independence of sample size discussed in Section 3. If not considering this property, one +might be able to obtain an even smaller uncertainty set with a probabilistic guarantee using existing +methods, e.g., [49] and [76], which are mostly approximations of Wasserstein distributionally robust +chance constraints. However, the existing methods can yield an uncertainty set that does not include +all the historical samples. As we cannot apply an affine policy for samples outside the uncertainty +set, the scalability issue may still exist in this case. Therefore, we develop the new method for +building Ω, which includes all the samples. +Problem (27) can be solved by combining Benders decomposition or the C&CG algorithm for +addressing (8), which consists of many equalities, with the cutting-plane algorithm for solving +(3). +In this article, we choose the C&CG algorithm, which is reportedly faster than Benders +decomposition [31]. In what follows, we explain the resulting iterative algorithm for (27). Some +symbols used to describe the cutting-plane algorithm for (3) in Section 3 may be re-defined. +Based on the discussions in the previous section, we first reformulate (27) as +min +x1∈X1,µ∈M(A), +(A,a)∈Av(x1,Ω) +c⊤ +1 x1 + c⊤ +2 +� +A˜ξ + a +� ++ c⊤ +3,Ωµ +s.t. +(8). +(28) +Subsequently, by introducing a wait-and-see decision vector xf +2q ∈ Rn2 associated with the qth +vertex ξf +q of Ξ for each q ∈ Q := {1, . . . , |V (Ξ)|} to deal with (8), we reformulate (28) as the +finite-dimensional MILP +min +x1∈X1,µ∈M(A),xf +2q, +(A,a)∈Av(x1,Ω) +c⊤ +1 x1 + c⊤ +2 +� +A˜ξ + a +� ++ c⊤ +3,Ωµ +s.t. +Ain +1 x1 + Ain +2 xf +2q + Ain +3 ξf +q ≤ bin +∀q ∈ Q. +(29) +Similar to (3), we decompose the large-scale problem (29) into a master problem and subproblems +that are iteratively solved. For initialization, we select any ξl1 ∈ V (Ω) and define Ωv +l1 := {ξl1} for + +14 +each l ∈ L. We also select any ξf +1 ∈ V (Ξ) and let Q1 := {Q1} with Q1 = 1. At each iteration +P ≥ 1, we solve the master problem +min +x1∈X1,µ∈M(A),xf +2q, +(A,a)∈Av +P(x1,Ωv +P) +c⊤ +1 x1 + c⊤ +2 +� +A˜ξ + a +� ++ c⊤ +3,Ωµ +s.t. +Ain +1 x1 + Ain +2 xf +2q + Ain +3 ξf +q ≤ bin +∀q ∈ QP +(30) +where Ωv +P := (Ωv +1P , . . . , Ωv +LP ). +Let (x1P , AP , aP ) and LP denote the solution corresponding to +(x1, A, a) and optimal value of (30), respectively. Subsequently, we solve the first subproblem +max +ξ∈V(Ξ) ff (x1P , ξ) +(31) +whose solution and optimal value are denoted by ξf +QP +1 and V f +P , respectively. If V f +P > ρ, implying +that (8) is violated, we let QP+1 := QP + 1, QP+1 := QP ∪ {QP+1}, and Ωv +P+1 := Ωv +P . Then, the +iteration step increases and we solve (30) again. Otherwise, we let QP+1 := QP and QP+1 := QP . +Further, we solve the second subproblem (23) for each l ∈ L. The rest of this algorithm works +similarly to the algorithm for (3). Specifically, with ξlP and FlP denoting the solution and optimal +value of (23), respectively, we define Ωv +l(P+1) := Ωv +lP ∪ {ξlP }, if FlP > ρ, and Ωv +l(P+1) := Ωv +lP , +otherwise. +If FP := maxl∈L FlP ≤ ρ, the algorithm stops and (x∗ +1, A∗, a∗) = (x1P , AP , aP ) is +returned as a solution to (27). Otherwise, the iteration step increases and we solve (30) again. In +Algorithm 2, we provide a pseudocode of the algorithm for (27). +Meanwhile, it should be noted that the first subproblem (31) is a max-min problem that is not +easy to handle. To solve (31), we reformulate it as an MILP using the Big M method [77,78]. First, +we rewrite (31) as +max +ζ∈{0,1}m ff � +x1P , ξ + ζ ◦ +� +ξ − ξ +�� +(32) +where each binary vector ζ ∈ {0, 1}m corresponds to a vertex of Ξ. Subsequently, we take the dual +formulation of the inner problem (7) for ξ = ξ + ζ ◦ (ξ − ξ) to obtain a maximization problem with +a bilinear objective function in terms of the dual decision variables and ζ. Finally, we linearize the +bilinear terms by introducing auxiliary integer variables to obtain the MILP equivalent to (31). +As a result, we can solve the master problem and two subproblems of (27) as a finite-dimensional +MILP or LP problem. +Problem (30) is a relaxation of (27) for any iteration step P such that V f +P > 0 or FP > 0. Thus, +LP monotonically converges to the optimal value of (27). Moreover, as |V (Ξ)| = |V (Ω)| < ∞, the +algorithm yields a solution optimal within the optimality tolerance of the off-the-shelf MILP solver +in a finite number of iterations. +In the following section, we examine the applicability and effectiveness of the 2-DRLP formu- +lation (27) using an affine policy for a practical decision-making problem. +5 +Application to Unit Commitment +In this section, we develop a UC model in the form of (27) for power systems under the uncertainty +of renewable energy generation (REG). As a fundamental planning problem for conventional gen- +erators, the UC problem is solved on a daily basis to optimize their commitment status as well as +economic dispatch policies (i.e., the tertiary controllers) given a forecast of the REG and demand. +In the following subsections, we first present a deterministic UC model without considering any +uncertainty to introduce basic decision variables and constraints. Subsequently, we explain how to + +15 +Algorithm 2 Algorithm for (27) +Input: Feasibility tolerance ρ ≥ 0, any ξf +1 ∈ V (Ξ) and ξl1 ∈ V (Ω) for each l ∈ L +Output: Solution (x∗ +1, A∗, a∗) to (27) +for l ← 1 to L do +Ωv +l1 ← {ξl1} +end for +Q1 ← 1, Q1 ← {Q1}, F1 ← ∞, P ← 1 +while FP > ρ do +Solve (30), (x∗ +1, A∗, a∗) ← (x1P , AP , aP ), +Solve (31) +if V f +P > ρ then +QP+1 ← QP + 1, QP+1 ← QP ∪ {QP+1}, +Ωv +P+1 ← Ωv +P , FP+1 ← FP +else +for l ← 1 to L do +Solve (23) +if FlP > ρ then +Ωv +l(P+1) ← Ωv +lP ∪ {ξlP } +else +Ωv +l(P+1) ← Ωv +lP +end if +end for +FP+1 ← maxl FlP +end if +P ← P + 1 +end while +formulate our UC model. Finally, we discuss the results of numerical experiments. Some symbols +used in the previous sections may be re-defined. +5.1 +Deterministic UC +We consider the UC problem for a transmission system of I buses connected by L transmission +lines over a planning horizon of T time periods, the indices of which are denoted by i, l, and t, +respectively. Each bus has a conventional generator, a load, and an REG system, all with the +same index. The demand of the load at each bus in each time period is known a priori, while the +REG is uncertain. We define the forecast error of REG at bus i in time period t as a random +variable ξit. Let ξ denote a vector of ξit for all (i, t). The REG curtailment and demand shedding +are fully allowed with penalties. The transmission network is represented by a DC power flow +model. Assuming that the realization of ξ is given, we formulate a deterministic UC model in this +subsection. +The decision variables of the deterministic UC model are binary variables uo +it, uu +it, and ud +it, +denoting the on/off, start-up, and shut-down status of generator i in time period t, respectively, +and real variables xg +it, xr +it, and xd +it, denoting the conventional generation, REG curtailment, and +demand shedding at bus i in time period t, respectively. Let u ∈ {0, 1}3IT and x ∈ R3IT denote +vectors of the binary and real decision variables, respectively. +The objective of the deterministic UC model is to minimize the total operating cost, i.e., a sum + +16 +of the fixed cost c⊤ +1 u and the variable cost c⊤ +2 x. Here, c1 ∈ R3IT is defined by the no-load, start-up, +and shut-down costs of each generator. Further, c2 ∈ R3IT is defined by the marginal costs of +conventional generation, REG curtailment, and demand shedding at each bus in each time period. +We denote by U the feasible set of u, which is defined by logic constraints among the binary +variables as well as the minimum up and down time constraints of each generator. For a specific +formulation of U, the reader is referred to [1]. The other decision vector x should meet the following +constraints (33)–(37) for all the associated indices (i, l, t). First, the conventional generation is +chosen under the capacity constraint +Xiuo +it ≤ xg +it ≤ Xiuo +it +(33) +where Xi and Xi denote the minimum and maximum possible output of generator i when it is in +operation, respectively, in addition to the ramping constraint +− Xrd +i uo +it − Xsd +i ud +it ≤ xg +it − xg +i(t−1) ≤ Xru +i uo +i(t−1) + Xsu +i uu +it +(34) +where Xrd +i , Xsd +i , Xru +i , and Xsu +i +denote the ramp-down, shut-down-ramp, ramp-up, and start-up- +ramp limits of generator i, respectively. The upper and lower limits of REG curtailment and those +of demand shedding are expressed by +0 ≤ xr +it ≤ wit + ξit, +0 ≤ xd +it ≤ dit +(35) +where wit and dit denote the forecast of REG and the demand at bus i in time period t, respectively. +Furthermore, x should never violate two system-wide constraints, i.e., the transmission capacity +constraint +− Fl ≤ +� +i Fil +� +xg +it + wit + ξit − xr +it − dit + xd +it +� +≤ Fl +(36) +where Fl and Fil denote the maximum possible power flow in transmission line l and the power shift +factor between bus i and transmission line l, respectively, and the power supply–demand balance +condition +� +i +� +xg +it + wit + ξit − xr +it − dit + xd +it +� += 0. +(37) +Compactly, the deterministic UC model is written as +min +u∈U,x∈R3IT +c⊤ +1 u + c⊤ +2 x +s.t. (33)–(37) +∀i, l, t, +which is an MILP that can be easily solved using off-the-shelf solvers. However, ξ is unknown a +priori in practice. To address the uncertainty of ξ, we use the 2-DRLP formulation (27) as explained +in the following subsection. +5.2 +Proposed Model +Our UC model is obtained by applying Wasserstein DRO and an affine policy to the two-stage +robust UC model in [79] modified for the transmission system of our interest. In this subsection, +we formulate the Wasserstein DRO counterpart of the model in [79]. Subsequently, we explain the +affine policy. The complete formulation of the proposed UC model is omitted to avoid redundancy. +We first provide the Wasserstein DRO counterpart of the UC model in [79] +min +u∈U,(xg,xg)∈X g(u) c⊤ +1 u + max +P∈Pε(Ξ) EP [f (xg, xg, ξ)] +(38) + +17 +where +f (xg, xg, ξ) := +min +x∈X(xg,xg,ξ) c⊤ +2 x +(39) +denotes the optimal value of the second-stage problem. The entries of xg ∈ RIT and xg ∈ RIT are +xg +it and xg +it for all (i, t), respectively. Here, xg +it and xg +it denote the allowable upper and lower limits +of conventional generation at bus i in time period t, respectively, which are introduced to enable +the non-anticipative operation of each conventional generator. Specifically, xg +it and xg +it are designed +so that any xg +it such that +xg +it ≤ xg +it ≤ xg +it +(40) +can be implemented independently of the conventional generation at bus i in any other time period +while satisfying the capacity and ramping constraints of generator i. Accordingly, X g (u) is defined +as a set of any (xg, xg) such that the following constraint hold: +� +� +� +� +� +� +� +Xiuo +it ≤ xg +it ≤ xg +it ≤ Xiuo +it +xg +it − xg +i(t−1) ≤ Xru +i uo +i(t−1) + Xsu +i uu +it +xg +i(t−1) − xg +it ≤ Xrd +i uo +it + Xsd +i ud +it +∀i, t. +Given (xg, xg) ∈ X g (u), the feasible set of x in (39) is defined as +X (xg, xg, ξ) := +� +x ∈ R3IT : (40),(35)–(37) +∀i, l, t +� +, +which, notably, encodes no dynamic constraint. Thus, for a fixed (xg, xg), solutions to (39) for +time period t depend only on the realization of forecast error in time period t. In other words, we +can optimize the conventional generation, REG curtailment and demand shedding at the second +stage non-anticipatively, i.e., not using the future realization of forecast error. If not relying on +(xg, xg), then the ramping constraint (34) may still be effective at the second stage. This implies +that we have to observe the future forecast error to determine the conventional generation, REG +curtailment and demand shedding in each time period, which is unrealistic. Thus, we introduce and +determine (xg, xg) at the first stage. Meanwhile, the support Ξ of ξ is defined using the forecast of +REG as well as the capacity of each REG system. +We now apply an affine policy to (38). In this study, we use xga +it +� +ξt +t +� +:= ag +itξt +t + bg +it, xda +it +� +ξt +t +� +:= +ad +itξt +t + bd +it, and xra +it +� +ξt +t +� +:= ar +itξt +t + br +it as decision rules for xg +it, xd +it, and xr +it for each (i, t), respectively, +where ξt +t := � +i ξit denotes the total forecast error in time period t. Although the coefficients of an +affine function can be arbitrary as discussed in Section 3, we employ these functions to reduce the +number of decision variables. Similar affine functions are frequently adopted in the literature on +two-stage optimization for power system operations [48,49]. +Applying the affine policy to (38) and, further, re-defining the Wasserstein ball over Ω ⊆ Ξ, we +can formulate our UC model in the form of (27). In the following subsection, we discuss simulation +results. +5.3 +Numerical Experiments +In this section, we compare the economic and computational performances of our UC model to +those of six existing models, SUC, RUC, MUC, KUC, NUC and CUC, on 6-, 24-, and 118-bus test +systems. Here, SUC and RUC are the SP and RO counterparts of (38), respectively. Moreover, +MUC, KUC, NUC and CUC are modifications of the UC models using DRO with ambiguity sets +based on the moment conditions, KL divergence, 1-norm distance, and CDF in [58], [59], [60], +and [61], respectively. + +18 +20 +40 +60 +80 +100 +7 +7.2 +7.4 +8 +8.3 +Cost ( +104) +20 +40 +60 +80 +100 +7 +7.1 +7.2 +20 +40 +60 +80 +100 +30 +31 +32 +33 +34 +Sample size +Figure 1: Average out-of-sample costs for different sample sizes. +The generator, load, and branch data of the 6- and 24-bus systems are from [80] and [81], +respectively. We locate one wind farm of capacity 80 MW at bus 2 of the 6-bus system, and three +wind farms, each of capacity 300 MW, at buses 3, 5, and 7 of the 24-bus system. For the 118- +bus system, the generator and load data are from [80], and we use the branch data from [82] to +accommodate five wind farms of capacities 40 MW, 75 MW, 120 MW, 250 MW, and 300 MW at +buses 24, 27, 31, 100, and 82, respectively, as well as five solar farms of capacities 700 MW, 330 +MW, 200 MW, 200 MW, and 150 MW at buses 32, 92, 54, 18, and 15, respectively. The penetration +levels of REG (i.e., the ratio of the total REG capacity to the peak demand) of the 6-, 24-, and +118-bus systems are 30.77%, 33.96%, and 35.38%, respectively. For all the test systems, we use a +planning horizon of T = 24 time periods of 1h. We assume that no more than 10 loads with the +highest total demand can be shed, while all the REG systems can be curtailed. The marginal costs +of demand shedding and REG curtailment are set to $3500/MWh and $20/MWh, respectively. We +run the simulations using MATLAB with MOSEK 9.3 for MUC and using CPLEX 12.10 for the +others on a PC with an Intel Core i7 3.70 GHz processor and 32 GB RAM. We discuss the results +in the following subsections. +5.3.1 +Comparison via Random Sampling +In the following, we compare our UC model to the six benchmark models on the 6- and 24-bus +systems via random sampling. +The simulation scheme is as follows: First, we model the true +distribution P⋆ of the wind power forecast error as a Pearson distribution based on the observation +data from [83]. +Randomly generating N = 20, 40, . . . , 100 samples according to P⋆, we build +empirical distributions of the forecast error and solve each UC model. We repeat this process 50 +times, i.e., with 50 independent sample sets, for statistical robustness. Then, we compare the UC +models in terms of the average out-of-sample cost and average computation time. For a here-and- +now decision (u, xg, xg) ∈ U × X g (u) obtained by solving any model, the out-of-sample cost is +defined as +J (u, xg, xg) := c⊤ +1 u + EP⋆ [f (xg, xg, ξ)] . +As exactly computing the out-of-sample cost is difficult, we use the sample average approximation +to estimate it with an additional 10,000 scenarios of the forecast error that are randomly generated +according to P⋆ independently of the N samples. For our UC model, we set β = 100 and use +the holdout method [18] to choose ε from 10−3, 10−2 and 10−1. For each benchmark model using +DRO, we set the parameter(s) of the ambiguity set as guided in the corresponding research paper + +19 +Table 1: Average computation time (in seconds): 6-bus system +N +20 +40 +60 +80 +100 +Prop. +8.04 +5.74 +7.81 +9.56 +6.40 +RUC +8.22 +SUC +1.96 +3.98 +7.69 +12.41 +19.55 +NUC +16.62 +45.78 +93.57 +156.22 +247.53 +MUC +310.79 +365.57 +372.34 +356.28 +343.87 +KUC +3834.08 +5966.40 +6106.50 +6665.26 +6761.11 +Table 2: Average computation time (in seconds): 24-bus system +N +20 +40 +60 +80 +100 +Prop. +77.60 +73.57 +76.11 +106.58 +79.63 +RUC +2.99 +CUC +9.44 +10.65 +11.04 +10.96 +10.85 +SUC +19.26 +193.01 +140.87 +184.96 +137.94 +NUC +79.68 +765.98 +761.52 +763.93 +618.36 +with its confidence level, if required, set to 0.99. We set a timeout limit of 1h only for the 24- +bus system. Further, we solve SUC, KUC and NUC on the 24-bus system with the fast-forward +selection method [84] to reduce the number of samples used for building empirical distributions, +thus avoiding time-out and memory-outage errors. +On the 6-bus system, MUC has no solution with one sample set for N = 20, while CUC +is infeasible with 35, 42, 45, 47, 49 sample sets for N = 20, . . . , 100, respectively. On the 24-bus +system, MUC and KUC with the first five sample sets for any N cannot be solved due to timeout +errors, neither of which we implement further. We illustrate the average out-of-sample costs of each +UC model in Fig. 1, except those of CUC for the 6-bus system and those of MUC and KUC for +the 24-bus system. We also report the average computation time on the 6- and 24-bus system in +Tables 1 and 2, respectively. +In Fig. 1, the proposed model shows the lowest average out-of-sample costs for N = 40, 60, 80 +and all N’s on the 6- and the 24-bus system, respectively. For N = 20 on the 6-bus system, RUC +leads to the lowest average out-of-sample cost. Thus, RUC, which is the most robust, can be an +alternative to our model when there are few samples. For N = 100 on the 6-bus system, SUC +and KUC perform better than the proposed model. In fact, SUC, KUC and NUC may incur lower +out-of-sample costs than our model when a huge number of samples are available. However, it +is highly likely that their computational performances are not satisfactory even for moderate-size +systems in such a case due to their poor scalability regarding sample size, as can be observed from +Fig. 1 (c). +Tables 1 and 2 verify that the computational load of our model is independent of sample size. +Although our model is not the most computationally tractable for every single case, the average +increase in computation time, if any, is acceptable given the accompanying decrease in the average +out-of-sample cost for most of the cases, compared to any benchmark model. + +20 +Table 3: Average cost ($106): 118-bus system +N +30 +60 +90 +120 +150 +180 +Prop. +1.34 +1.31 +1.33 +1.33 +1.34 +1.37 +SUC +3.01 +2.88 +3.05 +3.19 +3.00 +2.90 +RUC +1.33 +1.33 +1.35 +1.34 +1.35 +1.38 +(−0.51%) (1.77%) (1.59%) (1.49%) (1.21%) (0.99%) +NUC +3.00 +3.04 +3.07 +3.17 +3.09 +2.59 +CUC +1.49 +1.36 +- +- +- +- +Table 4: Average computation time (in seconds): 118-bus system +N +30 +60 +90 +120 +150 +180 +Prop. +418.59 +634.39 +457.30 369.49 405.62 364.09 +SUC +113.31 +100.42 +127.74 211.74 141.07 133.59 +RUC +120.80 +NUC +313.81 +217.56 +227.70 173.65 139.94 135.23 +CUC +2571.70 1597.13 +- +- +- +- +5.3.2 +Comparison Using Real Data +In the following, we further compare the UC models on the 118-bus system with a 365-day real- +world data set. +The data sets of wind and solar power forecast errors are from [85] and [86], +respectively. The simulation scheme is as follows: First, we construct SN pairs of training and +test distributions, both of which are empirical distributions obtained using N-day observation data +before and from day DN +k , k = 1, 2, . . . , SN of the year, respectively. +We set SN = 11, 9, . . . , 1 +for N = 30, 60, . . . , 180, respectively. +Further, day DN +k +corresponds to the first day of month +N/30+k, except for (N, k) = (60, 1), in which case we set D60 +1 = 61 to represent the 2nd of March. +We build the distribution pairs in this way so they consecutively cover almost all of the one-year +observation data. We solve each UC model for each training distribution and evaluate the “cost,” +i.e., the expected total operating cost with respect to the associated test distribution, as well as +the computation time. For SUC, KUC and NUC, we rebuild the training distributions with only +five samples obtained using the scenario reduction method. We set a timeout limit of 3h. +Tables 3 and 4 show the average costs and computation times of the UC models, except for +MUC and KUC, which face memory-outage and timeout errors for all cases, respectively. Moreover, +the results of CUC are only for six and three distribution pairs with N = 30, 60, respectively, except +when it is infeasible. The numbers in parentheses are the percentage increases from the average +costs of our model to those of RUC, which is the closest to our model in terms of the average cost. +The results indicate that the proposed model leads to the lowest average costs for all cases except +for the smallest N at the expense of acceptable increases in computation time, similar to the results +for the 6- and 24-bus systems, but on the larger-scale system with the real data set. +6 +Conclusions +In this article, we studied a generic class of 2-DRLPs over 1-Wasserstein balls using affine policies. +We showed that the problem of our interest has a tractable reformulation with a scale independent + +21 +of sample size. Subsequently, we proposed a method for refining the Wasserstein ball to reduce +the conservativeness of affine policies. 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Available: +https://www.nrel.gov/grid/solar-power-data.html + diff --git a/idAyT4oBgHgl3EQfXvfg/content/tmp_files/load_file.txt b/idAyT4oBgHgl3EQfXvfg/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ff5f35e3580f585683828c7dc067c0c75b745195 --- /dev/null +++ b/idAyT4oBgHgl3EQfXvfg/content/tmp_files/load_file.txt @@ -0,0 +1,1499 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf,len=1498 +page_content='Using Affine Policies to Reformulate Two-Stage Wasserstein Distributionally Robust Linear Programs to be Independent of Sample Size∗ Youngchae Cho Insoon Yang† Abstract Intensively studied in theory as a promising data-driven tool for decision-making under ambiguity, two-stage distributionally robust optimization (DRO) problems over Wasserstein balls are not necessarily easy to solve in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' This is partly due to large sample size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' In this article, we study a generic two-stage distributionally robust linear program (2-DRLP) over a 1-Wasserstein ball using an affine policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' The 2-DRLP has right-hand-side uncertainty with a rectangular support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' Our main contribution is to show that the 2-DRLP problem has a tractable reformulation with a scale independent of sample size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' The reformulated problem can be solved within a pre-defined optimality tolerance using robust optimization techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' To reduce the inevitable conservativeness of the affine policy while preserving independence of sample size, we further develop a method for constructing an uncertainty set with a probabilistic guarantee over which the Wasserstein ball is re-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' As an application, we present a novel unit commitment model for power systems under uncertainty of renewable energy generation to examine the effectiveness of the proposed 2-DRLP technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' Extensive numerical experiments demonstrate that our model leads to better out-of-sample performance on average than other state-of-the-art distributionally robust unit commitment models while staying computationally competent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' 1 Introduction Two-stage optimization is a popular tool for sequential decision-making under uncertainty, where the decision maker makes two kinds of decisions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=', here-and-now and wait-and-see decisions, be- fore and after observing the realization of uncertainty, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' Due to its generality, two-stage optimization has seen many applications in various research fields such as inventory management [2], workforce management [3], location planning [4], and power system operations [5,6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' In the present article, we consider a class of two-stage optimization problems based on distributionally robust optimization (DRO) with the Wasserstein metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content='1 Backgrounds Two-stage optimization approaches can be conveniently classified by the stochastic optimization method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' Among the most-studied stochastic optimization methods for two-stage optimization are ∗This work was supported in part by the National Research Foundation of Korea funded by MSIT(2020R1C1C1009766, 2021R1A4A2001824), the Information and Communications Technology Planning and Evaluation grant funded by MSIT(2022-0-00480), and Samsung Electronics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' A preliminary version of this work was presented at the 61st IEEE Conference on Decision and Control [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' †A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' Hakobyan, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' Yang are with the Department of Electrical and Computer Engineering and ASRI, Seoul National University, Seoul, 08826, South Korea {youngchaecho, insoonyang}@snu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content='kr 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content='00191v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content='OC] 31 Dec 2022 2 stochastic programming (SP), robust optimization (RO) and DRO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' A usual objective of SP is to minimize the expected total cost, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=', a sum of the deterministic cost associated with here- and-now decisions and the expected cost associated with wait-and-see decisions, with respect to a probability distribution of uncertainty [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' As the true distribution of uncertainty is difficult to obtain, an empirical distribution is used instead in most cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' For this reason, SP works well only with large sample datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' Without struggling to acquire the true distribution, RO uses worst-case analyses over an uncertainty set (a set of possible scenarios of uncertainty) with the common aim of minimizing the worst-case total cost, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=', a sum of the deterministic cost associated with here- and-now decisions and the worst-case cost associated with wait-and-see decisions [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' However, RO is often overly conservative as it ignores probabilistic features of uncertainty, which can be partially obtained through samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' To mitigate the disadvantages of SP and RO simultaneously, DRO uses worst-case analyses for an ambiguity set, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=', a family of probability distributions of uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' A typical goal of DRO is to minimize the expected total cost with respect to worst-case distributions in an ambiguity set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' Incorporating probabilistic features while hedging against the potential inappropriateness of any single pre-specified distribution, DRO better balances efficiency and robustness compared to SP and RO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' For details of general DRO problems, see, for example, [9] and the references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' Performances of DRO greatly depend on how the ambiguity set is chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' For example, ambigu- ity sets can be defined using f-divergences [10], e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=', the Kullback–Leibler (KL) divergence [11] and the total variation distance [12], as well as moment conditions [13,14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' However, these ambiguity sets have a few limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' First, ambiguity sets based on f-divergences may not be rich enough as they include only distributions that are absolutely continuous with respect to a nominal distribu- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' Moreover, the underlying assumption of moment information known a priori for DRO based on moment conditions hardly seems justifiable [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' Reportedly, moment-based DRO solutions may also be overly conservative [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' Ambiguity sets can be constructed using the Wasserstein metric as well [16,17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' A Wasserstein ball is defined as a statistical ball in the space of probability distributions, the radius of which is measured using the Wasserstein metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' Intuitively, the Wasserstein distance of two distributions is interpreted as the minimum cost of redistributing the probability mass from one distribution to the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' The center of a Wasserstein ball is mostly an empirical distribution constructed with a finite number of samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' As the elements of a Wasserstein ball are perturbations of the nominal distribution that are obtained considering the distance of uncertain scenarios, Wasserstein DRO does not suffer from the aforementioned drawbacks of DRO based on f-divergences or moment conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' Moreover, Wasserstein DRO offers a strong finite-sample performance guarantee [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' For these reasons, we focus on two-stage Wasserstein DRO in this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content='2 Related Work Research works providing solution methods for two-stage Wasserstein DRO in general forms includes [18–29] all of which, except for [18], consider linear costs of here-and-now and wait-and-see decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' Specifically, [19–26] deal with two-stage distributionally robust linear programs (2-DRLPs) over Wasserstein balls, where the second-stage problem to optimize wait-and-see decisions is a linear program (LP) while here-and-now decision variables can be integer or continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' In [19], it is briefly mentioned that 2-DRLPs over 1-Wasserstein balls can be reformulated as tractable semi- infinite or finite-dimensional optimization problems if the 1-, 2- or ∞-norm is used as the metric on the support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' In [20], decomposition algorithms are developed for solving exact reformulations of 2-DRLPs over 1-Wasserstein balls with the 1- and ∞-norm, assuming right-hand-side uncertainty and a rectangular uncertainty set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' The algorithms build on Benders decomposition [30] and the 3 column-and-constraint generation (C&CG) method [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' In [21], a second-order conic programming approach is employed to derive tractable reformulations of 2-DRLPs over 1-Wasserstein balls with the 2-norm, assuming that uncertainty appears in either the objective function or the constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' In [22], cutting-plane algorithms are used to exactly solve 2-DRLPs over 1-Wasserstein balls with either the generic p-norm for p ≥ 1 or a class of quadratic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' In [23], 2-DRLPs with the Wasserstein metric of order 2 are exactly solved using conic programming approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' In [24], 2-DRLPs over ∞- Wasserstein balls with the p-norm are approximately solved by applying multiple decision policies, one for an uncertainty set associated with each sample data point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' This scheme achieves optimality asymptotically, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=', as the number of samples goes to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' In [25], tractable reformulations of 2-DRLPs over ∞-Wasserstein balls with uncertainty in either the objective function or the constraints are presented for different continuity conditions on the uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' In [26], a sequential algorithm is developed for general two-stage DRO problems and applied to 2-DRLPs over 1- and ∞-Wasserstein balls for demonstration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' This algorithm creates at each iteration a Wasserstein ball using only a finite subset of the support as an approximation to the original ambiguity set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' With a new observation added at each iteration, the algorithm is proved to achieve asymptotic optimality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' References [18, 27–29] address more general classes of two-stage Wasserstein DRO problems than 2-DRLPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' In [27], two-stage distributionally robust conic LPs over 1-Wasserstein balls are considered, for which a cutting-plane algorithm based on Benders decomposition is suggested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' In [28] and [29], decomposition methods are developed assuming that both here-and-now and wait- and-see decisions are at least partially binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' The authors of [18] study a class of two-stage DRO problems over 1-Wasserstein balls where the costs of wait-and-see decisions are written as point- wise maximums of finitely many concave functions of uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' Using main results, tractable reformulations of 2-DRLPs over 1-Wasserstein balls with uncertainty in either the objective function or the right-hand-side of constraints are presented in [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' Notably, most of the existing solution methods for two-stage Wasserstein DRO problems, in- cluding those suggested in [18–29], have a scalability issue regarding sample size, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=', the number of historical sample data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' In other words, the existing solution methods require more computa- tional resources for more samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' This implies that two-stage Wasserstein DRO problems may not yield desired solutions that fully exploit historical data at hand when computational resources are limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content='3 Contributions In this article, we study a generic 2-DRLP over a 1-Wasserstein ball, which has right-hand-side uncertainty with a rectangular support, using an affine policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' Affine policies are a frequently used solution method for two-stage optimization problems which impose the linear dependence of wait-and-see decisions on uncertain parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' First developed in the context of SP [32–34], affine policies had been disregarded by the operations research community due to their intrinsic conservativeness that is hard to meaningfully quantify [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' A few decades later, however, affine policies have gained wide attention in the fields of not only SP [36] but also RO [8,37,38] as well as control theory for dynamical systems [39–44] due to their superior tractability and desirable properties related to cost performances such as robust invariance [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' Not only studied in theory, affine policies have seen many applications thereafter as well, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=', in portfolio management [46,47] and power system operations [48–51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' Furthermore, researchers have successfully extended these approaches by using piecewise affine [52,53], segregated affine [54,55] and polynomial [56] policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' The main contributions of this study are three-fold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' First, we show that the 2-DRLP of our interest has a tractable reformulation with a scale independent of sample size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' For this, we first recast the worst-case expectation problem nested in it, which is infinite-dimensional, as a finite 4 convex program with a scale that grows with sample size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' We then aggregate optimization vari- ables associated with different sample indices, which intuitively represent perturbation of samples, exploiting the fact that they have the same cost coefficient due to the affine policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' This yields an LP equivalent to the nested infinite-dimensional program, the scale of which is invariant with sample size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' Finally, using duality in LPs, we obtain a finite-dimensional mixed-integer LP (MILP) as an exact reformulation of the 2-DRLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' The reformulated problem can be solved up to a pre- defined precision by RO techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' We also present a cutting-plane algorithm for the reformulated problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' As a result, many samples can be efficiently exploited without relying on computation- ally expensive decomposition algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' To the best of our knowledge, our study is the first to reveal that affine policies can resolve the scalability issue regarding sample size in a general class of two-stage Wasserstein DRO problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content='1 Meanwhile, the optimality gap incurred by the affine policy can be arbitrarily large when the size of the Wasserstein ambiguity set is big enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' We assert that it is also true for any value of the radius, because the optimality gap as a function of the radius is a difference of two concave functions, which can be neither increasing nor decreasing in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' To reduce the inevitable conservativeness of the affine policy, we re-define the Wasserstein ball on an uncertainty set smaller than the support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' Our second main contribution is to design a data-driven method for constructing an uncertainty set with a bounded worst-case confidence level, over which the Wasserstein ball is rebuilt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' Since the feasibility of the affine policy is guaranteed on a smaller uncertainty set, more efficient solutions can be obtained by using our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' Unlike existing data-driven methods for building an uncertainty set with a similar probabilistic guarantee, our method ensures that the 2-DRLP does not depend on sample size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' Finally, to illustrate the applicability and effectiveness of the 2-DRLP approach using an affine policy for practical decision-making problems, we develop a novel UC model for power systems under the uncertainty of renewable generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' Extensive numerical experiments demonstrate that the proposed UC model outperforms not only classical models based on SP and RO but four state-of-the-art models based on DRO using ambiguity sets with the moment conditions [58], KL divergence [59], 1-norm distance [60] and cumulative density function (CDF) [61] in terms of out- of-sample performance, while staying computationally competent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' The rest of this article is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' In Section 2, we formulate the 2-DRLP of our interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' In Section 3, we show that the 2-DRLP has a tractable reformulation with a scale indepen- dent of sample size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' Furthermore, we provide a cutting-plane algorithm for solving the reformulated problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' In Section 4, we explain how to construct an uncertainty set with a probabilistic guaran- tee, over which we rebuild the Wasserstein ball to reduce conservativeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' In Section 5, we present the novel UC model based on the 2-DRLP approach using an affine policy and discuss simulation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' In Section 6, we give concluding remarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' We denote by R, R+, and R− the sets of all real numbers, non-negative real numbers, and non-positive real numbers, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' For a natural number n, we denote by 1n, 0n, In, and On the vector of ones, vector of zeros, identical matrix, and square zero matrix, respectively, all of dimension n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' Furthermore, [·]n represents the nth entry of a vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' We use | · | to denote the 1-norm of a vector or the cardinality of a finite set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' We also denote by (·)⊤, δ(·), E, ◦, (·)◦, and V (·) the transpose of a vector or matrix, Dirac delta distribution centered at a given point, expectation operator, entrywise product operator for two vectors, interior of a subset of a Euclidean space, and vertex set of a convex polytope, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' 1Although the scalability issue is addressed by [57] for the unit commitment (UC) problem, the method in [57] is applicable only when the cost of wait-and-see decisions calculated using an affine policy is univariate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' In contrast, we do not impose any special assumption on the affine policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' 5 2 Problem Formulation In this section, we formulate a two-stage Wasserstein DRO problem of our interest using an affine policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' To this end, we first consider the 2-DRLP min x1∈X1 c⊤ 1 x1 + max P∈Pε(Ξ) EP [f (x1, ξ)] (1) where f (x1, ξ) := min x2∈X2(x1,ξ) c⊤ 2 x2 (2) denotes the optimal cost of wait-and-see decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' Here, ξ ∈ Rm and Ξ ⊂ Rm denote a random vector and its support, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' The support Ξ is a bounded box that is known, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=', Ξ = [ξ, ξ] where ξ, ξ ∈ Rm can be obtained using a priori knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' We assume that N historical samples ξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' , ξN of ξ are available and denote the index set of samples by I := {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' , N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' In (1), x1 ∈ {0, 1}n11 ×Rn12 and c1 ∈ Rn1 with n1 := n11+n12 represent a here-and-now decision vector and its cost coefficient vector, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' The feasible set X1 of x1 is defined with finitely many linear inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' The symbol Pε (·) denotes a 1-Wasserstein ball on a given uncertainty set, which is a ball of radius ε > 0 centered at an empirical distribution Pe := 1 N � i∈I δξi in the space of probability distributions supported on the given uncertainty set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' Specifically, we let Pε (·) := {P ∈ P (·) : d (P, Pe) ≤ ε} where P (·) represents the family of all probability distributions supported on a given uncertainty set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' Furthermore, d denotes the Wasserstein metric of order 1 defined with the 1-norm, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=', d � P, P′� := inf π∈Π(P,P′) � Ξ×Ξ ��ξ − ξ′�� π � dξ, dξ′� where Π (·, ·) denotes the set of all joint distributions supported on Ξ × Ξ with marginals equal to two given distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' In (2), x2 ∈ Rn2 and c2 ∈ Rn2 represent a wait-and-see decision vector and its cost coefficient vector, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' The feasible set of x2 is defined as X2 (x1, ξ) := � x2 ∈ Rn2 : Ain 1 x1 + Ain 2 x2 + Ain 3 ξ ≤ bin� where Ain 1 ∈ RL×n1, Ain 2 ∈ RL×n2, Ain 3 ∈ RL×m, and bin ∈ RL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' In the above formulations, m, n11, n2, and L are natural numbers, while n12 is a non-negative integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' Throughout the study, we assume that (1) is feasible, as is standard in the literature [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' However, we do not impose the (relative) complete recourse condition, which is also usual (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=', [27,28]) but might be restrictive for some real-world problems [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' Problem (1) is general enough to model diverse decision-making problems in the real world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' For example, the biomass network design [62], unmanned aerial vehicle network design [63], and railway scheduling [64] problems have been addressed in the form of (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' However, it is often computationally demanding to exactly solve a two-stage optimization problem such as (1) [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' In this article, we focus on affine policies that approximately solve (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' Affine policies are a popular solution method for two-stage optimization problems, where wait-and-see decision variables are restricted to be affine functions of uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' Due to their computational efficiency, affine policies have been studied extensively for practical two-stage RO [48, 66, 67] and DRO [68–70] problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' Specifically, we use the affine function xa 2 (ξ) := Aξ + a 6 as our decision rule for x2, where A ∈ Rn2×m and a ∈ Rn2 are determined simultaneously with x1 at the first stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' Thus, the 2-DRLP of our interest is formulated as min x1∈X1,(A,a)∈A(x1,Ξ) c⊤ 1 x1 + hΞ (A, a) (3) where hΞ (A, a) := max P∈Pε(Ξ) EP � c⊤ 2 (Aξ + a) � (4) denotes the worst-case expected cost of wait-and-see decisions using the affine policy over Pε (Ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' To guarantee that xa 2 is feasible over Ξ, we define A (x1, Ξ) := � (A, a) ∈ Rn2×m × Rn2 : Ain 1 x1 + Ain 2 (Aξ + a) + Ain 3 ξ ≤ bin ∀ξ ∈ Ξ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' In this study, we assume that (3) is feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content='2 One reason (1) is hard to solve in practice is its scalability issue regarding sample size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' In- tractable in the current form due to the nested infinite-dimensional optimization problem, (1) can be rewritten in a tractable form using well-studied Wasserstein DRO techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' However, the scale of any tractable reformulation of (1) grows with sample size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' We present such a tractable reformulation in the following proposition, which can be proven by duality theory;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=', [18,19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' Problem (1) can be rewritten as the two-stage RO problem min x1∈X1,λ≥0,η∈RN c⊤ 1 x1 + λε + 1 N � i∈I [η]i s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' f (x1, ξ) − λ |ξ − ξi| ≤ [η]i ∀ξ ∈ Ξ, i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' (5) Problem (5) can be solved using decomposition algorithms such as Benders decomposition, the C&CG algorithm and variants of these methods [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' In these algorithms, (5) is decomposed into a master problem and two types of subproblems that are iteratively solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' Each of the master problem and subproblems is written as an MILP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' The scalability issue regarding sample size is problematic specifically for the following two reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' First, one of the two subproblems, which has a size independent of sample size, has to be solved for each sample at each iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' Second, a set of decision variables and/or constraints, the number of which is proportional to sample size, can be added to the master problem at each iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' As empirically shown in [71], this may well cause the actual computation time of the master problem to increase superlinearly with sample size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' Moreover, undoubtedly, the master problem with a scale increasing with sample size makes a decomposition algorithm for (5) susceptible to memory-outage errors when many samples are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' Considering the superior tractability of affine policies, one natural question arises: Does (3) suffer from the same scalability issue regarding sample size as (1)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' In the following section, we show that the answer is no, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=', (3) has a tractable reformulation with a scale independent of sample size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' The feasibility of (1) implies that any feasible point x1 should be such that X2 (x1, ξ) is non-empty for any ξ ∈ Ξ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=', ff (x1, ξ) = 0, ∀ξ ∈ Ξ (6) 2Unless m = 1, however, (3) might be infeasible even when (1) is feasible [45, 54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' In this case, the following discussions throughout the article do not apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' 7 where ff (x1, ξ) is equal to the optimal value of the LP min x2∈Rn2,y∈R+ y s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' Ain 1 x1 + Ain 2 x2 + Ain 3 ξ ≤ bin + Iky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' (7) In words, ff (x1, ξ) denotes the maximum violation of constraints in (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' By taking the dual for- mulation of (7), we observe that ff (x1, ξ) is convex in ξ for a fixed x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' Thus, (6) is rewritten as ff (x1, ξ) = 0, ∀ξ ∈ V (Ξ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' (8) We make explicit use of (8) in Section 4, where we construct a Wasserstein ball different from Pε (Ξ) and (6) may not be implied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' Problem (3) can also express a “multi-stage” DRLP over 1-Wasserstein balls using an affine policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' Specifically, we consider the multi-stage DRLP min x1∈X1 c⊤ 1 x1 + max P2∈P2ε (Ξ2) EP2 � min z2∈Z2(x1,ξ2) e⊤ 2 z2 + max P3∈P3ε (Ξ3) EP3 � min z3∈Z3(x1,z2,ξ2,ξ3) e⊤ 3 z3 + · · · + max PT ∈PT ε (ΞT ) EPT � min zT ∈ZT (x1,z2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=',zT −1,ξ2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=',ξT ) e⊤ T zT ��� (9) where ξt, Ξt and Pt ε � Ξt� denote a random vector, its rectangular support, and a 1-Wasserstein ball for each stage t = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' , T, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' Furthermore, zt, Zt, and et denote a real decision vector, its feasible set defined using a finite number of linear inequalities with right-hand-side uncertainty, and its cost coefficient vector for each stage t, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' Using the affine function za t � ξ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' , ξt� := t � τ=2 Aτξτ + aτ as a decision rule for zt in (9), which depends on the realization of uncertainty only up to stage t, we can formulate a multi-stage problem in the form of (3) for ξ = � ξ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' , ξT � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' Here, the matrices Aτ and vectors aτ to be determined at the first stage are of appropriate dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' However, it is unclear if affine policies for multi-stage DRLPs over Wasserstein balls with different assumptions and problem structures, possibly of greater practical importance, lead to (3) in a similar way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' Thus, we focus on the two-stage formulation (1) in this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' For details on general multi- stage DRO or distributionally robust dynamic programming problems, the reader is referred to, for example, [72–75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' 3 Independence of Sample Size Similar to (1), (3) is intractable in the current form as (4) is infinite-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' In this section, we prove that (3) has a tractable reformulation with a scale independent of sample size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' In particular, we derive a finite-dimensional MILP equivalent to (3), the scale of which is invariant with sample size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' Subsequently, we present a cutting-plane algorithm for solving the reformulated problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' To this end, we first prove the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' 8 Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' Problem (4) is rewritten as an LP with a scale independent of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' Specifically, we have hΞ (A, a) = max ˜q+,˜q−∈Rm + c⊤ 2 � A � ˜ξ + ˜q+ − ˜q−� + a � s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' 1⊤ m � ˜q+ + ˜q−� ≤ ε (10) ˜q+ ≤ ξ − ˜ξ (11) ˜q− ≤ ˜ξ − ξ (12) where ˜ξ := 1 N � i∈I ξi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' Let hΞ (A) := max P∈Pε(Ξ) EP � c⊤ 2 Aξ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' From Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content='4 in [18], it follows that hΞ (A) = max q∈RNm 1 N � i∈I c⊤ 2 A (ξi + qi) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' 1 N � i∈I |qi| ≤ ε (13) ξ ≤ ξi + qi ≤ ξ ∀i ∈ I where q := (q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' , qN) is a vector concatenating qi ∈ Rm + for all i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' Introducing auxiliary decision vectors q+ i , q− i ∈ Rm + such that qi = q+ i − q− i and q+ i ◦ q− i = 0m for each i ∈ I to linearize the norm constraint (13), we observe that hΞ (A) = max q+,q−∈RNm + 1 N � i∈I c⊤ 2 A � ξi + q+ i − q− i � s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' 1 N � i∈I 1⊤ m � q+ i + q− i � ≤ ε (14) q+ i ≤ ξ − ξi ∀i ∈ I (15) q− i ≤ ξi − ξ ∀i ∈ I (16) q+ i ◦ q− i = 0m ∀i ∈ I (17) where q+ := � q+ 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' , q+ N � and q− := � q− 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' , q− N � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' Note that the mutual exclusivity constraint (17) is redundant and thus can be omitted without affecting optimality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' Adding up the N inequalities in (15) and those in (16) respectively, we further have hΞ (A) ≤ max q+,q−∈RNm + 1 N � i∈I c⊤ 2 A � ξi + q+ i − q− i � s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' (14) � i∈I q+ i ≤ Nξ − � i∈I ξi (18) � i∈I q− i ≤ � i∈I ξi − Nξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' (19) 9 In what follows, we show that this holds as equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' For any q+′ = � q+′ 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' , q+′ N � ∈ � q+ ∈ RNm + : (18) � , there exists q+′′ = � q+′′ 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' , q+′′ N � ∈ � q+ ∈ RNm + : (15) � such that � i∈I q+′′ i = � i∈I q+′ i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAyT4oBgHgl3EQfXvfg/content/2301.00191v1.pdf'} +page_content=' For exam- ple, one such q+′′ can be obtained by letting q+′′ i := � � � � � � � � � � � � � � � min � � �ξ − ξi, � j∈I q+′ j � � � i = 1 min � � �ξ − ξi, � i∈I q+′ i − � j=, or *): using this range, the client specifies that all new provider releases are supported/accepted and +downloadable, even the ones with breaking changes. +– caret (^): with this range, the client specifies that all new provider releases that contain new features and bug +fixes are supported/accepted and downloadable; breaking changes must be avoided. This is the default range +used by npm when a dependency is installed. +– tilde range (�): this range specifies that all new provider releases that only contain bug fixes are support- +ed/accepted and downloadable; breaking changes and new features must be avoided. +– steady range: this range always resolves to a specific version and is also known as specific range. That is, the +versioning statement has no range on it but rather a specific version. npm allows installation with a steady +range using the command line option –save-exact. +• implicit and explicit update: an implicit update happens when the client receives a new provider version due +to the range version in the package.json. For a version statement defined with a range of versions, for example, +^4.10.6, an implicit update happens when npm installs a version 4.10.9 that matches the range. An explicit update +takes place when the client manually updates the versioning statement directly in the package.json. +• manifesting breaking changes are provider changes that manifest as a fault on the client package, ultimately +breaking the client’s build. The adopted definition of breaking change by the prior literature [3–6, 8, 15, 19, 21] +includes cases that are not considered breaking changes (e.g., a change in an API that is not effectively used +by a client package). Conversely, manifesting breaking changes include cases that are not covered by the prior +definitions of breaking change (e.g., because the provider package is used in a way that is not intended by the +provider developer, a semantic-version compliant change introduced by a new release of this provider causes an +expected error in the client package). +2.2 +Motivating Examples +We found the following two examples of manifesting breaking changes in our manual analysis (on each of the following +Listing, red lines have been removed from the source code whereas blue lines have been inserted into the source +code). Our manual analysis (Section 3.2.1) consists of executing the client tests suite for its releases and analyzing all +executions that run into an error. +The client assetgraph-builder@7.0.0 has a provider assetgraph@6.0.0 that has a provider terser@^4.0.0, but, due to a +range of versions, npm installed terser@4.6.10. Release 4.3.0 of terser introduces a change which, by default, enables +the wrapping of functions on parsing, as Listing 1.1 +Listing 1. Diff between terser@4.2.1 and terser@4.3.0 default behavior. +// terser@4 .2.1 without default wrapping behavior +foo(function (){}); +// terser@4 .3.0 default wrapping behavior +foo(( function (){})); +This change breaks the assetgraph-builder@7.0.0’s tests.2 Once this feature is turned a default behavior, the client +assetgraph-builder@8.0.0 adopts its test to make it compatible with the terser’s behavior, as Listing 2.3 +Listing 2. Diff with assetgraph@8.0.0 client’s tests adjusting to breaking change. +1https://github.com/terser/terser/compare/v4.2.1..v4.3.0 +2https://github.com/terser/terser/issues/496 +3https://github.com/assetgraph/assetgraph-builder/commit/e4140416e7feaa3d088cf3ad0229fd677ff36dbc +Manuscript submitted to ACM + +I depended on you and you broke me: An empirical study of manifesting breaking changes in client packages +5 +expect( +javaScriptAssets [0].text , +'to match', +- +/SockJS =[\s\S]* define \(" main",function \(\) \{\}\) ;/ ++ +/SockJS =[\s\S]* define \(" main ",\(? function \(\) \{\}\) +?\);/ +); +Sometimes, provider changes can break a client long after their introduction. This occurred in the client package +ember-cli-chartjs@2.1.1. In Figure 2, the release 1.0.4 of ember-cli-qunit (left-tree) introduced a change that did not +lead to a breaking change. However, almost three years later, ember-cli-qunit was used together with the release 1.3.1 +of the provider broccoli-plugin (middle-tree), and a breaking change manifested. +ember-cli-qunit@1.0.4 +ember-cli-chartjs@2.1.1 +ember-cli-chartjs@2.1.1 +Nov. 2015 +Aug. 2018 +Apr. 2020 +ember-cli-qunit@1.4.2 +broccoli-asset-rev@2.4.3 +broccoli-filter@1.2.3 +broccoli-plugin@1.2.1 +ember-cli-qunit@1.4.2 +broccoli-asset-rev@2.4.3 +broccoli-filter@1.3.0 +broccoli-plugin@1.3.1 +Fig. 2. The evolution of the dependency tree (resolved versions) for ember-cli-chartjs@2.1.1 when it was published (middle-tree) and +when the associated tests with the release were executed in our study (right-hand tree). +In November 2015, the provider ember-cli-qunit@1.0.4 fixed an error in its code, changing the returned object type of +function lintTree,4 as shown in Listings 3. Despite being a type change, it did not break the client when it was released, +and this fix was retained in further releases of ember-cli-qunit. +Listing 3. ember-cli-qunit@1.0.4 object type change. +lintTree: function(type , tree) { +// Skip if useLintTree === false. +if (this.options['ember -cli -qunit '] && ... ) { +- +return tree; ++ +// Fakes an empty broccoli tree ++ +return { inputTree: tree , rebuild: function () { return []; } }; +} +Almost three years later, on Aug. 2018, the provider broccoli-plugin@1.3.1 was released (middle-tree in Figure 2) to +fix a bug,5 as in Listing 4. +Listing 4. broccoli-plugin@1.3.1 validation function enhanced. +function isPossibleNode(node) { +- +return typeof node === 'string ' || +- +(node !== null && typeof node === 'object ') ++ +var type = typeof node; ++ +if (node === null) { ++ +return false; ++ +} else if (type === 'string ') { +... ++ +} else { +4https://github.com/ember-cli/ember-cli-qunit/commit/6fdfe7d +5https://github.com/broccolijs/broccoli-plugin/commit/3f9a42b +Manuscript submitted to ACM + +6 +Venturini, et al. ++ +return false; ++ +} +The release 1.3.1 of the broccoli-plugin package experienced a manifesting breaking change due to a fix in the +provider ember-cli-qunit@1.0.4,6 which was released almost three years prior. This manifesting breaking change +occurred because the ember-cli-chartjs’ dependency tree evolved over time due to the range versions, as shown in +Figure 2, causing the break. When the package ember-cli-chartjs@2.1.1 was installed on April 2020 (the date of our +analysis), the installation failed due to the integration of broccoli-plugin@1.3.1 changes into ember-cli-qunit. Fifteen +days later, ember-cli-qunit@1.4.3 fixed the issue when the ember-cli-qunit’s object type was changed again.7 During +the fifteen-day period when the manifesting breaking change remained unresolved, broccoli-plugin received about 384k +downloads from npm. This scenario shows that even popular and mature projects can be affected by breaking changes. +Although we recognize that the download count does not necessarily reflect the popularity of a package, we use this +metric as an illustrative example of how many client packages might have been impacted by a provider package. +3 +STUDY DESIGN +This section describes how we collected our data (Section 3.1) and the motivation and approach for each RQ (Section 3.2). +3.1 +Data Collection +3.1.1 +Obtaining metadata from npm packages. The first part of Figure 3 shows our approach for sampling the database. +We initially gathered all the metadata files (i.e., package.json files) from the published packages in the npm registry +between December 20, 2010 and April 01, 2020, accounting for 1,233,944 packages. This range refers to the oldest +checkpoint that we could retrieve and the most recent one when we started this study. We ignored packages that did +not have any providers in the package.json since they cannot be considered client packages and will therefore not suffer +breaking changes. After filtering packages without a provider, our dataset comprises 987,595 package.json metadata +files. For each release of each package, we recorded the timestamp of the release and the name of the providers with +their respective versioning statements. +We parsed all the versioning statements and determined the resolved provider version at the time of each client +release. Prior works have adopted similar approaches when studying dependency management [7, 29]. For each provider +in each client release, we retrieved the most recent provider version that satisfied the range specified by the client in +that release; i.e., the resolved version. Using this resolved version, we determined whether a provider changed its version +between the two client releases. In other words, we reproduced the adopted versions of all providers by resolving the +provider version at the release time of the client. +To further refine our sample, we analyzed two criteria in the associated package.json snapshot with the latest version +of the client packages in our dataset: +(1) The package.json snapshot should have a non-empty entry for the “script test” field, and the entry should differ +from the default: Error: no test specified. We specified this criterion in order to run the automated tests +that were part of our method to detect manifesting breaking changes. In total, 488,805 packages remained after +applying this criterion. +6https://github.com/broccolijs/broccoli-merge-trees/issues/65 +7https://github.com/ember-cli/ember-cli-qunit/commit/59ca6ad +Manuscript submitted to ACM + +I depended on you and you broke me: An empirical study of manifesting breaking changes in client packages +7 +(2) The package.json snapshot should have an entry containing the package’s repository URL, as we wanted to +retrieve information from the package codebase. After applying this criterion, 410,433 packages remained in our +dataset. +Clone the +repository +Restore the +next release +Update providers +into package.json +to resolved +version +NO +YES +Did any provider +change? +Change +the Node.js +version +Run +install/test +Save the result +Running install/test +1,233,944 metadata +files (packages) from +npm registry +Filtering client +packages: 987,595 +Sampling packages +Filtering valid test +scripts: 488,805 +Filtering valid +repositories URL: +410,805 +Breaking change detection +Restore client's releases +Sampling with 95% +confident level and ±5% +confident interval: +384 +Fig. 3. Approach to sampling the database and executing the associated tests with the client release. +3.1.2 +Running clients’ tests. Given the size of our dataset (more than 410,000 client packages), we ran tests on a random +sample. At a 95% confidence level and ±5% confidence interval, we randomly selected 384 packages. Our sample has +a median of 5.5 releases and 9 direct providers per package. We chose to study a random sample since our manual +analysis is slow to run over a large dataset (Section 3.1.3); we spent a month executing our method in our sample. We +did not ignore packages based on the number of releases or providers or any other metric. We performed a manual +check on all selected packages that had fewer than four releases (130 out of 384) by checking their repositories and +aiming to remove packages that are not real projects, lack tests, lack code, are example projects, etc. When we removed +one package, we sampled another one following the two criteria described above. +The second part of Figure 3 depicts our approach to running the test scripts for each release of the 384 clients. For +each client package, we cloned its repository – all client repositories are hosted on GitHub – and restored the work tree +of all releases using their respective release tags (e.g., “v1.0.0”). For releases that are not tagged, we used their provided +timestamp in the package.json metadata to restore the work tree (i.e., we matched the release timestamp and the closest +existing commit in the master branch). We conducted an analysis and verified that tags and timestamp point to the +same commit in 94% of releases with tags, thus checkout based on timestamps is reliable for untagged releases. +After restoring the work tree of a client release, we updated all versioning statements in the associated package.json +entry with the specific resolved provider version (see Section 3.1.1). We then excluded a file called package-lock.json, +which locks the providers and indirect providers versions. We also executed the associated tests on a release of the +client package whenever a provider package changed in that release, as this can potentially introduce a manifesting +breaking change. A provider change can be: 1) a provider added into the package.json; or 2) the resolved version of a +provider changed between the previous and current release of the client package. +We sought to reproduce the same build environment that existed when the provider changed. Therefore, before +executing the tests of the client packages, we performed a best-effort procedure to identify the Node.js that was adopted +Manuscript submitted to ACM + +8 +Venturini, et al. +by the client package at the time the provider changed. This was because every six months a new major version of +Node.js is released. 8 As we wanted to reproduce the test results with respect to the time when the client package +published its release, we changed the Node.js version before executing the client package tests. We selected the Node.js +version using two different approaches. Our preferred approach was to select the same Node.js version as the one +specified in the engines→node field of the package.json file.9 This field allows developers to manually specify the +Node.js version that runs the associated code with the build of a specific release. When this field was not set, we selected +the latest Node.js version available10 at the time of the client package release. Therefore, we changed the Node.js +version, executed the install script, and released tests using the npm install and npm test commands, respectively. If +the install or test commands failed due to incompatibilities with the selected Node.js version – or took more than 10 +minutes –, we changed to the previous major release of Node.js until the install and test commands succeeded. We used +the Node Version Manager (NVM) tool to exchange Node.js versions. Additionally, we also changed the npm version +according to the Node.js version. npm is the package manager to Node.js packages and executes the install and test +scripts. We performed the same procedure to select the npm version to use during the installation and test runs. Finally, +we executed the install/test scripts and saved the results (success or error) for each client release. +After executing the install/test scripts of the 384 client packages in our sample, we discarded 33 packages because +the errors did not allow the execution of the install/test script in any of their releases: 15 clients did not have one of the +required files; 11 had invalid test scripts (e.g., {"test": "no test"}); 4 listed some required files in the .gitignore file – +that specifies untracked files that git should ignore;11 2 required specific database configurations that could not be done; +and 1 package required a key to access a server. We randomly replaced these 33 packages following the aforementioned +criteria. +Table 1 shows the results of the execution of the install/test scripts of the 384 client packages and their 3,230 releases. +Since the associated providers’ version with 2,727 releases did not change, these tests’ releases were not executed. +Finally, we consider as possible manifesting breaking changes cases in which all client packages and releases failed the +install/test scripts. +Table 1. Results of execution of the install/test scripts. +Tests +Client Packages +Releases +Executed +384 +3,230 +Not executed +0 +2,727 +Success +181 +1,954 +Fails +203 +1,276 +Total +384 +5,957 +A replication package including our client packages sample, instruments, scripts, and identified manifesting breaking +changes is available for download at https://doi.org/10.5281/zenodo.5558085. +8https://github.com/nodejs/node#release-types +9https://docs.npmjs.com/files/package.json#engines +10https://nodejs.org/en/download/releases +11https://git-scm.com/docs/gitignore +Manuscript submitted to ACM + +I depended on you and you broke me: An empirical study of manifesting breaking changes in client packages +9 +3.1.3 +Manual check on failure cases: detecting manifesting breaking changes. For all failure cases (203 clients and 1,276 +releases) on the execution of install/test scripts, we manually analyzed which ones were true cases of manifesting +breaking changes. To identify breaking changes that manifest themselves in a client package, we leveraged the output +logs (logs generated by npm when executing the install and test scripts) generated as the result of executing the +method described in Section 3.1.2 (see the second part of Figure 3). For each failed test result, we obtained the error +description and the associated stack trace. We then differentiated failed test results caused by a related issue with the +client package (e.g., an introduced bug by the client) from those caused by a change in the provider package (e.g., a +change in the return type of a provider’s function). From the obtained stack traces, we determined whether any function +of a provider package was called and manually investigated the positive cases. During our manual investigation, we +sought to confirm that the test failure was caused by a manifesting breaking change introduced by the provider package. +The first author was responsible for running the tests and identifying the manifesting breaking changes and related +releases and commits. The first author also manually analyzed each of the manifesting breaking changes and recorded +the following information about each of them: the number of affected versions of the client; whether any documentation +mentions the manifesting breaking change; the responsible package for addressing the breaking change (provider +or client); the client version impacted by the manifesting breaking change; the provider version that introduced the +breaking change; and a textual description about the causes for the breaking change manifestation (e.g., "the provider +function was renamed by mistake", "The provider normalizeurl@1.0.0 introduce[d] a new function and the client +assetgraph use[d] it. But the client forgot to update the provider version in package.json.", "The provider inserts a " " in a +null body request"). During this process, several rounds of discussions were performed among the authors to refine the +analysis, using continuous comparison [22] and negotiated agreement [13]. In the negotiated agreement process, the +researchers discussed the rationale they used to categorize each code until reaching consensus [13]. More specifically, +we leveraged the recorded information about each manifesting breaking change to derive a consistent categorization of +the introduced breaking changes (RQ2 and RQ3) and to guide new iterations of the manual analysis. +More specifically, the following set of actions was performed during our manual investigation: +• Analyze the execution flow: To determine whether the associated function with the test failure occurred in +the provider or the client code, we leveraged the stack traces to identify which function was called when the test +failed. In particular, we instrumented the code of the provider and the client packages to output any necessary +information to analyze the execution flow. We analyzed the variable contents by adding a call to the console.log() +and console.trace() functions in each part of the code where the client package calls a function of the provider. +For example, suppose the following error appeared: “TypeError: myObject.callback is not a function”.To discover +the variable content, we use the command console.log(myObject) to check whether myObject variable was +changed, null, or received other values. +• Analyze the status of the Continuous Integration (CI) pipeline: We compared the status of the CI pipeline +between the originally built release and the status of CI pipeline at the time of our manual investigation. Since +the source code of the client package remains the same between the original release and the installed version +in our analysis, we use the difference between the status of the CI pipeline as additional evidence that the test +failure was caused by a provider version change. Not all clients had CI pipelines, but when they had, it was +helpful. +• Search for client fixing commits: We manually searched for recovering commits in the history of commits +between the installed and previous releases of the client package. Whenever a recovery commit was identified +Manuscript submitted to ACM + +10 +Venturini, et al. +(by reading the commit message), we determined whether the error was due to the client or the provider code. +For example, we observed cases in which a client updated a provider in the release with failed tests. We also +observed that, in the following commits, the provider was downgraded and the commit message was “downgrade +provider” or “fix breaking change”. In these cases, we considered the test failure as caused by a manifesting +breaking change. +• Search for related issue reports and pull requests: We hypothesized that a manifesting breaking change +would affect different clients that, in turn, would either issue a bug report or perform a fix followed by a pull +request to the codebase of the provider package. Therefore, we searched for issue reports and pull requests with +the same error message obtained in our stack trace. We then collected detailed information about the error to +confirm whether it was due to a manifesting breaking change introduced by the provider package. +• Previous and subsequent provider versions: If the test error was caused by a manifesting breaking change, +downgrading to the previous provider version or upgrading to a subsequent provider version might fix the error, +if the provider already fixed it. Subsequent provider versions means all provider versions that fit the versioning +statement and are greater than the provider version that introduced the manifesting breaking change (i.e., the +adopted provider version when the test failed). In this case, we uninstalled the current version and installed +the previous and subsequent versions, and executed the test scripts again. For example, if the client specified +a provider p as {"p": "^1.0.2"} that brought about a breaking change in the version, for example, 1.0.4, we +installed p@1.0.2, p@1.0.3, and p@1.0.5 to verify whether the error persisted for those versions. +3.2 +Research questions: motivation, approach +This section contains the motivation and the approach for each of the research questions. +3.2.1 +RQ1. To what extent do manifesting breaking changes manifest in client packages? +Motivation: By default, npm sets the caret range as a default versioning statement that automatically updates minor and +patch releases. Hence, manifesting breaking changes that are introduced in minor and patch releases can inadvertently +cause downtime in packages that are downloaded hundreds of thousands of times per day, affecting a large body of +software developers. Understanding the prevalence of manifesting breaking changes in popular software ecosystems +such as npm is important to help developers assess the risks of accepting automatic minor and patch updates. Although +prior studies have focused on the frequency of API breaking changes [3], breaking changes can occur for different +reasons. Determining the prevalence of a broader range of breaking change types remains an open research problem. +Approach: For all cases that resulted in an error on the install/test script, we determined the type of error (client, +provider, not discovered). We calculated, out of the 384 packages and 3,230 releases, the percentage of cases that we +confirmed as manifesting breaking change. Considering all the providers on the client’s latest releases, we calculated +the percentage of providers that introduced manifesting breaking changes. In addition, we calculated how many times +(number of releases) each provider introduced at least one manifesting breaking change. +3.2.2 +RQ2. What problems in the provider package cause a manifesting breaking change? +Motivation: Prior studies about breaking changes in the npm ecosystem are restricted to APIs’ breaking changes +[14]. However, other issues that provider packages can introduce in minor and patch releases can manifest a breaking +Manuscript submitted to ACM + +I depended on you and you broke me: An empirical study of manifesting breaking changes in client packages +11 +change. To support developers to reason about manifesting breaking changes, it is important to understand their root +causes. +Approach: In this RQ, we analyzed the type of changes introduced by provider packages that bring about a manifesting +breaking change. With the name and version of the provider packages, we manually analyzed the provider’s repository +to find the exact change that caused a break. We used the following approaches to find the specific changes introduced +by providers: +• Using diff tools: We used diff tools to analyze the introduced change between two releases of a provider. For +example, suppose that a manifesting breaking change was introduced in the release provider@1.2.5. In this case, +we retrieved the source code of previous versions, e.g., provider@1.2.4, and performed the diff between these +versions to manually inspect the changed code. +• Analyzing provider’s commits: We used the provider’s commits to analyze the changes between releases. For +a manifesting breaking change in the provider p, we verified its repository and manually analyzed the commits +ahead or behind the release tag commit that introduced a manifesting breaking change. +• Analyzing changelogs: Changelogs contain information on all relevant changes in the history of a package. +We used these changelogs to understand the introduced changes in a release of a client package and to verify +whether any manifesting breaking change fix was described. +We also looked at issue reports and pull requests for explanations of the causes of manifesting breaking changes. +After discovering the provider changes that introduced breaking changes, we analyzed, categorized, and grouped +common issues. For example, all related issues to changing object types were grouped into a category called Object type +changed. Furthermore, we analyzed the Semantic Version level that introduced and fixed/recovered the manifesting +breaking changes both in the provider and client packages to verify the relationship between manifesting breaking +changes and non-major releases. +We analyzed the version numbering of releases that fixed a manifesting breaking change and where manifesting +breaking changes were documented (changelogs, issue reports, etc.). Furthermore, we analyzed the depth of the +dependency tree of the provider that introduced a manifesting breaking change, since 25% of npm packages had at least +95 transitive dependencies in 2016 [10]. +3.2.3 +RQ3. How do client packages recover from a manifesting breaking change? +Motivation: A breaking change may impact the client package through an implicit or explicit update. A client recovery +is identified by an update to its code, by waiting for a new provider’s release, or by performing a downgrade/upgrade +in the provider’s version. Breaking changes may be caused either by a direct or indirect provider since the client +packages depend on a few direct providers and many indirect ones [11]. A breaking change may cascade to transitive +dependencies if it remains unfixed. Even if the client packages can recover from the breaking change by upgrading to a +newer version of the provider package, the client packages can manually resolve incompatibilities that might exist [12]. +Understanding how breaking changes manifest in client packages can help developers understand how to recover from +them. +Approach: We retrieved all information for this RQ from the clients’ repositories. We searched for information about +the error and how the client packages recovered from the manifesting breaking change. The following information was +analyzed: +Manuscript submitted to ACM + +12 +Venturini, et al. +• Commits: We manually checked the subsequent commits of the client packages that were pushed to their +repositories after the provider release that introduced the respective manifesting breaking change. In particular, +we searched for commits that touched the package.json file. In the file history, we checked if the provider was +downgraded, upgraded, replaced, or removed. +• Changelogs: We analyzed the client changelogs and release notes looking for mentions of provider updates/- +downgrades. About 48% of clients maintained a changelog or release notes in their repositories. +• Pull requests/Issue reports: We searched for pull requests and issue reports in the client repository that +contained information about the manifesting breaking changes. For example, we found pull requests and issue +reports with “Update provider” and “Fix provider error” in the title. +For each manifesting breaking change case, we recovered the provider’s dependency tree. For example, in our second +motivating example (Section 2), we recovered the dependency tree from the client to the package that introduced the +manifesting breaking change, which resulted in broccoli-asset-rev→broccoli-filter→broccoli-plugin (Figure 2). We +investigated how many breaking change cases were introduced by direct and indirect providers, when the manifesting +breaking change was introduced and fixed/recovered, which package fixed/recovered from it, and how it was fixed/re- +covered. We also verified how client packages changed the provider’s versions and how the associated documentation +with manifesting breaking changes related to the time to fix it. +3.3 +Scope and Limitations +As our definition of manifesting breaking changes includes cases that are not included by the prior definitions of +breaking changes (see Section 2.1), this paper does not intend to provide a direct comparison between these two +phenomena. As a result, the stated research questions do not indicate the proportion of manifest breaking changes +that are, in fact, breaking changes as defined by prior literature (e.g., an API change by the provider). In addition, since +provider packages are rarely accompanied by any formal specification of their intended behavior, it is impossible at +the scale of our study to differentiate errors that manifest in the client package due to breaking changes from those +that manifest due to an idiosyncratic usage of the provider by the client package. Therefore, the results of the stated +RQs cannot be used to assess whether a client package could fix its build by simply updating to a newer version of the +provider. +4 +RESULTS +This section presents the associated findings for each RQ. +4.1 +RQ1. How often do manifesting breaking changes occur in the client package? +Finding 1: 11.7% of the client packages (regardless of their releases) and 13.9% of the client releases were +impacted by a manifesting breaking change. From all 384 client packages, 45 (11.7%) suffered a failing test from a +manifesting breaking change in at least one release. From 3,230 client releases for which the tests were executed, 1,276 +failed, and all errors were manually analyzed. In 450 (13.9%) releases, the error was raised by the provider packages, +characterizing a manifesting breaking change. On 86 (2.7%) releases, we could not identify which package raised the +error. +We detected that 261 (8.1%) releases suffered a particular error type that we call breaking due to external change. +These releases used a provider that relied on data/resources from an external API/service (e.g., Twitter) that were no +Manuscript submitted to ACM + +I depended on you and you broke me: An empirical study of manifesting breaking changes in client packages +13 +longer available, impacting all client’s releases. The provider cannot fix this error, because it does not own the resource. +These cases imply that detecting manifest breaking changes by running the clients’ tests can introduce false positives, +which we simply ignored during our manual analyses. We also considered cases in which a provider package was +removed from npm as breaking due to external change. Table 2 shows the results of analyses by releases. +Table 2. Results of releases’ analyses. +Results +Releases (#) +(%) +Success +1954 +60.5 +Fail +Client’s errors +479 +14.8 +manifesting breaking changes +450 +13.9 +Breaking due to external changes +261 +8.1 +Errors not identified +86 +2.7 +Total +3230 +100 +Finding 2: 92.2% providers introduced a single manifesting breaking change. In our sample, 47 providers +(92.2%) of 51 introduced a single release with a manifesting breaking change, and four providers introduced two releases +with manifesting breaking changes. We detected 55 unique manifesting breaking change cases introduced by providers, +some of which impacted multiple clients. For example, the breaking change exhibited in the Incompatible Providers +Versions classification (Finding 3) impacted six clients. Therefore, 64 manifesting breaking change cases manifested in +the client packages. Finally, there were 1,909 providers on all clients’ latest versions, and the percentage of providers +that introduced manifesting breaking change was 2.6% (51 of 1909). +• About 11.7% of clients and 13.9% of their releases suffered from manifesting breaking changes. +• We detected failing tests due to 2% of the providers with changes. +• Over 90% of those that introduced manifesting breaking changes did so through just a single release with a +manifesting breaking change. +4.2 +RQ2. What issues in the provider package caused a breaking change to manifest? +Finding 3: We found 8 categories of issues. We grouped each manifesting breaking change into eight categories, +depending on its root cause (issue). Table 3 presents each category, the number of occurrences, and the number of +impacted client releases. +Manuscript submitted to ACM + +14 +Venturini, et al. +Table 3. The identified categories of manifesting breaking changes. +Category +Cases +Releases +(#) +(%) +(#) +(%) +Feature change +25 +39.1 +101 +22.4 +Incompatible providers versions +15 +23.4 +64 +14.2 +Object type changed +9 +14.1 +213 +47.3 +Undefined object +5 +7.8 +28 +6.2 +Semantically wrong code +5 +7.8 +14 +3.1 +Failed provider update +2 +3.1 +24 +5.3 +Renamed function +2 +3.1 +2 +0.4 +File not found +1 +1.6 +4 +0.9 +Total +64 +450 +In the following, we describe each category and present an example that we found during our manual analysis. +• Feature change: manifesting breaking changes in this category are related to modifications of provider features +(e.g., the default value of variables). An example happens in request@2.17.0 – this version was removed from +npm, but the introduced change remained in the package – when developers introduced a new decision rule into +their code12 as shown in Listing 5. +Listing 5. Example of a manifesting breaking change categorized as feature change. +debug('emitting complete ', self.uri.href) ++ if(response.body == undefined && !self._json) { ++ +response.body = ""; ++ } +self.emit('complete ', response , response.body) +In Listing 5, the provider request assigns an empty string to the response.body variable, instead of preserving +response.body with its default undefined value. +• Incompatible providers versions: In this category, the client breaks because of a change in an indirect provider. +An example happens in the packages babel-eslint and escope, where escope is an indirect provider of babel-eslint. +Listing 6. Incompatible provider’s versions example. +} +- +}, +- +visitClass: { ++ }, { ++ +key: 'visitClass ', +value: function visitClass(node) { +The release escope@3.4 introduced the presented change in Listing 6. This change impacted the package babel- +eslint,13 even though the escope had not been a direct provider to babel-eslint.14 This manifesting breaking +change remained unresolved for a single day, during which babel-eslint received about 80k downloads from npm. +12https://github.com/request/request/commit/d05b6ba +13https://github.com/babel/babel-eslint/issues/243 +14https://github.com/estools/escope/issues/99#issuecomment-178151491 +Manuscript submitted to ACM + +I depended on you and you broke me: An empirical study of manifesting breaking changes in client packages +15 +• Object type changed: We detected 9 (14.06%) cases in which the provider changed the type of an object, resulting +in a breaking change in the client packages. +Listing 7. Object type changed example. +this.setup(); +- this.sockets = []; ++ this.sockets = {}; +this.nsps = {}; +this.connect Buffer = []; +} +var socket = nsp.add(this , function () { +- +self.sockets.push(socket); ++ +self.sockets[socket.id] = socket; +self.nsps[nsp.name] = socket; +In Listing 7, the provider socket.io@1.4.0 turned an array into an object,15. This simple change broke many of +socket.io’s clients, even the package karma,16 a browser test runner, which was forced to update its code17 and +publish karma@0.13.19. During the single day, the manifesting breaking change remained unresolved, karma +was downloaded about 146k times from npm. +• Undefined object: In this category, an undefined object causes a runtime exception that breaks the provider, +which throws the exception to the client package. +Listing 8. Undefined object code example. ++ app.options = app.options || {}; +app.options.babel = app.options.babel || {}; +app.options.babel.plugins = app.options.babel.plugins || []; +This error happened in the provider ember-cli-htmlbars-inline-precompile@0.1.3, which solved it as shown in +Listing 818. +• Failed provider update: In this category, provider A updates its provider B, but provider A does not update its +code to work with the new provider B. We detected two cases of this category. In addition to an explicit update, +one provider A from this category specified its provider B as an accept-all range (>=). Over time, its provider +B published a major release that introduced a manifesting breaking change. Despite provider A specifying an +accept all range, it did not consider the implicit update of provider B and the client suffered an error. +• Semantically wrong code: manifesting breaking changes in this category happen when the provider writes +a semantically wrong code, generating an error in its runtime process19 and affecting the client. These errors +could be caught in compile-time in a compiled language, but in JavaScript these errors happen at runtime. This +occurred in the provider front-matter@0.2.0 and four other cases. +Listing 9. Semantically wrong code example. +const separators = [ '---', '= yaml ='] +- const pattern = pattern = '^(' ++ const pattern = '^(' ++ '((= yaml =)|(---))' +15https://github.com/socketio/socket.io/commit/b73d9be +16https://github.com/socketio/socket.io/issues/2368 +17https://github.com/karma-runner/karma/commit/3ab78d6 +18https://github.com/ember-cli/ember-cli-htmlbars-inline-precompile/pull/5/commits/b3faf95 +19https://hacks.mozilla.org/2017/02/a-crash-course-in-just-in-time-jit-compilers/ +Manuscript submitted to ACM + +16 +Venturini, et al. +On Listing 9, the provider repeated the variable name (pattern) on its declaration, which generated a semantic +error. Although this error can be easily detected and fixed, as the provider did20 in Listing 9, the provider took +almost one year to fix it (front-matter@0.2.2). Meanwhile, front-matter received about 366 downloads in that +period. +• Renamed function: The manifesting breaking changes in this category occur when functions are renamed. +Our analysis revealed 2 cases in which the functions were renamed. The renaming case is our first motivating +example (Section 2); we describe the second one below. +Listing 10. Renamed function code example. +- RedisClient.prototype.send_command = function (command , args , callback) { +- +var args_copy , arg , prefix_keys; ++ RedisClient.prototype.internal_send_command = function (command , args , callback) { ++ +var arg , prefix_keys; +The provider redis@2.6.0-1 renamed a function, as in Listing 10.21 However, this function was used in a client +package fakeredis,22, which broke with this change. Client package fakeredis@1.0.3 recovered from this error by +downgrading to redis@2.6.0-0.23 In the five days period within which the manifesting breaking change was not +fixed, fakeredis received about 2.3k downloads from npm. +• File not found: In the cases in this category, the provider removes a file or adds it to the version control ignore +list (.gitignore) and the client tries to access it. In the unique case of this category in our sample, the provider +referenced a file that was added to the ignore list. +Finding 4: manifesting breaking changes are often introduced in patch releases. As shown in Table 4, of the 64 +cases of manifesting breaking changes we analyzed, three cases were introduced in major releases, 26 in minor releases, +28 in patch releases, and 5 in pre-releases. Although we only analyzed manifesting breaking changes from minor and +patch releases, in three cases the manifesting breaking changes were introduced at major levels in an indirect provider, +which transitively affected client packages—as in the jsdom@16 case (see Section 2). +Table 4. manifesting breaking changes in each Semantic Version level. +Levels +(#) +(%) +Major +3 +4.7 +Minor +28 +43.75 +Patch +28 +43.75 +Pre-release +5 +7.8 +Total +64 +100 +Pre-releases precede a stable release and are considered unstable; anything may change until a stable version is +released.24 In all detected breaking changes in pre-releases, the providers introduced unstable changes in pre-releases +20https://github.com/jxson/front-matter/commit/f16fc01 +21https://github.com/NodeRedis/node-redis/commit/861749f +22https://github.com/NodeRedis/node-redis/issues/1030#issuecomment-205379483 +23https://github.com/hdachev/fakeredis/commit/01d1e99 +24https://semver.org/#spec-item-9 +Manuscript submitted to ACM + +I depended on you and you broke me: An empirical study of manifesting breaking changes in client packages +17 +and propagated these changes to stable versions. An example is the pre-release redis@2.6.0-1 (described in Section 3.2.2), +whose rename of a function propagated to the stable version and caused a failure in the client packages. +Finding 5: manifesting breaking change fixes/recoveries are introduced by both clients and/or providers. We +searched to identify which package fixed/recovered from the manifesting breaking changes – client or provider – and +at which level the fixed/recovered release was published, as depicted in Figure 4. +Breaking changes in minor +Fixed by provider (46.4%) +Breaking changes in patch +Fixed by provider (46.4%) +Patch (76.9%) +Fixed by client (42.9%) +Not fixed (10.7%) +Fixed by client (39.3%) +Not fixed (14.3%) +Others (7.7%) +Minor (15.4%) +Patch (61.5%) +Others (23.1%) +Minor (15.4%) +Fig. 4. Proportion of fixed/recovered manifesting breaking changes by provider and client packages and the respective Semantic +Version level of the fixing/recovering releases. +Figure 4 shows that client packages recover from nearly half of the manifesting breaking change introduced in minor +updates. In turn, 76.9% of the manifesting breaking changes that are introduced by providers in a minor release are +fixed in a patch release. Providers fix the majority of the manifesting breaking changes introduced in patch releases +(46.4% of the time), typically through a patch release (61.5%). +Finding 6: 21.9% of the manifesting breaking changes are not documented. Although clients and providers often +document the occurrence or repair of a manifesting breaking change in issue reports, pull requests, or changelogs, more +than one-fifth of the manifesting breaking changes are undocumented. +Table 5. Summary of the proportion of documented manifesting breaking changes when they are introduced and fixed/recovered. +Documentation +Introduced +Fixed/recovered +Proportion (%) +Issue report +– +32 +64 +Pull request +5 +15 +44 +Changelog +23 +16 +78 +Table 5 shows that client and provider packages documented manifesting breaking changes in 78.1% of all manifesting +breaking changes. Out of all cases that have documentation, 70% have more than one type of documentation. For +example, the provider received an issue report, fixed the manifesting breaking change, and documented it in a changelog. +Documenting manifesting breaking changes and their fixes supports client recovery (Section 3.2.3). +Finding 7: 57.8% of the manifesting breaking changes are introduced by an indirect provider. Indirect providers +might also introduce manifesting breaking changes, which can then propagate to the client. Table 6 shows the depth +level in the dependency tree of each provider that introduced a manifesting breaking change. About 42.2% of manifesting +Manuscript submitted to ACM + +18 +Venturini, et al. +breaking changes are introduced by a direct provider in the client’s package.json. These providers are the ones the client +directly installs and that perform function calls in their own code; they are in the first depth level of the dependency +tree. +Table 6. How deep the provider package that raised a manifesting breaking change is from the client in the dependency tree. +Depth +(#) +(%) +1 +27 +42.2 +2 +30 +46.9 +>3 +7 +10.9 +Total +64 +100 +Manifesting breaking changes introduced by indirect providers in the depth level greater than one represent 57.8% of +the cases. Six cases are in the third depth level and a single one is in the fourth depth level. Clients do not install these +providers directly; rather, they come from the direct provider. In these cases, the manifesting breaking change may be +totally unclear to client packages, since they are typically unaware of such providers (or have no direct control over +their installation). +• The most frequent issues with provider packages that introduced manifesting breaking changes were +feature changes, incompatible providers, and object type changes. +• Provider packages introduced these manifesting breaking changes at similar rates in minor and patch +releases. +• Most of the fixed manifesting breaking changes by providers were fixed in patch releases. +• Manifesting breaking changes are documented in 78.1% of the cases, mainly on issue reports. +• Indirect providers introduced manifesting breaking changes in most cases. +4.3 +RQ3. How do client packages recover from a manifesting breaking change? +Finding 8: Clients and transitive providers recover from breaking changes in 39.1% of cases. In the dependency +tree, the transitive provider is located between the provider that introduced the manifesting breaking change and +the client where it manifested (See Section 2.1). Table 7 shows which package fixed/recovered from each manifesting +breaking change case. The provider packages fixed the majority of the manifesting breaking changes. Since they +introduced the breaking change, theoretically this was the expected behavior. Client packages recovered from the +manifesting breaking change in 20.3% of cases, and transitive providers recovered from manifesting breaking changes in +18.8% of cases. When the provider who introduced a manifesting breaking change does not fix it, the transitive provider +may fix it and solve the client’s issue. +Manuscript submitted to ACM + +I depended on you and you broke me: An empirical study of manifesting breaking changes in client packages +19 +Table 7. Packages fixing/recovering from the error. +Fixed by/Recovered from +(#) +(%) +Provider +32 +50 +Client +13 +20.3 +Transitive provider +12 +18.8 +Client + Transitive provider +25 +39.1 +Not fixed/recovered +7 +10.9 +Total +64 +100 +Since transitive providers are also clients of the providers that introduced the manifesting breaking change, clients +(clients and transitive providers) recovered from these breaking changes in 39.1% of cases. This observation suggests +that client packages occasionally have to work on a patch when a manifesting breaking change is introduced since in +39.1% of the cases clients and transitive providers need to take actions to recover from the manifesting breaking change. +Finding 9: Transitive providers fix manifesting breaking changes faster than other packages: When a mani- +festing breaking change is introduced, it should be fixed by either the provider who introduced it or a transitive provider. +In a few cases, the client package will also recover from it. Table 8 shows the time that each package takes to fix the +breaking change. In general, manifesting breaking changes are fixed in seven days by provider packages. Even in this +relatively short period of time, many direct and indirect clients are affected. +Table 8. Median of number days that each package spent to fix/recover from the manifesting breaking change. +Fixed by/Recovered from +Days +Provider +7 +Client +134 +Transitive provider +4 +Client + Transitive provider +82.4 +Transitive providers fix manifesting breaking changes faster than clients and even providers. Since the manifesting +breaking change only exists when it is raised in the client packages, transitive providers break first and need a quick +fix; transitive providers usually spent four days to fix a break. Meanwhile, providers that introduced the manifesting +breaking change take a median of 7 days to introduce a fix. In cases where the provider neglected to introduce a fix or +took longer than the client, client packages took a comparably lengthy 134 days (mean 286; SD 429) to recover from +a manifesting breaking change. According to Table 7, the direct providers and transitive providers fixed most of the +manifesting breaking changes, about 78.8%, because clients can be slow to recover. +However, because transitive providers are also clients, we can analyze the time that clients and transitive providers +spend to fix/recover from a manifesting breaking change. Clients and transitive providers recovered from a manifesting +breaking change in around 82 days. +Finding 10: Upgrading is the most frequent way to recover from a manifesting breaking change. Table 9 de- +scribes how clients recovered from breaking changes. In 48 cases, the provider version was changed. In most cases +Manuscript submitted to ACM + +20 +Venturini, et al. +(71.4%), client packages upgraded their providers’ version. We analyzed all cases where clients and transitive providers +recovered from the manifesting breaking change by changing the provider’s version before the provider fixed the error. +We observed an upgrade in 12 (52.2%) cases out of 23. Thus, in more than half of the cases where the client and transitive +providers fixed/recovered from the manifesting breaking change, the provider package had newer versions, but the +client was not using any follow-up releases from the provider packages. +Table 9. How client packages changed the provider’s version after a manifesting breaking change. +Changed by +Total +Upgrade +Downgrade +Replace +Remove +(#) +(%) +(#) +(%) +(#) +(%) +(#) +(%) +Client +28 +20 +71.4 +6 +21.4 +1 +3.6 +1 +3.6 +Transitive provider +20 +9 +45 +10 +50 +01 +5 +— +— +The number of downgrades in a transitive provider may explain why they recover from the manifesting breaking +change faster than the client packages. Since transitive providers are also providers, they should fix the manifesting +breaking change as soon as possible, avoiding the propagation of the error caused by the manifesting breaking change. +Consequently, the downgrade to a stable release of the provider is the most frequent way for transitive providers to +recover from a manifesting breaking change. Finally, the provider is replaced or removed in a small proportion when a +breaking change is raised—about 7.2% for both cases combined. +Finding 11: To recover from manifesting breaking changes, clients often change the adopted provider version +without changing the range of automatically accepted versions. When a breaking change manifests itself, clients +often update the provider’s version. Figure 5 shows when the clients and transitive providers updated their providers’ +versions. +steady +caret +all +tilde +steady (66.7%) +caret (33.3%) +caret (53.8%) +steady (38.5%) +all (7.7%) +caret (60.0%) +all (20.0%) +tilde (20.0 %) +tilde (75.0%) +caret (25.0%) +(a) Client +caret +tilde +all +caret (85.7%) +steady (7.1%) +tilde (7.1 %) +tilde (100.0%) +caret (66.7%) +tilde (33.3%) +(b) Transitive Provider +Fig. 5. Provider’s version changed by clients and transitive providers. On the left side of each figure, one can see the range level +where the manifesting breaking change was introduced and on the right side, one can see the range level where the same manifesting +breaking change was fixed. +Manuscript submitted to ACM + +I depended on you and you broke me: An empirical study of manifesting breaking changes in client packages +21 +We verified that transitive providers never set a steady version of their provider. When a breaking change manifests in +transitive providers, they use a range in the provider’s version. However, a single transitive provider changed the range +from a caret range to a steady one (e.g., ^1.2.1 → 1.2.1), to recover from the manifesting breaking change. Nevertheless, +when the clients used a caret range and a breaking change manifested, in 38.5% of the cases they downgraded the +provider to a steady version. +The majority of the manifesting breaking changes were introduced when the clients and transitive providers used +the caret range (^). It is the default range statement that npm inserts in the package.json when a provider is added as a +dependency of a client package. In more than half of the cases, these clients changed the provider’s version to another +caret range. The accept all ranges (>=, or *) were less commonly used and less common when updating. +Clients and the transitive provider in 60.5% of cases retained the range type and updated it. The range type (all, caret, +tilde, or steady) was kept, but the provider was updated/downgraded. For example, a client package specifies a provider +p@^1.2.0 and receives a breaking change in p@1.3.2. Whenever the provider fixes the code, the client package will +update it to, for example, p@^1.4.0, but will not change it for another range type, such as all, tilde, or steady range. +• Client packages recovered manifesting breaking changes in 39.1% of cases, including clients and transitive +providers. +• Providers fixed manifesting breaking changes faster than client packages recovered from manifesting +breaking changes by updating the provider, and clients preferred to update rather than downgrade their +providers. +• The provider’s range can be updated or downgraded after a breaking change, but in around 60% of cases, +they did not change the range type. +5 +DISCUSSION +This section discusses the implications of our findings for dependency management practices (Section 5.1) and the +best practices that clients and providers can follow to mitigate the impact caused by manifesting breaking changes +(Section 5.2). We also discuss the manifestation of breaking changes and the aspects of Semantic Versioning in the npm +ecosystem (Section 5.3). +5.1 +Dependency management +When managing dependencies, client packages can use dependency bots in GitHub, such as Snyk and Dependabot, +to receive automatic pull requests when there is a new provider’s release [27]. These bots continuously check for +new versions and providers’ bugs/vulnerabilities fixes. They open pull requests in the client’s repository, updating the +package.json, including changelogs and information about the provider’s new version. Mirhosseini and Parnin [16] show +that packages using such bots update their dependencies 1.6x faster than through manual verification. Additionally, +tools such as JSFIX [20] can be helpful when upgrading provider releases, especially those that include manifesting +breaking changes or major releases. The JSFIX tool was designed to adapt the client code to the new provider release, +offering a safe way to upgrade providers. +We verified that a small percentage of the clients recovered from manifesting breaking changes by removing or +replacing the provider (c.f., Finding 10), which may be difficult when several features or resources from the provider +package are used by the client [2]. Instead, client packages tend to temporarily downgrade to a stable provider version. To +ease the process to upgrade/downgrade providers and avoid surprises, clients should search in the provider changelogs +Manuscript submitted to ACM + +22 +Venturini, et al. +for significant changes. As we verified in Finding 6, most manifesting breaking changes are documented in changelogs, +issue reports, or pull requests. Dependency bots also could analyze the content of changelogs and issue reports to create +red flags, like notifications, about documentation that cites a manifesting breaking change. +Finally, client packages may use a package-lock.json file to better manage dependencies. We observed in Finding 7 +that indirect providers – the ones in depth two and three in the dependency tree – are responsible for 57.8% of the +manifesting breaking changes that affect a client package. Using a package-lock.json file, client packages can stay aware +of all of the providers’ versions of the latest successful build. When a provider is upgraded due to the range of versions +and the new release manifests a breaking change on the client side, the client can still install all of the providers’ +versions that successfully built the client. +5.2 +Best practices +Several issues found in our manual classification of manifesting breaking changes (Section 3.2.2) could be avoided +through the use of static analysis tools. Errors classified as Semantically Wrong Code and Rename function are typically +captured by such tools. Both client and provider developers can use such tools. For a dynamic language such as +JavaScript, these tools can help avoid some issues [26]. Options for JavaScript include jslint, jshint and standard. +Tómasdóttir et al. [26] and Tómasdóttir et al. [25] show that developers use linters mainly to prevent errors, bugs, and +mistakes. +Due to the dynamic nature of JavaScript, however, static analysis tools cannot verify inherited objects’ properties. +They do not capture errors classified as Change one rule, Object type change, and Undefined object, as well as Rename +Function in functions of object’s properties. Thus, developers should be concerned about creating test cases that run +their code along with the functionality of providers, as only then will they (client developers) find breaking changes +that affect their own code. Many available frameworks, such as mocha, chai, and ava, support these tasks. These tests +should also be executed on integrated environments every time the developer commits and pushes new changes. For +this case, several tools are available, such as Travis, Jenkins, Drone CI, and Codefresh. Using linters and continuous +integration systems, developers can catch most of these errors before releasing a new version. +Finally, a good practice for npm packages is to keep a changelog or to document breaking changes and their fixes +in issue reports and pull requests. This practice should continue and be more widely adopted, since currently around +a fifth of providers do not do it (c.f., Finding 6). This would also help the development of automated tools (e.g. bots) +for dealing with breaking changes. Providers could create issue reports and pull request templates to allow clients to +specify consistent descriptions of issues they found. +5.3 +Breaking changes manifestation and Semantic Versioning +Breaking changes often occur in the npm ecosystem and impact client packages (c.f., Finding 1). Most of the manifesting +cases come from indirect providers; that is, providers from the second level or deeper in the dependency tree. Findings +from Decan et al. [10] show that in 2016 half of the client packages in npm had at least 22 transitive dependencies +(indirect providers), and a quarter had at least 95 transitive dependencies. In this context, clients may face challenges +in diagnosing where the manifesting breaking changes came from, because when a manifesting breaking change is +introduced by an indirect provider, the client may not know this provider. +Our results show that provider packages introduce manifesting breaking changes in minor and patch levels, which +in principle should only contain backward-compatible updates according to the Semantic Versioning specification. +Semantic Versioning is a recommendation that providers can choose to use it or not [4, 8]. If providers do not comply +Manuscript submitted to ACM + +I depended on you and you broke me: An empirical study of manifesting breaking changes in client packages +23 +with Semantic Versioning, several errors might be introduced, as we observed in Finding 4 that all manifesting breaking +changes in pre-releases were propagated to stable releases (c.f., Finding 4). One hypothesis is that providers might be +unaware of the correct use of the Semantic Versioning rules, which may explain why they propagated the unstable +changes to stable releases. Finally, npm could provide badges where provider packages would be able to explicitly show +that they are aware of and adhere to the Semantic Versioning. Trockman [24] claims that developers use visible signals +(specifically on GitHub) like badges to indicate project quality. This way, clients could make a better choice about their +providers and prefer those aware of Semantic Versioning. +6 +RELATED WORK +This section describes related work regarding breaking changes in npm and other ecosystems. +Breaking changes in npm: Bogart et al. [5] presents a survey about the stability of dependencies in the npm and CRAN +ecosystem. The authors interviewed seven package maintainers about software changes. In this paper, interviewees +highlighted the importance of adhering to Semantic Versioning to avoid issues with dependency updates. More recently, +the authors investigated policies and practices in 18 software ecosystems, finding that all ecosystems share values such as +stability and compatibility, but differ on other values [4]. Kraaijeveld [14] studied API breaking changes in three provider +packages. The author uses 3k client packages, parsing the providers’ and clients’ files to detect API-breaking changes +and their impact on clients. This work identified that 9.8% to 25.8% of client releases are impacted by API-breaking +changes. +Mezzetti et al. [15] present a technique called type regression testing that verifies the type of a returned object from +an API and compares it with the returned type in another provider release. The authors chose the 12 most popular +provider packages and their major releases, applying the technique in all patch/minor releases belonging to the first +major update. They verified type regression in 9.4% of the minor or patch releases. Our research focused on any kind of +manifesting breaking changes and we analyzed both client and provider packages, with 13.9% of releases impacted by +manifesting breaking changes. +Mujahid et al. [19] focus on detecting break-inducing versions of third-party dependencies. The authors analyzed +290k npm packages. They flagged each downgrade in the provider version as a possible breaking change. These provider +versions were tested using client tests and the authors identified 4.1% of fails after an update, which resulted in a +downgrade. Similar to these authors, we resolved each client’s providers for a release, but we ran the tests whenever at +least one provider version changed. +Møller et al. [17] present a tool that uses breaking change patterns described by providers and fixes the client code. +They analyzed a dataset with ten of the most used npm packages and searched for breaking changes described in +changelogs. We can compare our classification (Finding 3) with theirs. They found 153 cases of breaking changes that +were introduced in major releases. They claim that most of the breaking changes (85%) are related to specific package +API points, such as modules, properties, and function changes. Considering our classification (Finding 3), feature changes, +object type changed, undefined object, and renamed function can also be classified as changes in the package API and, if +so, we claim that 64.06% of manifesting breaking changes are package API related. +Breaking changes in other ecosystems: Brito et al. [6] studied 400 providers from the Maven repository for 116 days. +The provider packages were chosen by popularity on GitHub and the authors looked for commits that introduced an +API-breaking change during that period. Developers were asked about the reasons for breaking changes that occurred. +Our paper presents similar results: the authors claim that New Feature is the most frequent way a breaking change +Manuscript submitted to ACM + +24 +Venturini, et al. +is introduced, while we claim that Feature Change is the main breaking change type (Finding 3). Also, the authors +similarly detected that breaking changes are frequently documented on changelogs (Finding 6). +Foo et al. [12] present a study about API breaking changes in the Maven, PyPI, and RubyGems ecosystems. The study +focuses on detecting breaking changes by computing a diff between the code of two releases. They found API-breaking +changes in 26% of provider packages, and their approach suggests automatic upgrades for 10% of the packages. Our +approach goes beyond API breaking changes; we found that 11.7% of the client packages are impacted by manifesting +breaking changes. +7 +THREATS TO VALIDITY +Internal validity: When a breaking change was detected, we verified the type of change that the provider package +introduced and collectively grouped the changes into categories. However, some cases might fall into more than one +category. For example, a provider package changes the type of an object to change/improve its behavior. This case +might fall into Feature change and Object type changed. So, we categorized the case in the category that most represents +the error. In this case, since the object is changed by a feature change, the most appropriate category would be Feature +change. +The error cases that we categorized as breaking due to external change are the ones in which the clients or providers +use – or depend on – external data/resources from sites and APIs that changed over time (see Finding 1). These cases +represent about 8.1% of the client’s releases, and, in these cases, we could not search for manifesting breaking changes +because we could not execute the release tests. After all, the data/resource needed by the test were no longer available. +So, about 8% of client releases might be impacted by breaking changes, but we could not analyze them. +Construct validity: In our approach to detecting breaking changes, we only performed an analysis when the client +tests failed. If a client used a provider version that had a breaking change, but the client did not call the function that +causes the breaking change or did not have tests to exercise that code, we could not detect the breaking change. This is +why we call all of our cases manifesting breaking changes. +Therefore, we might not have detected all API-breaking changes, as we were able to detect only API name changes +and API removal. Parameter changes may not be detected because JavaScript allows making a call to an API with any +number of parameters.25 +We restored the working tree index in the respective commit tagged by the developer for each release. We listed all +tags in the repository, and we used the checkout with the respective tag. However, for untagged releases we performed +a checkout in the timestamp referenced in the package.json. We trusted the timestamp once we verified that the tags +and timestamp point to the same commit in 94% of cases for tagged repositories. +Lastly, we did not mention the file npm-shrinkwrap.json in our study. This file is intended to work like the file +package-lock.json when controlling transitive dependency updates, but it may be published along with the package. +However, npm strongly recommend avoiding its use. Also, the existence of npm-shrinkwrap.json files does not play +any major role in our study, as they do not affect our results, based on our adopted research method. We did not include +them in our study. +External validity: We randomly selected client packages that varied in release numbers, clients, providers, and size. +However, since we only analyzed npm packages hosted at GitHub projects, our findings cannot be directly generalized +to other settings. It is also important to state that representativeness can also be limited because npm increases the +25https://eloquentJavaScript.net/03_functions.html#p_kzCivbonMM +Manuscript submitted to ACM + +I depended on you and you broke me: An empirical study of manifesting breaking changes in client packages +25 +number of packages and releases daily. Future work can replicate our study in other platforms and ecosystems. Finally, +since the number of projects in our sample is small, we do not have enough statistical power to perform hypothesis +tests around results that involve package-level comparisons. +Conclusion validity: Conclusion validity relates to the inability to draw statistically significant conclusions due to +the lack of a large enough data sample. However, as our research used a qualitative approach, we mitigate any potential +conclusion threat by conducting a sanity check on repositories of all client packages with fewer than four releases. +This guarantees that all packages are intended for use in production (Subsection 3.1.2). Finally, all of the manifesting +breaking changes that we claim in our work were manually analyzed to assure they are legitimate breaking changes +that impact clients in the real world (Subsection 3.1.3). +8 +CONCLUSIONS +Software reuse is a widely adopted practice, and package ecosystems such as npm support reusing software packages. +However, breaking changes are a negative side effect of software reuse. Breaking changes and their impacts are studied +in the literature in several software ecosystems [3, 6, 18, 28]. A few papers examine breaking changes in the npm +ecosystem from the client packages perspective, i.e., executing the client tests to verify the impact of breaking changes +[5, 15, 19]. In this work, we analyzed manifesting breaking changes in the npm ecosystem from the client and provider +perspectives, providing an empirical analysis regarding breaking changes in minor and patch levels. +From the client’s perspective, we analyzed the impact of manifesting breaking changes. We found that 11.7% of +clients are impacted by such changes and offer some advice to help clients and automated tools developers discover, +avoid, and recover from manifesting breaking changes. Clients can use dependency bots to accelerate the process +of upgrading their providers, and clients can look at changelog files for any non-desired updating, such as breaking +changes. From the provider’s perspective, we analyzed the most frequent causes of manifesting breaking changes. We +found that the most common causes were when providers changed some rules/behaviors on features that had been +stable over the last releases, when an object type changes, and when there were unintentionally undefined objects at +runtime. Maintainers should pay attention during code review phases regarding these issues. Future research can look +into the correlation among package characteristics and metrics with breaking change occurrence. +9 +ACKNOWLEDGMENTS +This work is partially supported by the National Science Foundation under Grant Number IIS-1815503, CNPq/MC- +TI/FNDCT (grant #408812/2021-4 ) and MCTIC/CGI/FAPESP (grant #2021/06662-1). +REFERENCES +[1] 2018. This year in JavaScript: 2018 in review and npm’s predictions for 2019. https://blog.npmjs.org/post/180868064080/this-year-in-javascript- +2018-in-review-and-npms.html +[2] Hussein Alrubaye and Mohamed Wiem Mkaouer. 2018. Automating the Detection of Third-Party Java Library Migration at the Function Level. 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The power +of bots: Characterizing and understanding bots in oss projects. Proceedings of the ACM on Human-Computer Interaction 2, CSCW (2018), 1–19. +[28] Jooyong Yi, Dawei Qi, Shin Hwei Tan, and Abhik Roychoudhury. 2013. Expressing and Checking Intended Changes via Software Change +Contracts. In Proceedings of the 2013 International Symposium on Software Testing and Analysis (ISSTA 2013). Lugano, Switzerland, 1–11. https: +//doi.org/10.1145/2483760.2483772 +[29] Ahmed Zerouali, Eleni Constantinou, Tom Mens, Gregorio Robles, and Jesus Gonzalez-Barahona. 2018. An Empirical Analysis of Technical Lag in +npm Package Dependencies. https://doi.org/10.1007/978-3-319-90421-4_6 +Manuscript submitted to ACM + diff --git a/idE3T4oBgHgl3EQfggra/content/tmp_files/load_file.txt b/idE3T4oBgHgl3EQfggra/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..00dc2b66bca011f3d62aabaafbe561ac79726644 --- /dev/null +++ b/idE3T4oBgHgl3EQfggra/content/tmp_files/load_file.txt @@ -0,0 +1,1392 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf,len=1391 +page_content='I depended on you and you broke me: An empirical study of manifesting breaking changes in client packages DANIEL VENTURINI, Federal University of Technology (UTFPR), Brazil FILIPE ROSEIRO COGO, Huawei Technologies, Canada IVANILTON POLATO, Federal University of Technology (UTFPR), Brazil MARCO A GEROSA, Northern Arizona University (NAU), United States IGOR SCALIANTE WIESE, Federal University of Technology (UTFPR), Brazil Complex software systems have a network of dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Developers often configure package managers (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=', npm) to automatically update dependencies with each publication of new releases containing bug fixes and new features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' When a dependency release introduces backward-incompatible changes, commonly known as breaking changes, dependent packages may not build anymore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' This may indirectly impact downstream packages, but the impact of breaking changes and how dependent packages recover from these breaking changes remain unclear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' To close this gap, we investigated the manifestation of breaking changes in the npm ecosystem, focusing on cases where packages’ builds are impacted by breaking changes from their dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' We measured the extent to which breaking changes affect dependent packages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Our analyses show that around 12% of the dependent packages and 14% of their releases were impacted by a breaking change during updates of non-major releases of their dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' We observed that, from all of the manifesting breaking changes, 44% were introduced both in minor and patch releases, which in principle should be backward compatible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Clients recovered themselves from these breaking changes in half of the cases, most frequently by upgrading or downgrading the provider’s version without changing the versioning configuration in the package manager.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' We expect that these results help developers understand the potential impact of such changes and recover from them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' CCS Concepts: • Software and its engineering → Software evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Additional Key Words and Phrases: breaking changes, semantic version, npm, dependency management, change impact ACM Reference Format: Daniel Venturini, Filipe Roseiro Cogo, Ivanilton Polato, Marco A Gerosa, and Igor Scaliante Wiese.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' I depended on you and you broke me: An empirical study of manifesting breaking changes in client packages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' ACM Meas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' 37, 4, Article 111 (October 2021), 26 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='5281/zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='5558085 1 INTRODUCTION Complex software systems are commonly built upon dependency relationships in which a client package reuses the functionalities of provider packages, which in turn depend on other packages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' To automate the process of installing, Authors’ addresses: Daniel Venturini, danielventurini@alunos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='utfpr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='br, Federal University of Technology (UTFPR), Campo Mourão, Paraná, Brazil;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Filipe Roseiro Cogo, filipe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='cogo@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='com, Huawei Technologies, Kingston, Canada;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Ivanilton Polato, ipolato@utfpr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='br, Federal University of Technology (UTFPR), Campo Mourão, Paraná, Brazil;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Marco A Gerosa, Marco.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='Gerosa@nau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='edu, Northern Arizona University (NAU), Arizona, Flagstaff, United States;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Igor Scaliante Wiese, igor@utfpr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='br, Federal University of Technology (UTFPR), Campo Mourão, Paraná, Brazil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Copyrights for components of this work owned by others than ACM must be honored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Abstracting with credit is permitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Request permissions from permissions@acm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' © 2021 Association for Computing Machinery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Manuscript submitted to ACM Manuscript submitted to ACM 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='04563v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='SE] 11 Jan 2023 2 Venturini, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' upgrading, configuring, and removing dependencies, package managers such as npm, Maven, pip, and Cargo are widely adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Despite the many benefits brought by the reuse of provider packages, one of the main risks client packages face is breaking changes [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Breaking changes are backward-incompatible changes performed by the provider package that renders the client package build defective (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=', a change in a provider’s API).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' When client packages configure package managers to automatically accept updates on a range of provider package versions, the breaking change will have the serious consequence of catching clients off guard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' For example, in npm, where most of the packages follow the Semantic Versioning specification [23], clients adopt configurations that automatically update minor and patch releases of their providers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' In principle, these release types should not contain any breaking changes, as the semantic version posits that only major updates should contain breaking changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' However, minor or patch releases occasionally introduce breaking changes and generate unexpected errors in the client packages when these breaking changes manifest on clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Due to the transitive nature of the dependencies in package managers, unexpected breaking changes can potentially impact a large proportion of the dependency network, preventing several packages from performing a successful build.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Research has shown that providers occasionally incorrectly use the Semantic Versioning specification [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' In the npm ecosystem, prior research has shown that provider packages indeed publish releases containing breaking changes [14, 15, 18, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' However, such studies provide limited information regarding the prevalence of these breaking changes, focusing on API breaking changes without clarifying how the client packages solve the problems they cause.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' In this paper, we fill this gap by conducting an empirical study of npm projects hosted on GitHub, verifying the frequency and types of the breaking changes that manifest as defects in client packages and how clients recover from them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' npm is the main package manager for the JavaScript programming language, with more than one million packages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' An estimated 97% of web applications come from npm [1], making it the most extensive dependency network [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' We employed mixed methods to identify and analyze the types of manifesting breaking changes – changes in a provider release that render the client’s build defective – and how client packages deal with them in their projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' This paper does not study cases in which a breaking change does not manifest itself in other projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Our research answers the following questions: RQ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' To what extent do breaking changes manifest themselves in client packages?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' We analyzed 384 packages selected using a random sampling approach (95% confidence level and ±5% confidence interval) to select client packages with at least one provider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' We found that manifesting breaking changes impacted 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='7% of all client packages (regardless of their releases) and 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='9% of their releases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' In addition, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='6% of providers introduced manifesting breaking changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' RQ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' What changes in the provider packages manifest a breaking change?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' The main causes of manifesting breaking changes were feature modifications, change propagation among dependencies, and data type modifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' We also verified that an equal proportion of manifesting breaking changes was introduced in minor and patch releases (approximately 44% in each release type).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Providers fixed most of the manifesting breaking change cases introduced in minor and patch releases (46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='4% and 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='5%, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Finally, manifesting breaking changes were documented in issue reports, pull requests, or changelogs in 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='1% of cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' RQ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' How do client packages recover from manifesting breaking changes?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Client packages recovered from manifesting breaking changes in 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='1% of the cases, and their recovery took about 134 days when providers did not fix the break or when clients recovered first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' When providers released a fix to a manifesting breaking change, they took a median of seven days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Upgrading the provider is the most frequent way client packages recover from a manifesting breaking change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Manuscript submitted to ACM I depended on you and you broke me: An empirical study of manifesting breaking changes in client packages 3 This paper contributes to the literature by providing quantitative and qualitative empirical evidence about the phenomenon of manifesting breaking changes in the npm ecosystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Our qualitative study may help developers understand the types of changes that manifest defects in client packages and which strategies are used to recover from breaking changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' We also provide several suggestions about how clients and providers can enhance the quality of their release processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' As an additional contribution, we created pull requests for real manifesting breaking change cases that had not yet been resolved, half of which were merged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' 2 DEFINITIONS, SCOPE AND MOTIVATING EXAMPLES This section defines terms used in this paper as well as describes motivating examples for our research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='1 Glossary definitions In the following, we describe the terms and definitions that we use in the paper, based on related work [7, 11, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' body-parser express ember-cli bytes Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' An example of a dependency tree, with clients and providers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' provider package release is the package release that provides features and resources for use by others packages releases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' In Figure 1, the package express is a provider of ember-cli, body-parser is a provider of express, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' We refer to a provider package 𝑃 as a transitive provider when we want to emphasize that 𝑃 has other provider packages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' For instance, in Figure 1, body-parser is a provider of express;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' body-parser also has bytes as a provider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' In this scenario, we consider body-parser to be a transitive provider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' client package release is the package release that uses features and resources exposed by provider package releases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' In Figure 1, express is a client of body-parser, body-parser is a client of bytes, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' direct provider release is the one directly used by its client, that is, the package that the client explicitly declares as a dependency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' In Figure 1, express is a direct provider of ember-cli, and bytes is a direct provider of body-parser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' indirect provider release is a package release that at least one of its providers uses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' In other words, it is a provider of at least one of the direct client’s providers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' In Figure 1, both body-parser and bytes are indirect providers of ember-cli, and bytes is an indirect provider of express.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' transitive provider release is the package release between the one that introduced a breaking change and the client.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' For example, if a breaking change is introduced by bytes, in Figure 1, and affects client ember-cli, both packages express and body-parser are transitive providers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' This is because the breaking change transited through these packages (body-parser and express) to arrive at client ember-cli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' The transitive providers are all also impacted by the breaking change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' version statement: a client can specify its provider’s versions on package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='json, a metadata file used by npm to specify providers and their versions, among other purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' The version statement contains the accepted version of a provider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' For example, the version statement in the following metadata {"dependencies": {"express": "^4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='6"}}, defines that the client requires express on version ^4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' version range: on the version statement a client can specify a range of versions/releases accepted by its provider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' There are three types of ranges: Manuscript submitted to ACM 4 Venturini, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' – all (>=, or *): using this range, the client specifies that all new provider releases are supported/accepted and downloadable, even the ones with breaking changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' – caret (^): with this range, the client specifies that all new provider releases that contain new features and bug fixes are supported/accepted and downloadable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' breaking changes must be avoided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' This is the default range used by npm when a dependency is installed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' – tilde range (�): this range specifies that all new provider releases that only contain bug fixes are support- ed/accepted and downloadable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' breaking changes and new features must be avoided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' – steady range: this range always resolves to a specific version and is also known as specific range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' That is, the versioning statement has no range on it but rather a specific version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' npm allows installation with a steady range using the command line option –save-exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' implicit and explicit update: an implicit update happens when the client receives a new provider version due to the range version in the package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='json.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' For a version statement defined with a range of versions, for example, ^4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='6, an implicit update happens when npm installs a version 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='9 that matches the range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' An explicit update takes place when the client manually updates the versioning statement directly in the package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='json.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' manifesting breaking changes are provider changes that manifest as a fault on the client package, ultimately breaking the client’s build.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' The adopted definition of breaking change by the prior literature [3–6, 8, 15, 19, 21] includes cases that are not considered breaking changes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=', a change in an API that is not effectively used by a client package).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Conversely, manifesting breaking changes include cases that are not covered by the prior definitions of breaking change (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=', because the provider package is used in a way that is not intended by the provider developer, a semantic-version compliant change introduced by a new release of this provider causes an expected error in the client package).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='2 Motivating Examples We found the following two examples of manifesting breaking changes in our manual analysis (on each of the following Listing, red lines have been removed from the source code whereas blue lines have been inserted into the source code).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Our manual analysis (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='1) consists of executing the client tests suite for its releases and analyzing all executions that run into an error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' The client assetgraph-builder@7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='0 has a provider assetgraph@6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='0 that has a provider terser@^4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='0, but, due to a range of versions, npm installed terser@4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Release 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='0 of terser introduces a change which, by default, enables the wrapping of functions on parsing, as Listing 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='1 Listing 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Diff between terser@4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='1 and terser@4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='0 default behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' // terser@4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='1 without default wrapping behavior foo(function (){});' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' // terser@4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='0 default wrapping behavior foo(( function (){}));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' This change breaks the assetgraph-builder@7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='0’s tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='2 Once this feature is turned a default behavior, the client assetgraph-builder@8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='0 adopts its test to make it compatible with the terser’s behavior, as Listing 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='3 Listing 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Diff with assetgraph@8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='0 client’s tests adjusting to breaking change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' 1https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='com/terser/terser/compare/v4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='.v4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='0 2https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='com/terser/terser/issues/496 3https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='com/assetgraph/assetgraph-builder/commit/e4140416e7feaa3d088cf3ad0229fd677ff36dbc Manuscript submitted to ACM I depended on you and you broke me: An empirical study of manifesting breaking changes in client packages 5 expect( javaScriptAssets [0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='text , \'to match\', /SockJS =[\\s\\S]* define \\(" main",function \\(\\) \\{\\}\\) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='/ + /SockJS =[\\s\\S]* define \\(" main ",\\(?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' function \\(\\) \\{\\}\\) ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='\\);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='/ );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Sometimes, provider changes can break a client long after their introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' This occurred in the client package ember-cli-chartjs@2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' In Figure 2, the release 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='4 of ember-cli-qunit (left-tree) introduced a change that did not lead to a breaking change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' However, almost three years later, ember-cli-qunit was used together with the release 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='1 of the provider broccoli-plugin (middle-tree), and a breaking change manifested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' ember-cli-qunit@1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='4 ember-cli-chartjs@2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='1 ember-cli-chartjs@2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='1 Nov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' 2015 Aug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' 2018 Apr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' 2020 ember-cli-qunit@1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='2 broccoli-asset-rev@2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='3 broccoli-filter@1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='3 broccoli-plugin@1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='1 ember-cli-qunit@1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='2 broccoli-asset-rev@2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='3 broccoli-filter@1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='0 broccoli-plugin@1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='1 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' The evolution of the dependency tree (resolved versions) for ember-cli-chartjs@2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='1 when it was published (middle-tree) and when the associated tests with the release were executed in our study (right-hand tree).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' In November 2015, the provider ember-cli-qunit@1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='4 fixed an error in its code, changing the returned object type of function lintTree,4 as shown in Listings 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Despite being a type change, it did not break the client when it was released, and this fix was retained in further releases of ember-cli-qunit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Listing 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' ember-cli-qunit@1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='4 object type change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' lintTree: function(type , tree) { // Skip if useLintTree === false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' if (this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content="options['ember -cli -qunit '] && ." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' ) { return tree;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' + // Fakes an empty broccoli tree + return { inputTree: tree , rebuild: function () { return [];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' } };' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' } Almost three years later, on Aug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' 2018, the provider broccoli-plugin@1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='1 was released (middle-tree in Figure 2) to fix a bug,5 as in Listing 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Listing 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' broccoli-plugin@1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='1 validation function enhanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=" function isPossibleNode(node) { return typeof node === 'string ' || (node !" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content="== null && typeof node === 'object ') + var type = typeof node;" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' + if (node === null) { + return false;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=" + } else if (type === 'string ') { ." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' + } else { 4https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='com/ember-cli/ember-cli-qunit/commit/6fdfe7d 5https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='com/broccolijs/broccoli-plugin/commit/3f9a42b Manuscript submitted to ACM 6 Venturini, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' + return false;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' + } The release 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='1 of the broccoli-plugin package experienced a manifesting breaking change due to a fix in the provider ember-cli-qunit@1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='4,6 which was released almost three years prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' This manifesting breaking change occurred because the ember-cli-chartjs’ dependency tree evolved over time due to the range versions, as shown in Figure 2, causing the break.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' When the package ember-cli-chartjs@2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='1 was installed on April 2020 (the date of our analysis), the installation failed due to the integration of broccoli-plugin@1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='1 changes into ember-cli-qunit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Fifteen days later, ember-cli-qunit@1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='3 fixed the issue when the ember-cli-qunit’s object type was changed again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='7 During the fifteen-day period when the manifesting breaking change remained unresolved, broccoli-plugin received about 384k downloads from npm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' This scenario shows that even popular and mature projects can be affected by breaking changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Although we recognize that the download count does not necessarily reflect the popularity of a package, we use this metric as an illustrative example of how many client packages might have been impacted by a provider package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' 3 STUDY DESIGN This section describes how we collected our data (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='1) and the motivation and approach for each RQ (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='1 Data Collection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='1 Obtaining metadata from npm packages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' The first part of Figure 3 shows our approach for sampling the database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' We initially gathered all the metadata files (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=', package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='json files) from the published packages in the npm registry between December 20, 2010 and April 01, 2020, accounting for 1,233,944 packages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' This range refers to the oldest checkpoint that we could retrieve and the most recent one when we started this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' We ignored packages that did not have any providers in the package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='json since they cannot be considered client packages and will therefore not suffer breaking changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' After filtering packages without a provider, our dataset comprises 987,595 package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='json metadata files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' For each release of each package, we recorded the timestamp of the release and the name of the providers with their respective versioning statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' We parsed all the versioning statements and determined the resolved provider version at the time of each client release.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Prior works have adopted similar approaches when studying dependency management [7, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' For each provider in each client release, we retrieved the most recent provider version that satisfied the range specified by the client in that release;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=', the resolved version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Using this resolved version, we determined whether a provider changed its version between the two client releases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' In other words, we reproduced the adopted versions of all providers by resolving the provider version at the release time of the client.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' To further refine our sample, we analyzed two criteria in the associated package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='json snapshot with the latest version of the client packages in our dataset: (1) The package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='json snapshot should have a non-empty entry for the “script test” field, and the entry should differ from the default: Error: no test specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' We specified this criterion in order to run the automated tests that were part of our method to detect manifesting breaking changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' In total, 488,805 packages remained after applying this criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' 6https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='com/broccolijs/broccoli-merge-trees/issues/65 7https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='com/ember-cli/ember-cli-qunit/commit/59ca6ad Manuscript submitted to ACM I depended on you and you broke me: An empirical study of manifesting breaking changes in client packages 7 (2) The package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='json snapshot should have an entry containing the package’s repository URL, as we wanted to retrieve information from the package codebase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' After applying this criterion, 410,433 packages remained in our dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Clone the repository Restore the next release Update providers into package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='json to resolved version NO YES Did any provider change?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Change the Node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content="js version Run install/test Save the result Running install/test 1,233,944 metadata files (packages) from npm registry Filtering client packages: 987,595 Sampling packages Filtering valid test scripts: 488,805 Filtering valid repositories URL: 410,805 Breaking change detection Restore client's releases Sampling with 95% confident level and ±5% confident interval: 384 Fig." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Approach to sampling the database and executing the associated tests with the client release.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='2 Running clients’ tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Given the size of our dataset (more than 410,000 client packages), we ran tests on a random sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' At a 95% confidence level and ±5% confidence interval, we randomly selected 384 packages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Our sample has a median of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='5 releases and 9 direct providers per package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' We chose to study a random sample since our manual analysis is slow to run over a large dataset (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' we spent a month executing our method in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' We did not ignore packages based on the number of releases or providers or any other metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' We performed a manual check on all selected packages that had fewer than four releases (130 out of 384) by checking their repositories and aiming to remove packages that are not real projects, lack tests, lack code, are example projects, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' When we removed one package, we sampled another one following the two criteria described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' The second part of Figure 3 depicts our approach to running the test scripts for each release of the 384 clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' For each client package, we cloned its repository – all client repositories are hosted on GitHub – and restored the work tree of all releases using their respective release tags (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=', “v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='0”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' For releases that are not tagged, we used their provided timestamp in the package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='json metadata to restore the work tree (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=', we matched the release timestamp and the closest existing commit in the master branch).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' We conducted an analysis and verified that tags and timestamp point to the same commit in 94% of releases with tags, thus checkout based on timestamps is reliable for untagged releases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' After restoring the work tree of a client release, we updated all versioning statements in the associated package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='json entry with the specific resolved provider version (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' We then excluded a file called package-lock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='json, which locks the providers and indirect providers versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' We also executed the associated tests on a release of the client package whenever a provider package changed in that release, as this can potentially introduce a manifesting breaking change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' A provider change can be: 1) a provider added into the package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='json;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' or 2) the resolved version of a provider changed between the previous and current release of the client package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' We sought to reproduce the same build environment that existed when the provider changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Therefore, before executing the tests of the client packages, we performed a best-effort procedure to identify the Node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='js that was adopted Manuscript submitted to ACM 8 Venturini, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' by the client package at the time the provider changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' This was because every six months a new major version of Node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='js is released.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' 8 As we wanted to reproduce the test results with respect to the time when the client package published its release, we changed the Node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='js version before executing the client package tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' We selected the Node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='js version using two different approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Our preferred approach was to select the same Node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='js version as the one specified in the engines→node field of the package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='json file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='9 This field allows developers to manually specify the Node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='js version that runs the associated code with the build of a specific release.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' When this field was not set, we selected the latest Node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='js version available10 at the time of the client package release.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Therefore, we changed the Node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='js version, executed the install script, and released tests using the npm install and npm test commands, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' If the install or test commands failed due to incompatibilities with the selected Node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='js version – or took more than 10 minutes –, we changed to the previous major release of Node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='js until the install and test commands succeeded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' We used the Node Version Manager (NVM) tool to exchange Node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='js versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Additionally, we also changed the npm version according to the Node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='js version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' npm is the package manager to Node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='js packages and executes the install and test scripts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' We performed the same procedure to select the npm version to use during the installation and test runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Finally, we executed the install/test scripts and saved the results (success or error) for each client release.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' After executing the install/test scripts of the 384 client packages in our sample, we discarded 33 packages because the errors did not allow the execution of the install/test script in any of their releases: 15 clients did not have one of the required files;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' 11 had invalid test scripts (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=', {"test": "no test"});' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' 4 listed some required files in the .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='gitignore file – that specifies untracked files that git should ignore;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='11 2 required specific database configurations that could not be done;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' and 1 package required a key to access a server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' We randomly replaced these 33 packages following the aforementioned criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Table 1 shows the results of the execution of the install/test scripts of the 384 client packages and their 3,230 releases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Since the associated providers’ version with 2,727 releases did not change, these tests’ releases were not executed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Finally, we consider as possible manifesting breaking changes cases in which all client packages and releases failed the install/test scripts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Results of execution of the install/test scripts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Tests Client Packages Releases Executed 384 3,230 Not executed 0 2,727 Success 181 1,954 Fails 203 1,276 Total 384 5,957 A replication package including our client packages sample, instruments, scripts, and identified manifesting breaking changes is available for download at https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='5281/zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='5558085.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' 8https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='com/nodejs/node#release-types 9https://docs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='npmjs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='com/files/package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='json#engines 10https://nodejs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='org/en/download/releases 11https://git-scm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='com/docs/gitignore Manuscript submitted to ACM I depended on you and you broke me: An empirical study of manifesting breaking changes in client packages 9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='3 Manual check on failure cases: detecting manifesting breaking changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' For all failure cases (203 clients and 1,276 releases) on the execution of install/test scripts, we manually analyzed which ones were true cases of manifesting breaking changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' To identify breaking changes that manifest themselves in a client package, we leveraged the output logs (logs generated by npm when executing the install and test scripts) generated as the result of executing the method described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='2 (see the second part of Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' For each failed test result, we obtained the error description and the associated stack trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' We then differentiated failed test results caused by a related issue with the client package (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=', an introduced bug by the client) from those caused by a change in the provider package (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=', a change in the return type of a provider’s function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' From the obtained stack traces, we determined whether any function of a provider package was called and manually investigated the positive cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' During our manual investigation, we sought to confirm that the test failure was caused by a manifesting breaking change introduced by the provider package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' The first author was responsible for running the tests and identifying the manifesting breaking changes and related releases and commits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' The first author also manually analyzed each of the manifesting breaking changes and recorded the following information about each of them: the number of affected versions of the client;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' whether any documentation mentions the manifesting breaking change;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' the responsible package for addressing the breaking change (provider or client);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' the client version impacted by the manifesting breaking change;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' the provider version that introduced the breaking change;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' and a textual description about the causes for the breaking change manifestation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=', "the provider function was renamed by mistake", "The provider normalizeurl@1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='0 introduce[d] a new function and the client assetgraph use[d] it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' But the client forgot to update the provider version in package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='json.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' ", "The provider inserts a " " in a null body request").' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' During this process, several rounds of discussions were performed among the authors to refine the analysis, using continuous comparison [22] and negotiated agreement [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' In the negotiated agreement process, the researchers discussed the rationale they used to categorize each code until reaching consensus [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' More specifically, we leveraged the recorded information about each manifesting breaking change to derive a consistent categorization of the introduced breaking changes (RQ2 and RQ3) and to guide new iterations of the manual analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' More specifically, the following set of actions was performed during our manual investigation: Analyze the execution flow: To determine whether the associated function with the test failure occurred in the provider or the client code, we leveraged the stack traces to identify which function was called when the test failed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' In particular, we instrumented the code of the provider and the client packages to output any necessary information to analyze the execution flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' We analyzed the variable contents by adding a call to the console.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='log() and console.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='trace() functions in each part of the code where the client package calls a function of the provider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' For example, suppose the following error appeared: “TypeError: myObject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='callback is not a function”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='To discover the variable content, we use the command console.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='log(myObject) to check whether myObject variable was changed, null, or received other values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Analyze the status of the Continuous Integration (CI) pipeline: We compared the status of the CI pipeline between the originally built release and the status of CI pipeline at the time of our manual investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Since the source code of the client package remains the same between the original release and the installed version in our analysis, we use the difference between the status of the CI pipeline as additional evidence that the test failure was caused by a provider version change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Not all clients had CI pipelines, but when they had, it was helpful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Search for client fixing commits: We manually searched for recovering commits in the history of commits between the installed and previous releases of the client package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Whenever a recovery commit was identified Manuscript submitted to ACM 10 Venturini, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' (by reading the commit message), we determined whether the error was due to the client or the provider code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' For example, we observed cases in which a client updated a provider in the release with failed tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' We also observed that, in the following commits, the provider was downgraded and the commit message was “downgrade provider” or “fix breaking change”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' In these cases, we considered the test failure as caused by a manifesting breaking change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Search for related issue reports and pull requests: We hypothesized that a manifesting breaking change would affect different clients that, in turn, would either issue a bug report or perform a fix followed by a pull request to the codebase of the provider package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Therefore, we searched for issue reports and pull requests with the same error message obtained in our stack trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' We then collected detailed information about the error to confirm whether it was due to a manifesting breaking change introduced by the provider package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Previous and subsequent provider versions: If the test error was caused by a manifesting breaking change, downgrading to the previous provider version or upgrading to a subsequent provider version might fix the error, if the provider already fixed it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Subsequent provider versions means all provider versions that fit the versioning statement and are greater than the provider version that introduced the manifesting breaking change (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=', the adopted provider version when the test failed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' In this case, we uninstalled the current version and installed the previous and subsequent versions, and executed the test scripts again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' For example, if the client specified a provider p as {"p": "^1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='2"} that brought about a breaking change in the version, for example, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='4, we installed p@1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='2, p@1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='3, and p@1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='5 to verify whether the error persisted for those versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='2 Research questions: motivation, approach This section contains the motivation and the approach for each of the research questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='1 RQ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' To what extent do manifesting breaking changes manifest in client packages?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Motivation: By default, npm sets the caret range as a default versioning statement that automatically updates minor and patch releases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Hence, manifesting breaking changes that are introduced in minor and patch releases can inadvertently cause downtime in packages that are downloaded hundreds of thousands of times per day, affecting a large body of software developers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Understanding the prevalence of manifesting breaking changes in popular software ecosystems such as npm is important to help developers assess the risks of accepting automatic minor and patch updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Although prior studies have focused on the frequency of API breaking changes [3], breaking changes can occur for different reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Determining the prevalence of a broader range of breaking change types remains an open research problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Approach: For all cases that resulted in an error on the install/test script, we determined the type of error (client, provider, not discovered).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' We calculated, out of the 384 packages and 3,230 releases, the percentage of cases that we confirmed as manifesting breaking change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Considering all the providers on the client’s latest releases, we calculated the percentage of providers that introduced manifesting breaking changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' In addition, we calculated how many times (number of releases) each provider introduced at least one manifesting breaking change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='2 RQ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' What problems in the provider package cause a manifesting breaking change?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Motivation: Prior studies about breaking changes in the npm ecosystem are restricted to APIs’ breaking changes [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' However, other issues that provider packages can introduce in minor and patch releases can manifest a breaking Manuscript submitted to ACM I depended on you and you broke me: An empirical study of manifesting breaking changes in client packages 11 change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' To support developers to reason about manifesting breaking changes, it is important to understand their root causes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Approach: In this RQ, we analyzed the type of changes introduced by provider packages that bring about a manifesting breaking change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' With the name and version of the provider packages, we manually analyzed the provider’s repository to find the exact change that caused a break.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' We used the following approaches to find the specific changes introduced by providers: Using diff tools: We used diff tools to analyze the introduced change between two releases of a provider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' For example, suppose that a manifesting breaking change was introduced in the release provider@1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' In this case, we retrieved the source code of previous versions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=', provider@1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='4, and performed the diff between these versions to manually inspect the changed code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Analyzing provider’s commits: We used the provider’s commits to analyze the changes between releases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' For a manifesting breaking change in the provider p, we verified its repository and manually analyzed the commits ahead or behind the release tag commit that introduced a manifesting breaking change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Analyzing changelogs: Changelogs contain information on all relevant changes in the history of a package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' We used these changelogs to understand the introduced changes in a release of a client package and to verify whether any manifesting breaking change fix was described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' We also looked at issue reports and pull requests for explanations of the causes of manifesting breaking changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' After discovering the provider changes that introduced breaking changes, we analyzed, categorized, and grouped common issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' For example, all related issues to changing object types were grouped into a category called Object type changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Furthermore, we analyzed the Semantic Version level that introduced and fixed/recovered the manifesting breaking changes both in the provider and client packages to verify the relationship between manifesting breaking changes and non-major releases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' We analyzed the version numbering of releases that fixed a manifesting breaking change and where manifesting breaking changes were documented (changelogs, issue reports, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Furthermore, we analyzed the depth of the dependency tree of the provider that introduced a manifesting breaking change, since 25% of npm packages had at least 95 transitive dependencies in 2016 [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='3 RQ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' How do client packages recover from a manifesting breaking change?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Motivation: A breaking change may impact the client package through an implicit or explicit update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' A client recovery is identified by an update to its code, by waiting for a new provider’s release, or by performing a downgrade/upgrade in the provider’s version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Breaking changes may be caused either by a direct or indirect provider since the client packages depend on a few direct providers and many indirect ones [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' A breaking change may cascade to transitive dependencies if it remains unfixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Even if the client packages can recover from the breaking change by upgrading to a newer version of the provider package, the client packages can manually resolve incompatibilities that might exist [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Understanding how breaking changes manifest in client packages can help developers understand how to recover from them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Approach: We retrieved all information for this RQ from the clients’ repositories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' We searched for information about the error and how the client packages recovered from the manifesting breaking change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' The following information was analyzed: Manuscript submitted to ACM 12 Venturini, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Commits: We manually checked the subsequent commits of the client packages that were pushed to their repositories after the provider release that introduced the respective manifesting breaking change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' In particular, we searched for commits that touched the package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='json file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' In the file history, we checked if the provider was downgraded, upgraded, replaced, or removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Changelogs: We analyzed the client changelogs and release notes looking for mentions of provider updates/- downgrades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' About 48% of clients maintained a changelog or release notes in their repositories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Pull requests/Issue reports: We searched for pull requests and issue reports in the client repository that contained information about the manifesting breaking changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' For example, we found pull requests and issue reports with “Update provider” and “Fix provider error” in the title.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' For each manifesting breaking change case, we recovered the provider’s dependency tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' For example, in our second motivating example (Section 2), we recovered the dependency tree from the client to the package that introduced the manifesting breaking change, which resulted in broccoli-asset-rev→broccoli-filter→broccoli-plugin (Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' We investigated how many breaking change cases were introduced by direct and indirect providers, when the manifesting breaking change was introduced and fixed/recovered, which package fixed/recovered from it, and how it was fixed/re- covered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' We also verified how client packages changed the provider’s versions and how the associated documentation with manifesting breaking changes related to the time to fix it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='3 Scope and Limitations As our definition of manifesting breaking changes includes cases that are not included by the prior definitions of breaking changes (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='1), this paper does not intend to provide a direct comparison between these two phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' As a result, the stated research questions do not indicate the proportion of manifest breaking changes that are, in fact, breaking changes as defined by prior literature (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=', an API change by the provider).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' In addition, since provider packages are rarely accompanied by any formal specification of their intended behavior, it is impossible at the scale of our study to differentiate errors that manifest in the client package due to breaking changes from those that manifest due to an idiosyncratic usage of the provider by the client package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Therefore, the results of the stated RQs cannot be used to assess whether a client package could fix its build by simply updating to a newer version of the provider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' 4 RESULTS This section presents the associated findings for each RQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='1 RQ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' How often do manifesting breaking changes occur in the client package?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Finding 1: 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='7% of the client packages (regardless of their releases) and 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='9% of the client releases were impacted by a manifesting breaking change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' From all 384 client packages, 45 (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='7%) suffered a failing test from a manifesting breaking change in at least one release.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' From 3,230 client releases for which the tests were executed, 1,276 failed, and all errors were manually analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' In 450 (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='9%) releases, the error was raised by the provider packages, characterizing a manifesting breaking change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' On 86 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='7%) releases, we could not identify which package raised the error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' We detected that 261 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='1%) releases suffered a particular error type that we call breaking due to external change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' These releases used a provider that relied on data/resources from an external API/service (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=', Twitter) that were no Manuscript submitted to ACM I depended on you and you broke me: An empirical study of manifesting breaking changes in client packages 13 longer available, impacting all client’s releases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' The provider cannot fix this error, because it does not own the resource.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' These cases imply that detecting manifest breaking changes by running the clients’ tests can introduce false positives, which we simply ignored during our manual analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' We also considered cases in which a provider package was removed from npm as breaking due to external change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Table 2 shows the results of analyses by releases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Results of releases’ analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Results Releases (#) (%) Success 1954 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='5 Fail Client’s errors 479 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='8 manifesting breaking changes 450 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='9 Breaking due to external changes 261 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='1 Errors not identified 86 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='7 Total 3230 100 Finding 2: 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='2% providers introduced a single manifesting breaking change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' In our sample, 47 providers (92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='2%) of 51 introduced a single release with a manifesting breaking change, and four providers introduced two releases with manifesting breaking changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' We detected 55 unique manifesting breaking change cases introduced by providers, some of which impacted multiple clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' For example, the breaking change exhibited in the Incompatible Providers Versions classification (Finding 3) impacted six clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Therefore, 64 manifesting breaking change cases manifested in the client packages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Finally, there were 1,909 providers on all clients’ latest versions, and the percentage of providers that introduced manifesting breaking change was 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='6% (51 of 1909).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' About 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='7% of clients and 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='9% of their releases suffered from manifesting breaking changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' We detected failing tests due to 2% of the providers with changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Over 90% of those that introduced manifesting breaking changes did so through just a single release with a manifesting breaking change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='2 RQ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' What issues in the provider package caused a breaking change to manifest?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Finding 3: We found 8 categories of issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' We grouped each manifesting breaking change into eight categories, depending on its root cause (issue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Table 3 presents each category, the number of occurrences, and the number of impacted client releases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Manuscript submitted to ACM 14 Venturini, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' The identified categories of manifesting breaking changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Category Cases Releases (#) (%) (#) (%) Feature change 25 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='1 101 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='4 Incompatible providers versions 15 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='4 64 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='2 Object type changed 9 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='1 213 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='3 Undefined object 5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='8 28 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='2 Semantically wrong code 5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='8 14 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='1 Failed provider update 2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='1 24 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='3 Renamed function 2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='1 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='4 File not found 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='6 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='9 Total 64 450 In the following, we describe each category and present an example that we found during our manual analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Feature change: manifesting breaking changes in this category are related to modifications of provider features (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=', the default value of variables).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' An example happens in request@2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='0 – this version was removed from npm, but the introduced change remained in the package – when developers introduced a new decision rule into their code12 as shown in Listing 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Listing 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Example of a manifesting breaking change categorized as feature change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=" debug('emitting complete ', self." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='uri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='href) + if(response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='body == undefined && !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='_json) { + response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='body = "";' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' + } self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content="emit('complete ', response , response." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='body) In Listing 5, the provider request assigns an empty string to the response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='body variable, instead of preserving response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='body with its default undefined value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Incompatible providers versions: In this category, the client breaks because of a change in an indirect provider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' An example happens in the packages babel-eslint and escope, where escope is an indirect provider of babel-eslint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Listing 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Incompatible provider’s versions example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=" } }, visitClass: { + }, { + key: 'visitClass ', value: function visitClass(node) { The release escope@3." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='4 introduced the presented change in Listing 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' This change impacted the package babel- eslint,13 even though the escope had not been a direct provider to babel-eslint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='14 This manifesting breaking change remained unresolved for a single day, during which babel-eslint received about 80k downloads from npm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' 12https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='com/request/request/commit/d05b6ba 13https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='com/babel/babel-eslint/issues/243 14https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='com/estools/escope/issues/99#issuecomment-178151491 Manuscript submitted to ACM I depended on you and you broke me: An empirical study of manifesting breaking changes in client packages 15 Object type changed: We detected 9 (14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='06%) cases in which the provider changed the type of an object, resulting in a breaking change in the client packages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Listing 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Object type changed example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='setup();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='sockets = [];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' + this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='sockets = {};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='nsps = {};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='connect Buffer = [];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' } var socket = nsp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='add(this , function () { self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='sockets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='push(socket);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' + self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='sockets[socket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='id] = socket;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='nsps[nsp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='name] = socket;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' In Listing 7, the provider socket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='io@1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='0 turned an array into an object,15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' This simple change broke many of socket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='io’s clients, even the package karma,16 a browser test runner, which was forced to update its code17 and publish karma@0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' During the single day, the manifesting breaking change remained unresolved, karma was downloaded about 146k times from npm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Undefined object: In this category, an undefined object causes a runtime exception that breaks the provider, which throws the exception to the client package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Listing 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Undefined object code example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' + app.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='options = app.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='options || {};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' app.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='babel = app.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='babel || {};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' app.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='babel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='plugins = app.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='babel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='plugins || [];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' This error happened in the provider ember-cli-htmlbars-inline-precompile@0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='3, which solved it as shown in Listing 818.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Failed provider update: In this category, provider A updates its provider B, but provider A does not update its code to work with the new provider B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' We detected two cases of this category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' In addition to an explicit update, one provider A from this category specified its provider B as an accept-all range (>=).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Over time, its provider B published a major release that introduced a manifesting breaking change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Despite provider A specifying an accept all range, it did not consider the implicit update of provider B and the client suffered an error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Semantically wrong code: manifesting breaking changes in this category happen when the provider writes a semantically wrong code, generating an error in its runtime process19 and affecting the client.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' These errors could be caught in compile-time in a compiled language, but in JavaScript these errors happen at runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' This occurred in the provider front-matter@0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='0 and four other cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Listing 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Semantically wrong code example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=" const separators = [ '---', '= yaml ='] const pattern = pattern = '^(' + const pattern = '^(' + '((= yaml =)|(---))' 15https://github." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='com/socketio/socket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='io/commit/b73d9be 16https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='com/socketio/socket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='io/issues/2368 17https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='com/karma-runner/karma/commit/3ab78d6 18https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='com/ember-cli/ember-cli-htmlbars-inline-precompile/pull/5/commits/b3faf95 19https://hacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='mozilla.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='org/2017/02/a-crash-course-in-just-in-time-jit-compilers/ Manuscript submitted to ACM 16 Venturini, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' On Listing 9, the provider repeated the variable name (pattern) on its declaration, which generated a semantic error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Although this error can be easily detected and fixed, as the provider did20 in Listing 9, the provider took almost one year to fix it (front-matter@0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Meanwhile, front-matter received about 366 downloads in that period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Renamed function: The manifesting breaking changes in this category occur when functions are renamed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Our analysis revealed 2 cases in which the functions were renamed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' The renaming case is our first motivating example (Section 2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' we describe the second one below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Listing 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Renamed function code example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' RedisClient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='prototype.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='send_command = function (command , args , callback) { var args_copy , arg , prefix_keys;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' + RedisClient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='prototype.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='internal_send_command = function (command , args , callback) { + var arg , prefix_keys;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' The provider redis@2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='0-1 renamed a function, as in Listing 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='21 However, this function was used in a client package fakeredis,22, which broke with this change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Client package fakeredis@1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='3 recovered from this error by downgrading to redis@2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='0-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='23 In the five days period within which the manifesting breaking change was not fixed, fakeredis received about 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='3k downloads from npm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' File not found: In the cases in this category, the provider removes a file or adds it to the version control ignore list (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='gitignore) and the client tries to access it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' In the unique case of this category in our sample, the provider referenced a file that was added to the ignore list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Finding 4: manifesting breaking changes are often introduced in patch releases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' As shown in Table 4, of the 64 cases of manifesting breaking changes we analyzed, three cases were introduced in major releases, 26 in minor releases, 28 in patch releases, and 5 in pre-releases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Although we only analyzed manifesting breaking changes from minor and patch releases, in three cases the manifesting breaking changes were introduced at major levels in an indirect provider, which transitively affected client packages—as in the jsdom@16 case (see Section 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' manifesting breaking changes in each Semantic Version level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Levels (#) (%) Major 3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='7 Minor 28 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='75 Patch 28 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='75 Pre-release 5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='8 Total 64 100 Pre-releases precede a stable release and are considered unstable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' anything may change until a stable version is released.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='24 In all detected breaking changes in pre-releases, the providers introduced unstable changes in pre-releases 20https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='com/jxson/front-matter/commit/f16fc01 21https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='com/NodeRedis/node-redis/commit/861749f 22https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='com/NodeRedis/node-redis/issues/1030#issuecomment-205379483 23https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='com/hdachev/fakeredis/commit/01d1e99 24https://semver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='org/#spec-item-9 Manuscript submitted to ACM I depended on you and you broke me: An empirical study of manifesting breaking changes in client packages 17 and propagated these changes to stable versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' An example is the pre-release redis@2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='0-1 (described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='2), whose rename of a function propagated to the stable version and caused a failure in the client packages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Finding 5: manifesting breaking change fixes/recoveries are introduced by both clients and/or providers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' We searched to identify which package fixed/recovered from the manifesting breaking changes – client or provider – and at which level the fixed/recovered release was published, as depicted in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Breaking changes in minor Fixed by provider (46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='4%) Breaking changes in patch Fixed by provider (46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='4%) Patch (76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='9%) Fixed by client (42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='9%) Not fixed (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='7%) Fixed by client (39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='3%) Not fixed (14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='3%) Others (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='7%) Minor (15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='4%) Patch (61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='5%) Others (23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='1%) Minor (15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='4%) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Proportion of fixed/recovered manifesting breaking changes by provider and client packages and the respective Semantic Version level of the fixing/recovering releases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Figure 4 shows that client packages recover from nearly half of the manifesting breaking change introduced in minor updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' In turn, 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='9% of the manifesting breaking changes that are introduced by providers in a minor release are fixed in a patch release.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Providers fix the majority of the manifesting breaking changes introduced in patch releases (46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='4% of the time), typically through a patch release (61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='5%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Finding 6: 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='9% of the manifesting breaking changes are not documented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Although clients and providers often document the occurrence or repair of a manifesting breaking change in issue reports, pull requests, or changelogs, more than one-fifth of the manifesting breaking changes are undocumented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Summary of the proportion of documented manifesting breaking changes when they are introduced and fixed/recovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Documentation Introduced Fixed/recovered Proportion (%) Issue report – 32 64 Pull request 5 15 44 Changelog 23 16 78 Table 5 shows that client and provider packages documented manifesting breaking changes in 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='1% of all manifesting breaking changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Out of all cases that have documentation, 70% have more than one type of documentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' For example, the provider received an issue report, fixed the manifesting breaking change, and documented it in a changelog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Documenting manifesting breaking changes and their fixes supports client recovery (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Finding 7: 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='8% of the manifesting breaking changes are introduced by an indirect provider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Indirect providers might also introduce manifesting breaking changes, which can then propagate to the client.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Table 6 shows the depth level in the dependency tree of each provider that introduced a manifesting breaking change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' About 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='2% of manifesting Manuscript submitted to ACM 18 Venturini, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' breaking changes are introduced by a direct provider in the client’s package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='json.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' These providers are the ones the client directly installs and that perform function calls in their own code;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' they are in the first depth level of the dependency tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' How deep the provider package that raised a manifesting breaking change is from the client in the dependency tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Depth (#) (%) 1 27 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='2 2 30 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='9 >3 7 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='9 Total 64 100 Manifesting breaking changes introduced by indirect providers in the depth level greater than one represent 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='8% of the cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Six cases are in the third depth level and a single one is in the fourth depth level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Clients do not install these providers directly;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' rather, they come from the direct provider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' In these cases, the manifesting breaking change may be totally unclear to client packages, since they are typically unaware of such providers (or have no direct control over their installation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' The most frequent issues with provider packages that introduced manifesting breaking changes were feature changes, incompatible providers, and object type changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Provider packages introduced these manifesting breaking changes at similar rates in minor and patch releases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Most of the fixed manifesting breaking changes by providers were fixed in patch releases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Manifesting breaking changes are documented in 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='1% of the cases, mainly on issue reports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Indirect providers introduced manifesting breaking changes in most cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='3 RQ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' How do client packages recover from a manifesting breaking change?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Finding 8: Clients and transitive providers recover from breaking changes in 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='1% of cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' In the dependency tree, the transitive provider is located between the provider that introduced the manifesting breaking change and the client where it manifested (See Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Table 7 shows which package fixed/recovered from each manifesting breaking change case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' The provider packages fixed the majority of the manifesting breaking changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Since they introduced the breaking change, theoretically this was the expected behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Client packages recovered from the manifesting breaking change in 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='3% of cases, and transitive providers recovered from manifesting breaking changes in 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='8% of cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' When the provider who introduced a manifesting breaking change does not fix it, the transitive provider may fix it and solve the client’s issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Manuscript submitted to ACM I depended on you and you broke me: An empirical study of manifesting breaking changes in client packages 19 Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Packages fixing/recovering from the error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Fixed by/Recovered from (#) (%) Provider 32 50 Client 13 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='3 Transitive provider 12 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='8 Client + Transitive provider 25 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='1 Not fixed/recovered 7 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='9 Total 64 100 Since transitive providers are also clients of the providers that introduced the manifesting breaking change, clients (clients and transitive providers) recovered from these breaking changes in 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='1% of cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' This observation suggests that client packages occasionally have to work on a patch when a manifesting breaking change is introduced since in 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='1% of the cases clients and transitive providers need to take actions to recover from the manifesting breaking change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Finding 9: Transitive providers fix manifesting breaking changes faster than other packages: When a mani- festing breaking change is introduced, it should be fixed by either the provider who introduced it or a transitive provider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' In a few cases, the client package will also recover from it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Table 8 shows the time that each package takes to fix the breaking change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' In general, manifesting breaking changes are fixed in seven days by provider packages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Even in this relatively short period of time, many direct and indirect clients are affected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Median of number days that each package spent to fix/recover from the manifesting breaking change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Fixed by/Recovered from Days Provider 7 Client 134 Transitive provider 4 Client + Transitive provider 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='4 Transitive providers fix manifesting breaking changes faster than clients and even providers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Since the manifesting breaking change only exists when it is raised in the client packages, transitive providers break first and need a quick fix;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' transitive providers usually spent four days to fix a break.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Meanwhile, providers that introduced the manifesting breaking change take a median of 7 days to introduce a fix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' In cases where the provider neglected to introduce a fix or took longer than the client, client packages took a comparably lengthy 134 days (mean 286;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' SD 429) to recover from a manifesting breaking change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' According to Table 7, the direct providers and transitive providers fixed most of the manifesting breaking changes, about 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='8%, because clients can be slow to recover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' However, because transitive providers are also clients, we can analyze the time that clients and transitive providers spend to fix/recover from a manifesting breaking change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Clients and transitive providers recovered from a manifesting breaking change in around 82 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Finding 10: Upgrading is the most frequent way to recover from a manifesting breaking change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Table 9 de- scribes how clients recovered from breaking changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' In 48 cases, the provider version was changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' In most cases Manuscript submitted to ACM 20 Venturini, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' (71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='4%), client packages upgraded their providers’ version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' We analyzed all cases where clients and transitive providers recovered from the manifesting breaking change by changing the provider’s version before the provider fixed the error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' We observed an upgrade in 12 (52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='2%) cases out of 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Thus, in more than half of the cases where the client and transitive providers fixed/recovered from the manifesting breaking change, the provider package had newer versions, but the client was not using any follow-up releases from the provider packages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Table 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' How client packages changed the provider’s version after a manifesting breaking change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Changed by Total Upgrade Downgrade Replace Remove (#) (%) (#) (%) (#) (%) (#) (%) Client 28 20 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='4 6 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='4 1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='6 1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='6 Transitive provider 20 9 45 10 50 01 5 — — The number of downgrades in a transitive provider may explain why they recover from the manifesting breaking change faster than the client packages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Since transitive providers are also providers, they should fix the manifesting breaking change as soon as possible, avoiding the propagation of the error caused by the manifesting breaking change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Consequently, the downgrade to a stable release of the provider is the most frequent way for transitive providers to recover from a manifesting breaking change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Finally, the provider is replaced or removed in a small proportion when a breaking change is raised—about 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='2% for both cases combined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Finding 11: To recover from manifesting breaking changes, clients often change the adopted provider version without changing the range of automatically accepted versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' When a breaking change manifests itself, clients often update the provider’s version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Figure 5 shows when the clients and transitive providers updated their providers’ versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' steady caret all tilde steady (66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='7%) caret (33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='3%) caret (53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='8%) steady (38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='5%) all (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='7%) caret (60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='0%) all (20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='0%) tilde (20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='0 %) tilde (75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='0%) caret (25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='0%) (a) Client caret tilde all caret (85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='7%) steady (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='1%) tilde (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='1 %) tilde (100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='0%) caret (66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='7%) tilde (33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='3%) (b) Transitive Provider Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Provider’s version changed by clients and transitive providers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' On the left side of each figure, one can see the range level where the manifesting breaking change was introduced and on the right side, one can see the range level where the same manifesting breaking change was fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Manuscript submitted to ACM I depended on you and you broke me: An empirical study of manifesting breaking changes in client packages 21 We verified that transitive providers never set a steady version of their provider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' When a breaking change manifests in transitive providers, they use a range in the provider’s version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' However, a single transitive provider changed the range from a caret range to a steady one (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=', ^1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='1 → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='1), to recover from the manifesting breaking change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Nevertheless, when the clients used a caret range and a breaking change manifested, in 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='5% of the cases they downgraded the provider to a steady version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' The majority of the manifesting breaking changes were introduced when the clients and transitive providers used the caret range (^).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' It is the default range statement that npm inserts in the package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='json when a provider is added as a dependency of a client package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' In more than half of the cases, these clients changed the provider’s version to another caret range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' The accept all ranges (>=, or *) were less commonly used and less common when updating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Clients and the transitive provider in 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='5% of cases retained the range type and updated it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' The range type (all, caret, tilde, or steady) was kept, but the provider was updated/downgraded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' For example, a client package specifies a provider p@^1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='0 and receives a breaking change in p@1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Whenever the provider fixes the code, the client package will update it to, for example, p@^1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='0, but will not change it for another range type, such as all, tilde, or steady range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Client packages recovered manifesting breaking changes in 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='1% of cases, including clients and transitive providers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Providers fixed manifesting breaking changes faster than client packages recovered from manifesting breaking changes by updating the provider, and clients preferred to update rather than downgrade their providers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' The provider’s range can be updated or downgraded after a breaking change, but in around 60% of cases, they did not change the range type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' 5 DISCUSSION This section discusses the implications of our findings for dependency management practices (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='1) and the best practices that clients and providers can follow to mitigate the impact caused by manifesting breaking changes (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' We also discuss the manifestation of breaking changes and the aspects of Semantic Versioning in the npm ecosystem (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='1 Dependency management When managing dependencies, client packages can use dependency bots in GitHub, such as Snyk and Dependabot, to receive automatic pull requests when there is a new provider’s release [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' These bots continuously check for new versions and providers’ bugs/vulnerabilities fixes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' They open pull requests in the client’s repository, updating the package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='json, including changelogs and information about the provider’s new version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Mirhosseini and Parnin [16] show that packages using such bots update their dependencies 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='6x faster than through manual verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Additionally, tools such as JSFIX [20] can be helpful when upgrading provider releases, especially those that include manifesting breaking changes or major releases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' The JSFIX tool was designed to adapt the client code to the new provider release, offering a safe way to upgrade providers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' We verified that a small percentage of the clients recovered from manifesting breaking changes by removing or replacing the provider (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=', Finding 10), which may be difficult when several features or resources from the provider package are used by the client [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Instead, client packages tend to temporarily downgrade to a stable provider version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' To ease the process to upgrade/downgrade providers and avoid surprises, clients should search in the provider changelogs Manuscript submitted to ACM 22 Venturini, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' for significant changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' As we verified in Finding 6, most manifesting breaking changes are documented in changelogs, issue reports, or pull requests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Dependency bots also could analyze the content of changelogs and issue reports to create red flags, like notifications, about documentation that cites a manifesting breaking change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Finally, client packages may use a package-lock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='json file to better manage dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' We observed in Finding 7 that indirect providers – the ones in depth two and three in the dependency tree – are responsible for 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='8% of the manifesting breaking changes that affect a client package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Using a package-lock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='json file, client packages can stay aware of all of the providers’ versions of the latest successful build.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' When a provider is upgraded due to the range of versions and the new release manifests a breaking change on the client side, the client can still install all of the providers’ versions that successfully built the client.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='2 Best practices Several issues found in our manual classification of manifesting breaking changes (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='2) could be avoided through the use of static analysis tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Errors classified as Semantically Wrong Code and Rename function are typically captured by such tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Both client and provider developers can use such tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' For a dynamic language such as JavaScript, these tools can help avoid some issues [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Options for JavaScript include jslint, jshint and standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Tómasdóttir et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' [26] and Tómasdóttir et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' [25] show that developers use linters mainly to prevent errors, bugs, and mistakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Due to the dynamic nature of JavaScript, however, static analysis tools cannot verify inherited objects’ properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' They do not capture errors classified as Change one rule, Object type change, and Undefined object, as well as Rename Function in functions of object’s properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Thus, developers should be concerned about creating test cases that run their code along with the functionality of providers, as only then will they (client developers) find breaking changes that affect their own code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Many available frameworks, such as mocha, chai, and ava, support these tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' These tests should also be executed on integrated environments every time the developer commits and pushes new changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' For this case, several tools are available, such as Travis, Jenkins, Drone CI, and Codefresh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Using linters and continuous integration systems, developers can catch most of these errors before releasing a new version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Finally, a good practice for npm packages is to keep a changelog or to document breaking changes and their fixes in issue reports and pull requests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' This practice should continue and be more widely adopted, since currently around a fifth of providers do not do it (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=', Finding 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' This would also help the development of automated tools (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' bots) for dealing with breaking changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Providers could create issue reports and pull request templates to allow clients to specify consistent descriptions of issues they found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='3 Breaking changes manifestation and Semantic Versioning Breaking changes often occur in the npm ecosystem and impact client packages (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=', Finding 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Most of the manifesting cases come from indirect providers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' that is, providers from the second level or deeper in the dependency tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Findings from Decan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' [10] show that in 2016 half of the client packages in npm had at least 22 transitive dependencies (indirect providers), and a quarter had at least 95 transitive dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' In this context, clients may face challenges in diagnosing where the manifesting breaking changes came from, because when a manifesting breaking change is introduced by an indirect provider, the client may not know this provider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Our results show that provider packages introduce manifesting breaking changes in minor and patch levels, which in principle should only contain backward-compatible updates according to the Semantic Versioning specification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Semantic Versioning is a recommendation that providers can choose to use it or not [4, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' If providers do not comply Manuscript submitted to ACM I depended on you and you broke me: An empirical study of manifesting breaking changes in client packages 23 with Semantic Versioning, several errors might be introduced, as we observed in Finding 4 that all manifesting breaking changes in pre-releases were propagated to stable releases (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=', Finding 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' One hypothesis is that providers might be unaware of the correct use of the Semantic Versioning rules, which may explain why they propagated the unstable changes to stable releases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Finally, npm could provide badges where provider packages would be able to explicitly show that they are aware of and adhere to the Semantic Versioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Trockman [24] claims that developers use visible signals (specifically on GitHub) like badges to indicate project quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' This way, clients could make a better choice about their providers and prefer those aware of Semantic Versioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' 6 RELATED WORK This section describes related work regarding breaking changes in npm and other ecosystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Breaking changes in npm: Bogart et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' [5] presents a survey about the stability of dependencies in the npm and CRAN ecosystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' The authors interviewed seven package maintainers about software changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' In this paper, interviewees highlighted the importance of adhering to Semantic Versioning to avoid issues with dependency updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' More recently, the authors investigated policies and practices in 18 software ecosystems, finding that all ecosystems share values such as stability and compatibility, but differ on other values [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Kraaijeveld [14] studied API breaking changes in three provider packages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' The author uses 3k client packages, parsing the providers’ and clients’ files to detect API-breaking changes and their impact on clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' This work identified that 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='8% to 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='8% of client releases are impacted by API-breaking changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Mezzetti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' [15] present a technique called type regression testing that verifies the type of a returned object from an API and compares it with the returned type in another provider release.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' The authors chose the 12 most popular provider packages and their major releases, applying the technique in all patch/minor releases belonging to the first major update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' They verified type regression in 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='4% of the minor or patch releases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Our research focused on any kind of manifesting breaking changes and we analyzed both client and provider packages, with 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='9% of releases impacted by manifesting breaking changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Mujahid et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' [19] focus on detecting break-inducing versions of third-party dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' The authors analyzed 290k npm packages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' They flagged each downgrade in the provider version as a possible breaking change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' These provider versions were tested using client tests and the authors identified 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='1% of fails after an update, which resulted in a downgrade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Similar to these authors, we resolved each client’s providers for a release, but we ran the tests whenever at least one provider version changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Møller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' [17] present a tool that uses breaking change patterns described by providers and fixes the client code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' They analyzed a dataset with ten of the most used npm packages and searched for breaking changes described in changelogs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' We can compare our classification (Finding 3) with theirs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' They found 153 cases of breaking changes that were introduced in major releases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' They claim that most of the breaking changes (85%) are related to specific package API points, such as modules, properties, and function changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Considering our classification (Finding 3), feature changes, object type changed, undefined object, and renamed function can also be classified as changes in the package API and, if so, we claim that 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='06% of manifesting breaking changes are package API related.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Breaking changes in other ecosystems: Brito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' [6] studied 400 providers from the Maven repository for 116 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' The provider packages were chosen by popularity on GitHub and the authors looked for commits that introduced an API-breaking change during that period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Developers were asked about the reasons for breaking changes that occurred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Our paper presents similar results: the authors claim that New Feature is the most frequent way a breaking change Manuscript submitted to ACM 24 Venturini, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' is introduced, while we claim that Feature Change is the main breaking change type (Finding 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Also, the authors similarly detected that breaking changes are frequently documented on changelogs (Finding 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Foo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' [12] present a study about API breaking changes in the Maven, PyPI, and RubyGems ecosystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' The study focuses on detecting breaking changes by computing a diff between the code of two releases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' They found API-breaking changes in 26% of provider packages, and their approach suggests automatic upgrades for 10% of the packages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Our approach goes beyond API breaking changes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' we found that 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='7% of the client packages are impacted by manifesting breaking changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' 7 THREATS TO VALIDITY Internal validity: When a breaking change was detected, we verified the type of change that the provider package introduced and collectively grouped the changes into categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' However, some cases might fall into more than one category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' For example, a provider package changes the type of an object to change/improve its behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' This case might fall into Feature change and Object type changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' So, we categorized the case in the category that most represents the error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' In this case, since the object is changed by a feature change, the most appropriate category would be Feature change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' The error cases that we categorized as breaking due to external change are the ones in which the clients or providers use – or depend on – external data/resources from sites and APIs that changed over time (see Finding 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' These cases represent about 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='1% of the client’s releases, and, in these cases, we could not search for manifesting breaking changes because we could not execute the release tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' After all, the data/resource needed by the test were no longer available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' So, about 8% of client releases might be impacted by breaking changes, but we could not analyze them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Construct validity: In our approach to detecting breaking changes, we only performed an analysis when the client tests failed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' If a client used a provider version that had a breaking change, but the client did not call the function that causes the breaking change or did not have tests to exercise that code, we could not detect the breaking change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' This is why we call all of our cases manifesting breaking changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Therefore, we might not have detected all API-breaking changes, as we were able to detect only API name changes and API removal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Parameter changes may not be detected because JavaScript allows making a call to an API with any number of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='25 We restored the working tree index in the respective commit tagged by the developer for each release.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' We listed all tags in the repository, and we used the checkout with the respective tag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' However, for untagged releases we performed a checkout in the timestamp referenced in the package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='json.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' We trusted the timestamp once we verified that the tags and timestamp point to the same commit in 94% of cases for tagged repositories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Lastly, we did not mention the file npm-shrinkwrap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='json in our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' This file is intended to work like the file package-lock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='json when controlling transitive dependency updates, but it may be published along with the package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' However, npm strongly recommend avoiding its use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Also, the existence of npm-shrinkwrap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='json files does not play any major role in our study, as they do not affect our results, based on our adopted research method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' We did not include them in our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' External validity: We randomly selected client packages that varied in release numbers, clients, providers, and size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' However, since we only analyzed npm packages hosted at GitHub projects, our findings cannot be directly generalized to other settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' It is also important to state that representativeness can also be limited because npm increases the 25https://eloquentJavaScript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='net/03_functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='html#p_kzCivbonMM Manuscript submitted to ACM I depended on you and you broke me: An empirical study of manifesting breaking changes in client packages 25 number of packages and releases daily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Future work can replicate our study in other platforms and ecosystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Finally, since the number of projects in our sample is small, we do not have enough statistical power to perform hypothesis tests around results that involve package-level comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Conclusion validity: Conclusion validity relates to the inability to draw statistically significant conclusions due to the lack of a large enough data sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' However, as our research used a qualitative approach, we mitigate any potential conclusion threat by conducting a sanity check on repositories of all client packages with fewer than four releases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' This guarantees that all packages are intended for use in production (Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Finally, all of the manifesting breaking changes that we claim in our work were manually analyzed to assure they are legitimate breaking changes that impact clients in the real world (Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' 8 CONCLUSIONS Software reuse is a widely adopted practice, and package ecosystems such as npm support reusing software packages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' However, breaking changes are a negative side effect of software reuse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Breaking changes and their impacts are studied in the literature in several software ecosystems [3, 6, 18, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' A few papers examine breaking changes in the npm ecosystem from the client packages perspective, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=', executing the client tests to verify the impact of breaking changes [5, 15, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' In this work, we analyzed manifesting breaking changes in the npm ecosystem from the client and provider perspectives, providing an empirical analysis regarding breaking changes in minor and patch levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' From the client’s perspective, we analyzed the impact of manifesting breaking changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' We found that 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='7% of clients are impacted by such changes and offer some advice to help clients and automated tools developers discover, avoid, and recover from manifesting breaking changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Clients can use dependency bots to accelerate the process of upgrading their providers, and clients can look at changelog files for any non-desired updating, such as breaking changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' From the provider’s perspective, we analyzed the most frequent causes of manifesting breaking changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' We found that the most common causes were when providers changed some rules/behaviors on features that had been stable over the last releases, when an object type changes, and when there were unintentionally undefined objects at runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Maintainers should pay attention during code review phases regarding these issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Future research can look into the correlation among package characteristics and metrics with breaking change occurrence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' 9 ACKNOWLEDGMENTS This work is partially supported by the National Science Foundation under Grant Number IIS-1815503, CNPq/MC- TI/FNDCT (grant #408812/2021-4 ) and MCTIC/CGI/FAPESP (grant #2021/06662-1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' REFERENCES [1] 2018.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Adding Sparkle to Social Coding: An Empirical Study of Repository Badges in the npm Ecosystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' In 2018 IEEE/ACM 40th International Conference on Software Engineering: Companion (ICSE-Companion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' 524–526.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' [25] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' F.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='1109/TSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='2871058 [26] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Tómasdóttir, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Aniche, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=', Delft University of Technology, Delft, Netherlands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' [27] Mairieli Wessel, Bruno Mendes De Souza, Igor Steinmacher, Igor S Wiese, Ivanilton Polato, Ana Paula Chaves, and Marco A Gerosa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' The power of bots: Characterizing and understanding bots in oss projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Proceedings of the ACM on Human-Computer Interaction 2, CSCW (2018), 1–19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' [28] Jooyong Yi, Dawei Qi, Shin Hwei Tan, and Abhik Roychoudhury.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Expressing and Checking Intended Changes via Software Change Contracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' In Proceedings of the 2013 International Symposium on Software Testing and Analysis (ISSTA 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' Lugano, Switzerland, 1–11.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} +page_content='1007/978-3-319-90421-4_6 Manuscript submitted to ACM' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfggra/content/2301.04563v1.pdf'} diff --git a/jNFLT4oBgHgl3EQfbi-Q/content/2301.12079v1.pdf b/jNFLT4oBgHgl3EQfbi-Q/content/2301.12079v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..6bbb9879641ec73d9dac70136f724a97e9fdd97d --- /dev/null +++ b/jNFLT4oBgHgl3EQfbi-Q/content/2301.12079v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid 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+Paterna 46980, Spain. +e-mail: samuel.manas@uv.es, eugenio.coronado@uv.es + +KEYWORDS: 2D magnets, van der Waals heterostructures, twistronics, spintronics, +nanomagnetism, nanodevices, quantum materials, CrSBr + +The advent of twisting-engineering in two-dimensional (2D) crystals enables the design of +van der Waals (vdW) heterostructures exhibiting emergent properties.1–4 In the case of +magnets, this approach can afford artificial magnets with tailored spin arrangements that +do not exist in nature. Here, we fabricate an artificial magnet by twisting 90 degrees two +CrSBr ferromagnetic monolayers with an easy-axis in-plane anisotropy, thus forming an +‘orthogonally-twisted bilayer’. The magneto-transport properties reveal multistep spin +switching with a magnetic hysteresis opening, which is absent in the pristine case. By tuning +the magnetic field, we modulate the remanent state and coercivity and select between +hysteretic and non-hysteretic magneto-resistance scenarios. This complexity is +fundamentally different from that achieved in Moiré superlattices. Our results highlight the +control over the magnetic properties in vdW heterostructures, leading to a variety of field- +induced phenomena and opening a fruitful playground for creating artificial magnetic +symmetries and manipulating non-collinear magnetic textures. +Metamagnets and their field-induced phase transitions offer a plethora of counterintuitive +phenomenology, as already quoted by Kramers,5 with a direct competition between magnetic +anisotropy, exchange, and dipolar energies.6 In absence of magnetic field, these materials show +zero net magnetization that suddenly increases until its saturation -thus, resembling a +ferromagnet- above a certain magnetic field threshold.5 A good example of an A-type metamagnet +is offered by the layered vdW semiconductor CrSBr. The spins in every single layer (ab plane) +couple ferromagnetically between them (TC ~150 K), pointing along the easy b axis, whereas the +layers couple between them antiferromagnetically (TN ~ 140 K).7 By applying a magnetic field, +it is possible to flip the layers’ magnetization in a parallel fashion via a spin reversal and to induce +a spin reorientation along the magnetic field direction. This transition does not present any +hysteresis.8–14 In bulk, the saturation fields at 2 K are 0.6 T, 1 T and 2 T for the easy (b), +intermediate (a) and hard (c) magnetic axis, respectively.14 This vdW material can be thinned +down to the monolayer limit and integrated into electronic nano-devices. Upon the field-induced +spin switching, the magneto-resistance (MR) is large and negative from bulk down to the bilayer +case, with a reduction of the saturation field along the easy-axis (from 0.6 T in bulk to 0.2 T in +the bilayer at 2 K).12–16 The monolayer limit is characterized by the absence of MR for fields +applied along the easy axis and small and positive MR for fields applied along the intermediate +and hard axis.14,15 +The ability for isolating, manipulating and twisting 2D crystals adds a new degree of +control in vdW heterostructures, affording emergent new properties, like superconductivity in +twisted bilayer graphene.1 As far as the 2D magnetic materials are concerned, twisting is much +less explored. Still, it has allowed the creation of new magnetic ground states. For example, by +twisting small angles the 2D magnet CrI3, a modulation of the spin-reversal by magneto-optical +techniques has been reported.2–4 This twisting not only produces a Moiré superlattice but can also + +induce Moiré magnetic exchange interactions, in which unique spin textures like magnetic +skyrmion bubbles have been theoretically predicted.17–20 However, no 2D twisted-magnets have +been incorporated into electronic devices so far, remaining the magneto-transport effects in +twisted-magnets fully unexplored. +Here, we twist by ca. 90 degrees two CrSBr ferromagnetic monolayers, thus forming an +orthogonally-twisted bilayer, and integrate them in a vertical vdW heterostructure formed by +either few-layers graphene or metallic NbSe2 thin-layers for inspecting its magneto-transport +properties (Figure 1.a-b; see Methods). 21–24 Note that, in stark contrast with CrI3, where the spins +are out-of-plane, in CrSBr the spins are in-plane pointing along the easy b-axis. Therefore, this +orthogonal configuration yields to a complex spin scenario in which the application of an external +magnetic field (Zeeman energy), the inter-layer magnetic interactions (which favors an +antiparallel orientation of both spin layers) and the local spin anisotropy (which tends to orient +the spins along their easy magnetic axes) are competing since the easy and intermediate magnetic +axis of both monolayers are perpendicular, while keeping the hard magnetic axis of both layers +parallel (out-of-plane direction). This case is different from the common Moiré patterns in twisted +bilayers, where the twisting angle is small.1 +An example of an orthogonally-twisted-CrSBr heterostructure is shown in Figure 1.a-b. +In this vertical geometry, the MR can be rationalized within a spin-valve picture, considering a +two-current channel model: when the magnetization of both layers is antiparallel (parallel), there +is a higher (lower) resistance across the heterostructure.14,25,26 The field-dependence of the MR at +10 K is presented in Figure 1.c for in-plane magnetic fields aligned along the easy-axis of one of +the layers (in this case, the top layer; θ = φ = 0º). Starting at high negative fields (red curve in +Figure 1.c), the MR is negative and field independent down to -1 T; then, it increases until a +maximum positive MR is observed at ca. +0.16 T. Above this field, it decreases again until +reaching a saturation value above +1 T. This value coincides with that observed for the spin +reorientation along the intermediate magnetic axis, a, thus suggesting that this is determined by +the spin anisotropy. Reversing the magnetic field yields to a symmetrical curve that exhibits the +maximum in MR at ca. -0.16 T (blue curve in Figure 1.c). These two curves cross at zero field +showing a hysteretic behavior when the field modulus is kept below ca. 0.32 T. For an easier +visualization of the hysteresis, we present as a top panel in Figure 1.c the increment value, defined +as ΔX = X+B→-B – X-B→+B, where X states either for the resistance (R) or the MR while decreasing +(+B→-B; blue curve in Figure 1.c) or increasing (-B→+B; red curve in Figure 1.c) the external +magnetic field (B). Then, non-zero ΔX values indicate a hysteretic effect. As well, a zoom of the +hysteretic region is presented in Figure 1.d, showing several resistance drops and plateaus and +two lower limiting MR branches (with positive —red— and negative —blue— slopes) crossing +at ZF. No relevant influence of the field sweeping rate is observed (Supplementary Figure 1). +A qualitative understanding of the MR behavior is as follows: at high fields, the +magnetization of both layers is parallel and aligned in the direction of the external magnetic field, +thus presenting a minimum value. On the contrary, the two maxima in MR are indicative of +antiparallel alignment of the layers’ magnetization. The shift of these maxima with respect to ZF +is in agreement with the twisted structure of the bilayer. In fact, assuming negligible inter-layer +magnetic interactions, the spins are expected to lie along their respective magnetic easy-axes, thus +being mutually orthogonal. This behavior is in sharp contrast with that of the pristine bilayer, +which shows a single maximum of MR at zero field (ZF), as a result of the antiparallel orientation +between the two layers, and no hysteretic effects. 14,15 Finally, we consider the in-plane angular +dependence (Supplementary Figure 2). All the curves exhibit the same general trend discussed +above but with different coercivity fields and ΔX values. Thus, the field orientation allows for a +fine tuning and control of the hysteretic parameters. Note the asymmetry between 0º and 90º, +suggesting that the underlying spin dynamics are dominated by one of the layers —probably, the + +one with larger area—. Regarding magnetic fields applied along the hard-magnetic axis c (out- +of-plane direction), a hysteretic behavior is manifested as well, but with a significantly broader +maximum of MR (Supplementary Figure 3). In this case, the MR curves are saturating for fields +above 2 T, which, as for the in-plane case, coincides with the field needed to reorient the spins +along the magnetic field direction (c in the present case). Similar results are observed in different +orthogonally-twisted bilayer CrSBr heterostructures, underlying the robustness of the +phenomenology (Supplementary Figure 4), although the exact switching magnetic values differ +between the different devices, probably due to slightly different twisting angles. + +Figure 1.- Magnetic field dependence of the magneto-resistance (MR) in orthogonally-twisted bilayer CrSBr. a) +Optical image of a vertical van der Waals heterostructure consisting on twisted CrSBr monolayers (black dashed lines) +in between few-layers graphene (blue dashed lines). Different insulating h-BN layers (green dashed lines) are employed +both for avoiding shortcuts and protecting the heterostructure. Red arrows indicate the easy-magnetic axis (b) of every +CrSBr monolayer, being the intermediate-magnetic axis (a) perpendicular to it. The hard-magnetic axis (c) corresponds +to the out-of-plane direction. Scale bar: 5 µm. b) Artistic view of the heterostructure (not to scale), highlighting the +twisted CrSBr monolayers (pink, yellow and cyan balls correspond to bromine, sulfur and chromium atoms, +respectively; red arrows represent the spin lying along the easy-magnetic axis, assuming negligible inter-layer magnetic +interactions) placed in between few-layers graphene or NbSe2 thin layers (blue color) on top of pre-patterned electrodes +(gold color) together with a sketch of the electrical measurement configuration. c, d) Field-dependence of the resistance +and MR (bottom panel) as well as its increment (top panel), defined as ΔX = X+B→-B – X-B→+B, where X states either +for the resistance or the MR at T = 10 K and θ = φ = 0º. Sweeping up (down) trace is depicted in red (blue). Red/blue +arrows indicate the sweeping direction of the magnetic field. Black arrows sketch the relative configuration of both +layers’ magnetization. MR is defined as MR (%) = 100·[R(B) – R(0)]/R(0). +Next, we consider both the field and temperature dependence of the MR (Figure 2). We +observe that the behavior resembles that reported for the pristine bilayer.14 Thus, upon cooling +down, a negative MR start developing below 200 K due to the onset of short-range interactions +within the layers, then MR reaches a broad plateau at ca. 150 K, near Tc, and below 100 K it +increases again (Figure 2.a). However, some differences with the pristine bilayer are observed. +First, in the pristine bilayer a minimum in MR, instead of a plateau, is observed at 150 K, followed +at 100 K by a decrease. Second, a hysteretic behavior is observed from temperatures below TN +(second to fourth panels in Figure 2), increasing the coercive field and ΔMR upon cooling down, +while no hysteresis is observed in the pristine bilayer. Similar trends are observed for fields +applied along different directions (Supplementary Figure 5). + +b +C +d +△MR (%) +5 +500 +△R +(%) +5 +500 +AR +0 +0 +AMR +0 +0 +-500 +-5 +-500 +B+B +2 +B ++B +8.0 +0 +8.0 +7.8 +0 +R (10°2) +(%) +MR (%) +2 +7.8 +-4 +7.6 +MR +7.4 +-4 +7.6 +-8 +7.2 +-6 +7.4 +-3 +-2 +-1 +0 +1 +2 +3 +-0.4 +-0.2 +0.0 +0.2 +0.4 +B (T) +B (T) + +Figure 2.- Field and temperature dependence of the MR in orthogonally-twisted bilayer CrSBr. a) Temperature +dependence of the MR at saturated fields (B = 3 T). b, c) Field and temperature dependence of the MR while sweeping +from negative (positive) to positive (negative) fields. d) Field and temperature dependence of ΔMR. MR is defined as +MR (%) = 100·[R(B) – R(0)]/R(0), being R(0) the resistance obtained at zero field and ΔMR= MR+B→-B – MR-B→+B. θ += φ = 0º. +To further explore the irreversibility of the observed hysteresis in Figure 1, we perform +a series of first-order reversal curves (FORC). The FORC analysis lies behind the Preisach +model.27,28 We increment sequentially the maximum applied magnetic field (Bmax) in steps of 20 +mT, after an initial saturation at negative fields (sequence: -0.6 T → +Bmax → -0.6 T). Selected +curves are shown in Figure 3.a (see the Supplementary Video for the whole data set). For +sweeping fields below ca. 0.1 T (first panel in Figure 3.a), the resistance increases/decreases +upon increasing/decreasing B following the behavior already observed in Figure 1.d when +sweeping from negative fields (limiting branch with positive slope). No hysteresis is observed for +this loop, being the MR curve symmetric (ΔMR = 0 at ZF). A more interesting scenario is offered +when this field threshold is overcome (second panel in Figure 3.a). In this case, the resistance +increases (red curve) upon increasing B, as before, until a sharp drop occurs at ca. 0.1 T. Then, +upon decreasing B (blue curve), the resistance decreases but with a smaller slope until a second +drop is observed at ca. -0.1 T, when it returns to the initial path (limiting branch with positive +slope). This behavior results in the emergence of an asymmetric hysteresis (ΔMR ≠ 0 at ZF). +Similar asymmetric curves with successive drops in the resistance, giving rise to steps and +plateaus at well-defined magnetic fields, are observed upon increasing the maximum sweeping +magnetic field value (third panel in Figure 3, Supplementary Video and Supplementary Figure +6). Interestingly, each step observed for positive fields is characterized by a different slope while +returning to ZF. This slope decreases until a saturation field is reached (0.32 T in the present +case). For B > 0.32 T, the limiting branch with negative slope is reached and the hysteresis loop +becomes fully symmetric with respect to the R axis (fourth panel in Figure 3.a). Interestingly, +when coming from positive saturated fields (sequence: +0.6 T → -Bmax → +0.6 T; Figure 3.b and +Supplementary Video), the same phenomenology is observed but reversing the modulus of the +switching fields (mirror image with respect to the R axis). +Therefore, this magnetic system is formed by two ground states that are degenerated at +ZF but that evolve with opposite MR slopes in presence of B. Thus, an initial saturation at negative +fields leads to a state defined by the MR branch with positive slope (Figure 3.a). This state is +sketched as a set of blue circles in the Figure. Conversely, when coming from positive fields, a +different state is obtained (set of red circles in Figure 3) leading to the MR branch with negative +slope. For B values within the range ±0.32 T the system evolves hysteretically and selectively +towards one of these two ground states and only for higher |B| values a change of ground state is +possible. This allows to select at will the ground state of the system. Furthermore, in the hysteretic +region such evolution takes place through successive steps at specific fields that may be associated +to intermediate states. This multistep phenomenology can be related with the Preisach model. +Thus, starting from one of the two MR branches, each one of the resistance drops observed in the + +a +b +MR(%) +c +MR(%) +p +AMR% +-25-20-15-10-50 +25-20-15-10-50 +-4 +0 +4 +0 +300 +300 +300 +B=3T +-B +3+B ++B→-B +-5 +250 +250 +250 +-10 +200- +200 +200- +150 +150 +R +-15 +> +100 +100 +100 +20 +50 +50 +50 ++0 +0- +0- +150300 +-3 +-2 +-101 +2 +3 +-3 +-2 +2-101 +2 +3 +-3-2-10 +1 +2 +3 +T (K) +B (T) +B (T) +B (T)hysteresis curves is associated to the switch of an individual hysteron, leading to each one of the +intermediate states postulated above. In applied terms, every one of these switches could be +potentially employed as a bit of information. This is schematically sketched in Figure 3 by +sweeping red/blue bytes. Importantly, there is also magnetic memory at ZF since we can select +between hysteretic and non-hysteretic MR scenarios depending on the initial ground state of the +system. In the Supplementary Figure 7, we consider different magnetic field sweep protocols +and, for example, in the sequence ZF → + Bmax → ZF we observe hysteresis only after an initial +saturation in negative magnetic fields. Therefore, the magnetic history allows us to control the +appearance or not of hysteresis. Regarding the physical origin of this multistep spin switching, +we can tentatively attribute it to the stabilization of different domain configurations or spin +textures (as the different skyrmions or vortices phases observed in metallic magnetic multilayered +thin film heterostructures, like Pt/Co),6,29,30 thus motivating future magnetic imaging experiments +in these CrSBr twisted bilayers to identify the nature of these states. To manifest the robustness +of these results, we present in Supplementary Figure 8 the study for other orthogonally-twisted +CrSBr bilayers. Overall, similar trends, although at different switching fields, are observed under +different in-plane field orientations (Supplementary Figure 9) and temperatures +(Supplementary Figure 10). + +Figure 3.- Multistep spin switching with magnetic memory in orthogonally-twisted bilayer CrSBr. First-order +reversal curves considering the sequence a) -0.6 T → +Bmax → -0.6 T and b) +0.6 T → -Bmax → +0.6 T at 10 K and θ += φ = 0 °. Bmax is incremented sequentially in steps of 20 mT and selected curves are shown (see Supplementary Video +for the whole dataset). The saturated state at negative (positive) magnetic fields is schematically sketched as a set of +blue (red) circles configuration, being every spin switch related to the change of one individual hysteron (squared +hysteresis operator characterized by a coercive field and a field shift from zero) within the Preisach model. MR is +defined as MR (%) = 100·[R(B) – R(0)]/R(0), being R(0) the resistance obtained at zero field in the symmetric case. + +In conclusion, we have shown that twisting engineering of magnetic 2D materials is a +fruitful platform for the emergence of new correlated phases in synthetic metamagnets, as +exemplified here by the appearance of multistep spin switching accompanied by hysteretic MR +effects in orthogonally-twisted bilayer CrSBr. These field-induced features can be controlled by +playing with the modulus and direction of the applied magnetic field, being absent in pristine +CrSBr mono- and bi-layers. Overall, our results pinpoint twisted bilayer CrSBr as a versatile and +rich platform for controlling and addressing the magnetic information on 2D magnets —of special + +a +606 +:IBmaxl = 0.06 T 主 +:IBmaxl = 0.16 T +:IBmaxl = 0.18 T +IBmaxl = 0.50 T +400 +0 +-400 +2 +T+ +B +8 +N +-8.0 +0 +8 +R +8 +2 +7.8 +(10 +.4 +7.6 +a +-6 +8 +8 +8 +8 +8 +7.4 +-8 +0.4-0.2 0.0 0.20.4 +-0.4-0.2 0.00.20.4 ++-0.4-0.2 0.00.20.4 +-0.4-0.2 0.00.20.4 +B (T) +B (T) +B (T) +B (T) +b +g09 +IBmaxl = 0.06 T 主 +:IBmaxl = 0.16 T +:IBmaxl = 0.18 T ± +[Bmaxl = 0.50 T +400 +0 +-400 +2 ++ +B +M +E8.0 +0 +R +88 +2 +7.8 +(10° +.4 +-7.6 +b +-6 +8 +8 +8 +8 +7.4 +-8 +-0.4 -0.2 0.0 0.2 0.4 +-0.4-0.2 0.0 0.2 +0.4 +-0.4 -0.2 0.0 0.2 0.4 +-0.4 -0.2 0.0 0.2 0.4 +B (T) +B (T) +B (T) +B (T)relevance in areas such as spintronics or magnonics31—, as well as for motivating a new +playground for fundamental studies. In particular, this orthogonally-twisted bilayer CrSBr, may +offer a promising route for the creation and manipulation of non-colinear magnetic textures, like +vortices or topologically protected skyrmions and merons.18,19,32,33 On the other hand, the +controlled stacking of 2D magnetic monolayers under defined angles opens new avenues to +artificially increase the magnetic symmetry in the plane, thereby reducing the anisotropy energy. +Of special interest is to reach the crossover from easy-axis to easy-plane anisotropy, since easy- +plane (XY) systems34 are predicted to host dissipationless spin transport. 35,36 +Methods +Crystal growth: +CrSBr crystals were synthesized by chemical vapor transport and characterized by powder and +crystal X-Ray diffraction, energy dispersive X-Ray analysis (EDX), high-resolution TEM, +SQUID magnetometry and temperature-dependent single crystal, as reported in our previous +work.14 +van der Waals heterostructure fabrication: +2D layers were obtained by mechanical exfoliation from their bulk counterparts under strict inert +conditions (argon glovebox) since CrSBr monolayers degrade in air.15,24 The obtained flakes were +examined by optical microscopy (NIKON Eclipse LV-100 optical microscope under normal +incidence) as a fast tool for identifying the number of layers and compared with our previously +calibrated values.14 Typical CrSBr flakes exhibit a ribbon shape, being the long (short) direction +associated with the a (b) axis and being the c axis the out-of-plane direction, as previously verified +by optical contrast, Raman spectroscopy and selected area electron diffraction patterns.14 The van +der Waals heterostructures were fabricated by assembling the different layers by the deterministic +assembly of the flakes using polycarbonate, as reported by Wang et al.,37 using a +micromanipulator. Thus, the twisted-monolayers were placed between top and bottom few-layers +metallic NbSe2 or few-layers graphene, where several insulating h-BN layers were inserted both +for avoiding possible short-cuts and protecting the whole heterostructure from degradation. The +stack of 2D materials was placed on top of pre-lithographed electrodes (5 nm Ti/50 nm Au on +285 nm SiO2/Si from NOVA Electronic Materials, LCC). The whole process was performed +under inert atmosphere conditions. +A total of three orthogonally-twisted CrSBr bilayers were fabricated (device 1 -data shown in the +main text- is based on metallic NbSe2 thin-layers while device 2 and 3 -data shown in the +Supplementary Information- are based on few-layers graphene), observing a consistent +phenomenology between all of them. Note that, in the case of using few-layers graphene, the +intrinsic MR arising from the few-layers graphene is observed as well (in special, for out-of-plane +applied magnetic fields), yielding to a finite positive value of the MR even at room temperature. +Nonetheless, the magnetic fingerprints of the twisted-CrSBr are well noticeable, clearly +developing below TN. +In particular, device 1 is formed by a top (bottom) CrSBr monolayer of 77.2 µm2 (53.3 µm2), with +an overlap area of 9.3 µm2 and a twisted angle of 92.5º. Device 2 is formed by a top (bottom) +CrSBr monolayer of 190.1 µm2 (117.3 µm2), with an overlap area of 15.9 µm2 and a twisted angle +of 89.3º. Device 3 is formed by a top (bottom) CrSBr monolayer of 206.6 µm2 (121.1 µm2), with +an overlap area of 7.9 µm2 and a twisted angle of 87.0º. +Magneto-transport measurements: +Electrical measurements were performed in a Quantum Design PPMS-9 cryostat with a 4-probe +geometry, where a DC current was passed by the outer leads and the DC voltage drop was + +measured in the inner ones. DC voltages and DC currents were measured (MFLI from Zurich +Instruments) using an external resistance of 1 MΩ, i.e., a resistance much larger than the sample. +Temperature sweeps were performed at 1 K·min−1, field sweeps at 200 Oe/s, rotation sweeps at 3 +°/s and the current bias was 1 µA, unless otherwise explicitly specified. Magneto-resistance (MR) +is defined as: MR = 100[R(B) – R(0)]/R(0), where B is the external magnetic field and R(0) is the +resistance at zero field in the symmetric case (see text). +Supporting Information +Supporting Information is available from the authors. + +Acknowledgements +The authors acknowledge the financial support from the European Union (ERC AdG Mol-2D +788222, FET OPEN SINFONIA 964396), the Spanish MCIN (2D-HETEROS PID2020- +117152RB-100, co-financed by FEDER, and Excellence Unit “María de Maeztu” CEX2019- +000919-M) and the Generalitat Valenciana (PROMETEO Program, PO FEDER Program +IDIFEDER/2018/061, a Ph.D fellowship to C.B.-C., and the postdoctoral fellow APOSTD- +CIAPOS2021/215 to S.M.-V). This study forms part of the Advanced Materials program and was +supported by MCIN with funding from European Union NextGenerationEU (PRTR-C17.I1) and +by Generalitat Valenciana. We thank Á. López-Muñoz for his constant technical support and +fundamental insights, A. 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One-dimensional electrical contact to a two-dimensional material. +Science 342, 614–7 (2013). + + + + +Supplementary Information for +Multistep spin switching in orthogonally-twisted ferromagnetic monolayers. +Carla Boix-Constant, Samuel Mañas-Valero*, Eugenio Coronado.* +Instituto de Ciencia Molecular (ICMol) - Universitat de València, Catedrático José Beltrán 2, +Paterna 46980, Spain. +e-mail: samuel.manas@uv.es, eugenio.coronado@uv.es + + + +Supplementary Figure 1.- Magnetic field dependence of the magneto-resistance (MR) in orthogonally-twisted +bilayer CrSBr for different devices and field sweep rates. a) Device based on NbSe2 vertical van der Waals +heterostructure (T = 10 K). b-c) Devices based on few-layers graphene vertical van der Waals heterostructures (T = 2 +K). The field is applied in-plane (φ = 0 º) along the easy-axis of the CrSBr monolayer with smaller area, corresponding +to θ = 0º (for a and b) and θ = 90º (for c). MR is defined as MR (%) = 100·[R(B) – R(0)]/R(0), being R(0) the resistance +obtained at zero field. + + + +Supplementary Figure 2.- In-plane magnetic field dependence of the magneto-resistance (MR) in orthogonally- +twisted bilayer CrSBr (device 1). a) 2D plot of ΔMR. b) Selected MR/resistance hysteresis loops (bottom panel) and +its increment (top panel) at selected angles. Measurements corresponds to an orthogonally-twisted CrSBr bilayer based +on metallic NbSe2 thin-layers vertical van der Waals heterostructure. MR is defined as MR (%) = 100·[R(B) – +R(0)]/R(0), being R(0) the resistance obtained at zero field. + +a +b +c +(%) +6 +Device 1 +400 +202 +Device 2 +200> +1.5- +Device 3 +100 +AR +R +0 +AMR +0 +R +△MR +B +AMR +-400. +-100 +6. +8.2 +-1.5 +TTTTTTT +1 +0.5 +8.0 +9.3 +0 +0 +0.0 +11.70 +(%) +7.8 +(%) +9.2 +(10 +-0.5 - +7.6 +-1 +R +MR +MR +11.60 +4 +9.1 +p +-1.0 - +M +-2 +10:0e/s +Oe/s +-1.5 - +10.0e/s +7.2 +50:0e/s +50 8e/s +11.50 +0e/s +3 +9.0 +1000e/s +： +-8 +100:0e/s +100Qe/s +-200:0e/s +-2.0 - +200:0e/s +200 0e/s +-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 +-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 +-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 +B (T) +B (T) +B (T)a +△MR (%) +b +.8 +-6 +-4 +-2 +0 +4 +6 +8 +AMR (%) +5 +T = 10 K +400 +$=0% +200 +360 +T = 10 K +0 +$ = 0° +-200F +315 +5 +-400 +270 +B++B +0=00 +2 +0 = 45 ° +-4 = 90 0 +225 +8.0 +0 +180 +2 +-7.8刀 +(10) +135 +- 7.6P +90 +-6 +7.4 +45 +-8 +0 +7.2 +-0.6 +-0.4 +-0.2 +0.0 +0.2 +0.4 +0.6 +-0.6 +-0.4 +-0.2 +0.0 +0.2 +0.4 +0.6 +B (T) +B (T) + +Supplementary Figure 3.- Out-of-plane magnetic field dependence of the magneto-resistance (MR) in +orthogonally-twisted bilayer CrSBr (device 1). a,b) Field-dependence of the resistance and MR (bottom panel) as +well as its increment (top panel), defined as ΔX = X+B→-B – X-B→+B, where X states either for the resistance or the MR +(T = 10 K, θ = 0º and φ = 90º). Sweeping up (down) trace is depicted in red (blue). + + +b +1 +100 +(%) +100 +△R +△R +0 +0 +0 +-100 +-100 +0 +0.0 +8.95 +-0.5 - +-2 +8.8 +8.90 +-1.0 - +R +(%) +8.85 +R +-4 +(10° +MR +MR +-2.0 +8.80 +a +-6 +8.4 +-2.5 +一 +8.75 +-8 +-3.0 +8.70 +8.2 +B++ E +-3 +-2 +-1 +0 +1 +2 +3 +-0.8 +-0.4 +0.0 +0.4 +0.8 +B (T) +B (T) +Supplementary Figure 4.- Magnetic field dependence of the magneto-resistance (MR) in orthogonally-twisted +bilayer CrSBr based on few-layers graphene van der Waals heterostructures. Panels a-d (e-h) correspond to device +2 (3). a, b, e, f) Field-dependence of the resistance and MR (bottom panel) as well as its increment (top panel), defined +as ΔX = X+B→-B – X-B→+B, where X states either for the resistance or the MR for in-plane (a, e panels) and out-of-plane +(b, f panel) fields. Sweeping up (down) trace is depicted in red (blue). Red/blue arrows indicate the sweeping direction +of the magnetic field. MR is defined as MR (%) = 100·[R(B) – R(0)]/R(0). c, g) 2D plot of ΔMR. d, h) Selected MR +and resistance hysteresis loops (bottom panel) and its increment (top panel) at selected angles. Note that the intrinsic +MR arising from the few-layers graphene is observed as well (in special, for out-of-plane applied magnetic fields), +yielding to a finite positive value of the MR even at room temperature. Nonetheless, the magnetic fingerprints of the +twisted-CrSBr are well noticeable. + +Device 2 +b +[%] +2 - +50 += 90 ° +AR (Q) +AMR ( +0 +0 +AMR +0.0 +0 +-2 1 += 2 K +L +-0.5 - +T = 35 K +-50 ++ +-B=±+B +6.36 +1- +0.4 +F0 +T +6.34 +R (103 +(%) +R +MR ( +E 9.5 +-2 } +-0.4 +1 +9.4 +-6.28 +-0.8- +F 9.3 +LLLLL +-0.6 +-0.4 +-0.2 +0.0 +0.2 +0.4 +0.6 +-1.0 +-0.5 +0.0 +0.5 +1.0 +B (T) +B (T) +C +AMR (%) +d +3 +-2 +1 +2 +200 +△R (Q) +-200 +270 +m +8=46: +225 +R(103 +-33 +45- +-4} +E9.1 +0 +-0.6 +-0.4 +0.0 +0.2 +0.4 +0.6 +-0.6 +-0.2 +0.0 +0.2 +0.4 +B (T) +B (T) +Device 3 +200 +△MR (%) +△R (Q2) +0=0° +E 50 +T =2K +0.0 - +E0 +卡-200 +T= 35K +12- +8+ +-B=+B +10 +E 9.2 +0.0} +-卡11.70 +11.65 2 +9.0- +MR (%) +E11.552 +E8.6 +E +11.50 +-2.0→ +8.4 +L +0.6-0.4-0.2 +0.0 +0.2 +0.4 +0.6 +-1.5 +-1.0 +-0.5 +0.0 +0.5 +1.0 +1.5 +B (T) +B (T) +AMR (%) +h +3 +-1 +1 +2 +(%) +2 J +360FT=2.K +△R (Q) +AMR +315 +-2 - +-200 +270 +TLLL +0.53 +11.8 +225- +-0.5↓ +8：11.7 +R (103 +MR +-1.53 +0) +E 06 +-2.0 月 +45于 +-2.5 3 +11.5 +0 +LLLLLLLLI +-0.6 +-0.4 +-0.2 +0.0 +0.2 +0.4 +0.6 +-0.6 +-0.4 +-0.2 +0.0 +0.2 +0.4 +0.6 +B (T) +B (T) +Supplementary Figure 5.- Temperature and magnetic field dependence in orthogonally-twisted bilayer CrSBr. +a-c) In-plane field orientations. d) Out-of-plane orientation. First panel: Temperature dependence of the MR in the +saturated state (B = 3 T). Second (third) panel: field and temperature dependence of the MR while sweeping from +negative (positive) to positive (negative) fields. Fourth panel: field and temperature dependence of ΔMR. Magneto- +resistance (MR) is defined as MR (%) = 100·[R(B) – R(0)]/R(0), being R(0) the resistance obtained at zero field. + +a +0=00 +MR (% +△MR (%) +-25-20-15-10°-5 +-25-20-15-10° +-5 +A +Φ=00 +300- +300 ++B→-B +300- +-B→+B +-5# +250 +250 +250 +-10 +200 +200 +200 +150 +Z150 +100 +100- +100 +-20- +50 +50 +50 +01 +01 +0+ +150300 +-3 +-2 +-10 +1 +3 +-3 +-2 +-10 +1 +2 +3 +-3 +-21 +0 +1 +2 +3 +T (K) +B (T) +B (T) +B (T) +b +0=450 +MR (%) +MR (%) +AMR (%) +-20-15-10-5 +05 +Φ=0。 +-20 +-10 +0 +-8 +8 +300- +300- +300- +B=3T +-B→+B ++B→B +-5 +250 +250 +250 +-10 +200 +200 +200 +150 +R +-15 +100 +100 +100 +-20- +50 +50 +50 +01 +01 +0 +150300 +-3 +-2-1 0 +1 +2 +3 +-3 +-2 +-10 +1 +2 +3 +-3 +-2 +-1 +0 +1 +2 +3 +T (K) +B (T) +B (T) +B (T) +c +0=90° +MR (%) +MR (%) +△MR (%) +Φ=0° +-20-15-10-5 +0 +-20-15-10 +-5. +-4 +-2 +300 +300 +300- +-B→+B ++B→-B +-5 +250 +250 +250 +-10 +200 +200 +200 +150 +100手 +100 +100 +-20 +50 +50 +50 +0- +0 ++0 +0 +150300 +-3-2-1012 +3 +-3-2-1012 +3 +-3 -2 -1 +0 +1 +2 +3 +T (K) +B (T) +B (T) +B (T) +d +0=0° +MR (%) +MR (%) +△MR (%) +-20-15 +-10 +-5 +-20-15-10° +Φ=900 +0 +-2 +-1 +0 +2 +B=3T +300- +300 +300- +-B→+B ++B→-B +-5 +250 +250 +250 +0-101 +200 +200- +200 +150 +150 +100 +100 +100 +-20- +50 +50 +50 +01 +01 +150300 +-3 -2 -1 0 +1 +2 +3 +-3 -2 -1. 0 +2 +3 +-3-2-10 +1 +2 +3 +T (K) +B (T) +B (T) +B (T) +Supplementary Figure 6.- Hysteresis opening in orthogonally-twisted bilayer CrSBr (device 1). a-e) Field- +dependence of the resistance and MR (bottom panel) as well as its increment (top panel), defined as ΔX = X+B→-B – X- +B→+B, where X states either for the resistance or the MR after sweeping up to different selected magnetic fields at 10 K +and θ = φ = 0°, being the magnetic field applied in plane along the easy (intermediate) magnetic axis of the top (bottom) +CrSBr monolayer. f) ΔR 2D plot. The magnetic sweep protocol is as follows: after a first saturation at negative fields, +we perform the sequence ZF → Bmax →-Bmax→ ZF, increasing in every cycle the maximum field in 20 mT step. +Sweeping up (down) trace is depicted in red (blue) in a-d. Red/blue arrows indicate the sweeping direction of the +magnetic field. MR is defined as MR (%) = 100·[R(B) – R(0)]/R(0), being R(0) the resistance obtained at zero field in +the symmetric case. + +a +b +C +% +0.1 +(%) +Bmaxl = 0.12 T +100 +AR +200 +4 +AR +AR +0 +A +0 +AMR +0 +0 +AMR +0 +0 +(2) +0.1 +4 +-1 +-2 +-100 +200 +2 +8.1 +2 +2 +1. +8.0元 +-8.0元 +8.0刀 +(%) +(10 +(10 +(10° +MR +0 +MR +7.9 +MR +7.9 +b +.1 +-2 +7.8 +21 +2 ++ B +F7.8 +B +B +3 +B +-0.05 +0.00 +0.05 +-0.1 +0.0 +0.1 +-0.1 +0 +0.1 +B (T) +B (T) +B (T) +e +f +AMR (%) +400 +5 +400 +△R (2) +4 +:|Bmaxl = 0.30 +AR +R +-400 +0 +400 +0 +0 +AMR +0 +0.6 +-4 +b +400 +-5 +0.5 +2 +2- +8.0 +-8.0 +E +0.4 +0 +0 +R +(%) +R +0.3 +2 +MR +0.2 +7.6 +7.6 +Bmax +0.087 +0.1 +0.12 +2+ B +0.30T +0.18 +0.40 T +7.4 +0.0 +. +-0.2 +0.0 +0.2 +-0.4 +0.0 +0.4 +-0.4 +0.0 +0.4 +B (T) +B (T) +B (T) +Supplementary Figure 7.- Multistep spin switching with magnetic memory in orthogonally-twisted CrSBr under +different magnetic field sweep protocols. T = 10 K and θ = φ = 0 ° (device 1). a, b) Sequence ZF → +Bmax → ZF. c, +d) Sequence ZF → -Bmax → ZF. e, f) Sequence ZF → +Bmax → ZF → -Bmax → ZF. g, h) Sequence ZF → -Bmax → ZF +→ +Bmax → ZF. i) Sequence +0.6 T → -Bmax → + 0.6 T. j) Sequence -0.6 T → +Bmax → - 0.6 T. Panels a, c, e, g and i +(b, d, f, h and j) correspond to an initial saturation at positive (negative) magnetic fields. In every field sweep, Bmax is +incremented in steps of 20 mT. + +a +Sequence: ZF → Bmax → ZF +b +Sequence: ZF → Bmax → ZF +Initial saturation at positive fields. +Initial saturation at negative fields. +0.0 +7800- +0.1 +8000 } +70.0 +0.2 +0.1 ++0.3菱 +7800 +0.2 +F0.3 +0.4 3 +R +0.5 +R 7600- +7400- +0.6 +0.5 +7400- +0.6 +7200 +7200 +-0.6 +-0.4 +-0.2 +0.0 +0.2 +0.4 +0.6 +-0.6 +-0.4 +-0.2 +0.0 +0.2 +0.4 +0.6 +B (T) +B (T) +c +Sequence:ZF +→ ZF +d +Sequence: ZF +→ ZF +Initial saturation at positive fields. +Initial saturation at negative fields. +8000 - + 0.0 +0.0 +0.1 +7800- +0.1 +0.2 +-0.2 ++0.3菱 +-0.3 菱 +-0.4 3 +R +-0.4 3 +R 7600- +0.5 + 0.5 +-0.6 +7400 +7400- +-0.6 +-0.6 +-0.4-0.2 +0.0 +0.2 +0.4 +0.6 +-0.6-0.4-0.2 +0.0 +0.2 +0.4 +0.6 +B (T) +B (T) +e +f +Sequence: ZF → + Bmax +Initial saturation at positive fields. +Initial saturation at negative fields. +8000- +0.0 +8000 } +-0.0 +0.1 ++ 0.1 +0.2 +0.2 +7800 ++0.3菱 ++0.3 +0.4 3 +Eto +R 7600 +R 7600 +0.5 +0.5 +7400- +-0.6 +0.6 +7400 +-0.6 +-0.4 +-0.2 +0.0 +0.2 +0.4 +0.6 +-0.6 +-0.4 +-0.2 +0.0 +0.2 +0.4 +0.6 +B (T) +B(T) +g +Sequence: ZF +→ ZF +h +Sequence: ZF +→ZF. +→ ZF +Initial saturation at positive fields. +Initial saturation at negative fields. +8200 +0.0 +8000- +0.0 +8000- +0.1 +0.1 +0.2 +0.2 +7800 +0.3菱 +-0.3菱 +0.4 3 +-0.4 3 +R 7600 - +R 7600 +0.5 + 0.5 +7400- +-0.6 +7400- +0.6 +7200 +7200 +-0.6 +-0.4 +-0.2 +0.0 +0.2 +0.4 +0.6 +-0.6 +-0.4 +-0.2 +0.0 +0.2 +0.4 +0.6 +B (T) +B(T) +Sequence: + 0.6 T → - B, ++ 0.6 T +Sequence: - 0.6 T → + Br +-0.6T +Initial saturation at positive fields. +Initial saturation at negative fields. +8000 } + 0.0 +8000 } +70.0 +0.1 +0.1 +7800- +0.2 +0.2 +7800- +0.3菱 +(0) +- 0.3 显 +R 7600 +0.4 3 +-0.4 3 +R 7600 - +一 +0.5 +0.5 +7400 +0.6 +7400 +0.6 +7200 +7200 +-0.6 +-0.4 +-0.2 +0.0 +0.2 +0.4 +0.6 +-0.6 +-0.4 +-0.2 +0.0 +0.2 +0.4 +0.6 +B (T) +B (T) +Supplementary Figure 8.- Hysteresis opening in orthogonally-twisted bilayer CrSBr based on few-layers +graphene vertical van der Waals heterostructures. a-e) Field-dependence of the resistance and MR (bottom panel) +as well as its increment (top panel), defined as ΔX = X+B→-B – X-B→+B, where X states either for the resistance or the +MR after sweeping up to different selected magnetic fields at 2 K and θ = φ = 0 ° (device 2). f-g) ΔR 2D plot. The +magnetic sweep protocol is as follows: for panels a-e, after a first saturation at negative fields, we perform the sequence +ZF → Bmax →-Bmax→ ZF, increasing in every cycle the maximum field in 20 mT step. In panel f-g, the sequence is ++0.6 T → -Bmax →+0.6 and increasing in every cycle the maximum negative field in 20 mT step, for device 2 (g) and +device 3 (f). Sweeping up (down) trace is depicted in red (blue) in a-d. Arrows indicate the sweeping direction of the +magnetic field. Magneto-resistance (MR) is defined as MR (%) = 100·[R(B) – R(0)]/R(0), being R(0) the resistance +obtained at zero field in the symmetric case. + +a +Sweep up to |Bm + = 0.10 T +b +Sweep up to IBmaxl = 0.16 T +e +10 +(%) +△R +2.5- +Sweepupto: +0 +AMR +0.10 +200 +0 +0 +0.161 +-10 +0.30 +100 Z +-1 +-100 +0.60 +R +0.0 +0 +F9.7 +Device 2 +Device 2 +Device 2 +0.5 +9.7 +R +R +-200 +(%) +-2.5- +(103 +(103 +MR +0.0- +MR +0 +9.6 +1.5 +-9.6 +a +9.75 +0.5 +1.0 +9.70 +→+BL9.5 +B +B +0.5 # +-0.1 +0.0 +0.1 +-0.1 +0.0 +0.1 +9.65 +B (T) +B (T) +0.0- +9.60 +c +Sweep up to IB, +maxl=0.30T +d +Sweep up to IBm +maxl = 0.60 T +(%) +(%) ) +0.5- +9.557 +2 +Device2E +Device 2 +(103 +AMR +AMR +0 +0- +0 +-1.5 +F9.7 +9.45 +1 - +0 +-2.0 +(%) +R +(%) +9.6 +R +9.40 +0. +F9.6 +(103) +(103 +MR +-2.5 +MR +a +a +9.35 +9.5 +-21 +9.4 +-3.0 +-2 +9.30 +B±+ B [9.4 +-3.5 +-0.2 +0.0 +0.2 +-0.5 +0.0 +0.5 +-0.5 +0.0 +0.5 +B (T) +B (T) +B (T) +f +g +△R (Q) +△R (Q) +300 +-200 +-100 +0 +100 +200 +300 +-200 +-100 +0 +100 +200 +0.6 +0.6 +0.5 - +0.5. +TT +0.4 +0.4 +E +E +0.3- +0.3 +一 +TTT +0.2 +0.2 +0.1 - +0.1 +Device 2 +Device 3 +0.0 +0.0 +-0.6 +-0.4 +-0.2 +0.0 +0.2 +0.4 +0.6 +-0.6 +-0.4 +-0.2 +0.0 +0.2 +0.4 +0.6 +B (T) +B (T) +Supplementary Figure 9.- First-order reversal curves for different in-plane (φ = 0 °) magnetic fields at T = 10 K +(device 1). We consider the sequence +0.6 T → -Bmax → + 0.6 T. In every field sweep, Bmax is incremented in steps +of 20 mT. + +a +c +4Bmax 2 +.d +4Bmax .2 +0.6 +0.4 +0.0 +0.6 +0.4 +0.0 +0.6 +0.4 +0.0 +0.6 +0.4 +0.0 +N0= +=30g +8000 +8000- +8200 - +1大 +8000- +R 7600 +7600 +7400- +7400 +7400- +7400- +7200 +LL +TELLEE +-0.4_0.0 +0.4 +-0.4 _0.0.0.4 +-0.4 _0.0、0.4 +-0.4 _0.0. 0.4 +B (T) +B (T) +B (T) +B (T) +e +,IBmax I(T) +/Bmax (T) +h +0.6 +0.4 +0.0 +0.6 +0.4 +0.0 +0.6 +0.4 +0.0 +0.6 +0.2 +D.4 +0.0 +e= 60°1 +8200 = 90° +e=105° +8200 - +8000 - +8000 +8000 - +8000 - +7800 +a +7800 +7800 +7600 +7600 +R + 7600 +7600- +7400- +7400- +7400- +7400 +LLLL +-0.4 0.0 0.4 +-0.40.0 +0.4 +-0.4 0.0 0.4 +-0.4 0.0 0.4 +B (T) +B (T) +B (T) +B (T) +- +0.4 max T) +0.6 +0.0 +0.6 +0.4 +0.6 +0.0 +0.6 +0.0 +8200e = 120% +e=135R1 +=150° +8200 - + = 165 +8200 - +8200 - +8000 +.8000 - +8000 - +7800 +R +R +7600 +7600 +7600 +7400- +7400- +7400- +7400 - +-0.40.0 +0.4 +-0.40.0 +0.4 +-0.40.0 +0.4 +-0.4 0.0 0.4 +B(T) +B (T) +B (T) +B (T) +m +Bmax I(T) +↓Bmax T) +JBmax 4T2 +0.6 +04 +0.0 +0.6 +0.4 +02 +0.0 +0.6 +0.4 +0.0 +0.6 +0.0 +8200 - +e = 180° +8200 = 210° +1e=195°1 +8200 - +16 = 225 +8000 + 0008 +8000 +8000 +7800 +g7800 +7800 +7800 +R 7600 +R 7600 + 7600 + 7600 +7400 - +7400 +7400 +7400- +-0.4 0.0 0.4 +-0.4 0.0 0.4 +-0.4 0.0 0.4 +-0.4 0.0、0.4 +B (T) +B (T) +B (T) +B (T) +b +IBmax I(T) +IBmax I(T) +s +IBmax I(T) +JBmax I(T) +0.6 +0.4 +0.0 +0.6 +0.4 +0.0 +0.6 +0.4 +0.0 +0.6 +0.0 +0.2 +0.2 +e= 240° +8200 - +[e = 255° +8200=270 +Ve = 285° +8200 - +E 0008 +8000 - +8000 - +8000 +F 0082 +R +R 7600 +P 7600 +7600 +一 +一 +7400 +7400- +7400- +7400 +-0.40.0 +0.4 +-0.4 0.00.4 +-0.4 0.0 0.4 +-0.4 0.9、0.4 +B (T) +B (T) +B (T) +B (T) +u +IBmaxl (T) +[Bmaxl (T) +w +IBmaxl (T) +x +0.6 +0.4° +0.2 +0.0 +0.6 +0.4 +0.2 +0.0 +0.6 +0.4 +0.0 +0.6 +0.0 +8200 →= 300° : +e=315 +@ = 330° +8200 - +1=345 +8200 - +8200 - +8000 +8000 - +8000 - +E 0082 +R 7600 +7600 +7600 +R +0092 +7400 +7400 - +7400- +7400- +-0.40.0 +0.4 +-0.4 0.0 +0.4 +-0.40.0 +0.4 +-0.40.0 +0.4 +BT +B (T) +B (T) +B (T) +Supplementary Figure 10.- First-order reversal curves for in-plane (θ = φ = 0 °) magnetic fields at different +temperatures (device 1). We consider the sequence +0.6 T → -Bmax → + 0.6 T. In every field sweep, Bmax is +incremented in steps of 20 mT. + +a +[Bmaxl (1) +b +Bmaxl (1) +0.6 +0.4 +0.2 +0.0 +0.6 +0.4 +0.2 +0.0 +T = 10 K +6800 - +T = 20 K +8000 +7800 +6600 +7600 +R 6400 +一 +7400 +一 +6200 +7200 +-0.6 +-0.4 +-0.2 +0.0 +0.2 +0.4 +0.6 +-0.6 +-0.4 +-0.2 +0.0 +0.2 +0.4 +0.6 +B (T) +B(T) +c +IBmaxl (T) +d +[Bmaxl (T) +0.6 +0.4 +0.2 +0.0 +0.6 +0.4 +0.2 +0.0 +5400 +一 +T = 40 K +T = 60 K +4600 +5200 +4400 +5000 +4200 +R +R +4800 +一 +4000 +4600 +一 +3800 - +-0.6 +-0.4 +-0.2 +0.0 +0.2 +0.4 +0.6 +-0.6 +-0.4 +-0.2 +0.0 +0.2 +0.4 +0.6 +B(T) +B (T) +e +[Bmaxl (T) +f +(Bmaxl (T) +0.6 +0.4 +0.2 +0.0 +0.6 +0.4 +0.2 +0.0 +4000 +T = 100 K +4200 +一 +T = 140 K +3800. +一 +(u) +4000 +R +3600 +一 +R +3800 +3400 +一 +3600. +-0.6 +-0.4 +-0.2 +0.0. +0.2 +0.4 +0.6 +0.6 +-0.4 +-0.2 +0.0 +0.2 +0.4 +0.6 +B (T) +B (T) +g +IBmaxl (T) +0.6 +0.4 +0.2 +0.0 +5520 +T = 200 K +5500 +R +5480 +/ +- +-0.6 +-0.4 +-0.2 +0.0 +0.2 +0.4 +0.6 +B (T) \ No newline at end of file diff --git a/lNE5T4oBgHgl3EQfiQ-H/content/tmp_files/load_file.txt b/lNE5T4oBgHgl3EQfiQ-H/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9154b68b7b307584b2151e069aa478a2abc67406 --- /dev/null +++ b/lNE5T4oBgHgl3EQfiQ-H/content/tmp_files/load_file.txt @@ -0,0 +1,1326 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf,len=1325 +page_content='Multistep spin switching in orthogonally-twisted ferromagnetic monolayers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' Carla Boix-Constant, Samuel Mañas-Valero*, Eugenio Coronado.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' * Instituto de Ciencia Molecular (ICMol) - Universitat de València, Catedrático José Beltrán 2, Paterna 46980, Spain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' e-mail: samuel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='manas@uv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='es, eugenio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='coronado@uv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='es KEYWORDS: 2D magnets, van der Waals heterostructures, twistronics, spintronics, nanomagnetism, nanodevices, quantum materials, CrSBr The advent of twisting-engineering in two-dimensional (2D) crystals enables the design of van der Waals (vdW) heterostructures exhibiting emergent properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='1–4 In the case of magnets, this approach can afford artificial magnets with tailored spin arrangements that do not exist in nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' Here, we fabricate an artificial magnet by twisting 90 degrees two CrSBr ferromagnetic monolayers with an easy-axis in-plane anisotropy, thus forming an ‘orthogonally-twisted bilayer’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' The magneto-transport properties reveal multistep spin switching with a magnetic hysteresis opening, which is absent in the pristine case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' By tuning the magnetic field, we modulate the remanent state and coercivity and select between hysteretic and non-hysteretic magneto-resistance scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' This complexity is fundamentally different from that achieved in Moiré superlattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' Our results highlight the control over the magnetic properties in vdW heterostructures, leading to a variety of field- induced phenomena and opening a fruitful playground for creating artificial magnetic symmetries and manipulating non-collinear magnetic textures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' Metamagnets and their field-induced phase transitions offer a plethora of counterintuitive phenomenology, as already quoted by Kramers,5 with a direct competition between magnetic anisotropy, exchange, and dipolar energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='6 In absence of magnetic field, these materials show zero net magnetization that suddenly increases until its saturation -thus, resembling a ferromagnet- above a certain magnetic field threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='5 A good example of an A-type metamagnet is offered by the layered vdW semiconductor CrSBr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' The spins in every single layer (ab plane) couple ferromagnetically between them (TC ~150 K), pointing along the easy b axis, whereas the layers couple between them antiferromagnetically (TN ~ 140 K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='7 By applying a magnetic field, it is possible to flip the layers’ magnetization in a parallel fashion via a spin reversal and to induce a spin reorientation along the magnetic field direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' This transition does not present any hysteresis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='8–14 In bulk, the saturation fields at 2 K are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='6 T, 1 T and 2 T for the easy (b), intermediate (a) and hard (c) magnetic axis, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='14 This vdW material can be thinned down to the monolayer limit and integrated into electronic nano-devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' Upon the field-induced spin switching, the magneto-resistance (MR) is large and negative from bulk down to the bilayer case, with a reduction of the saturation field along the easy-axis (from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='6 T in bulk to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='2 T in the bilayer at 2 K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='12–16 The monolayer limit is characterized by the absence of MR for fields applied along the easy axis and small and positive MR for fields applied along the intermediate and hard axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='14,15 The ability for isolating, manipulating and twisting 2D crystals adds a new degree of control in vdW heterostructures, affording emergent new properties, like superconductivity in twisted bilayer graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='1 As far as the 2D magnetic materials are concerned, twisting is much less explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' Still, it has allowed the creation of new magnetic ground states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' For example, by twisting small angles the 2D magnet CrI3, a modulation of the spin-reversal by magneto-optical techniques has been reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='2–4 This twisting not only produces a Moiré superlattice but can also induce Moiré magnetic exchange interactions, in which unique spin textures like magnetic skyrmion bubbles have been theoretically predicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='17–20 However, no 2D twisted-magnets have been incorporated into electronic devices so far, remaining the magneto-transport effects in twisted-magnets fully unexplored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' Here, we twist by ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' 90 degrees two CrSBr ferromagnetic monolayers, thus forming an orthogonally-twisted bilayer, and integrate them in a vertical vdW heterostructure formed by either few-layers graphene or metallic NbSe2 thin-layers for inspecting its magneto-transport properties (Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='a-b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' see Methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' 21–24 Note that, in stark contrast with CrI3, where the spins are out-of-plane, in CrSBr the spins are in-plane pointing along the easy b-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' Therefore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' this orthogonal configuration yields to a complex spin scenario in which the application of an external magnetic field (Zeeman energy),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' the inter-layer magnetic interactions (which favors an antiparallel orientation of both spin layers) and the local spin anisotropy (which tends to orient the spins along their easy magnetic axes) are competing since the easy and intermediate magnetic axis of both monolayers are perpendicular,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' while keeping the hard magnetic axis of both layers parallel (out-of-plane direction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' This case is different from the common Moiré patterns in twisted bilayers, where the twisting angle is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='1 An example of an orthogonally-twisted-CrSBr heterostructure is shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='a-b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' In this vertical geometry, the MR can be rationalized within a spin-valve picture, considering a two-current channel model: when the magnetization of both layers is antiparallel (parallel), there is a higher (lower) resistance across the heterostructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='14,25,26 The field-dependence of the MR at 10 K is presented in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='c for in-plane magnetic fields aligned along the easy-axis of one of the layers (in this case, the top layer;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' θ = φ = 0º).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' Starting at high negative fields (red curve in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='c), the MR is negative and field independent down to -1 T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' then, it increases until a maximum positive MR is observed at ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='16 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' Above this field, it decreases again until reaching a saturation value above +1 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' This value coincides with that observed for the spin reorientation along the intermediate magnetic axis, a, thus suggesting that this is determined by the spin anisotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' Reversing the magnetic field yields to a symmetrical curve that exhibits the maximum in MR at ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='16 T (blue curve in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' These two curves cross at zero field showing a hysteretic behavior when the field modulus is kept below ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='32 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' For an easier visualization of the hysteresis, we present as a top panel in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='c the increment value, defined as ΔX = X+B→-B – X-B→+B, where X states either for the resistance (R) or the MR while decreasing (+B→-B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' blue curve in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='c) or increasing (-B→+B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' red curve in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='c) the external magnetic field (B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' Then, non-zero ΔX values indicate a hysteretic effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' As well, a zoom of the hysteretic region is presented in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='d, showing several resistance drops and plateaus and two lower limiting MR branches (with positive —red— and negative —blue— slopes) crossing at ZF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' No relevant influence of the field sweeping rate is observed (Supplementary Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' A qualitative understanding of the MR behavior is as follows: at high fields, the magnetization of both layers is parallel and aligned in the direction of the external magnetic field, thus presenting a minimum value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' On the contrary, the two maxima in MR are indicative of antiparallel alignment of the layers’ magnetization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' The shift of these maxima with respect to ZF is in agreement with the twisted structure of the bilayer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' In fact, assuming negligible inter-layer magnetic interactions, the spins are expected to lie along their respective magnetic easy-axes, thus being mutually orthogonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' This behavior is in sharp contrast with that of the pristine bilayer, which shows a single maximum of MR at zero field (ZF), as a result of the antiparallel orientation between the two layers, and no hysteretic effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' 14,15 Finally, we consider the in-plane angular dependence (Supplementary Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' All the curves exhibit the same general trend discussed above but with different coercivity fields and ΔX values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' Thus, the field orientation allows for a fine tuning and control of the hysteretic parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' Note the asymmetry between 0º and 90º, suggesting that the underlying spin dynamics are dominated by one of the layers —probably, the one with larger area—.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' Regarding magnetic fields applied along the hard-magnetic axis c (out- of-plane direction), a hysteretic behavior is manifested as well, but with a significantly broader maximum of MR (Supplementary Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' In this case, the MR curves are saturating for fields above 2 T, which, as for the in-plane case, coincides with the field needed to reorient the spins along the magnetic field direction (c in the present case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' Similar results are observed in different orthogonally-twisted bilayer CrSBr heterostructures, underlying the robustness of the phenomenology (Supplementary Figure 4), although the exact switching magnetic values differ between the different devices, probably due to slightly different twisting angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='- Magnetic field dependence of the magneto-resistance (MR) in orthogonally-twisted bilayer CrSBr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' a) Optical image of a vertical van der Waals heterostructure consisting on twisted CrSBr monolayers (black dashed lines) in between few-layers graphene (blue dashed lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' Different insulating h-BN layers (green dashed lines) are employed both for avoiding shortcuts and protecting the heterostructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' Red arrows indicate the easy-magnetic axis (b) of every CrSBr monolayer, being the intermediate-magnetic axis (a) perpendicular to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' The hard-magnetic axis (c) corresponds to the out-of-plane direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' Scale bar: 5 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' b) Artistic view of the heterostructure (not to scale), highlighting the twisted CrSBr monolayers (pink, yellow and cyan balls correspond to bromine, sulfur and chromium atoms, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' red arrows represent the spin lying along the easy-magnetic axis, assuming negligible inter-layer magnetic interactions) placed in between few-layers graphene or NbSe2 thin layers (blue color) on top of pre-patterned electrodes (gold color) together with a sketch of the electrical measurement configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' c, d) Field-dependence of the resistance and MR (bottom panel) as well as its increment (top panel), defined as ΔX = X+B→-B – X-B→+B, where X states either for the resistance or the MR at T = 10 K and θ = φ = 0º.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' Sweeping up (down) trace is depicted in red (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' Red/blue arrows indicate the sweeping direction of the magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' Black arrows sketch the relative configuration of both layers’ magnetization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' MR is defined as MR (%) = 100·[R(B) – R(0)]/R(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' Next, we consider both the field and temperature dependence of the MR (Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' We observe that the behavior resembles that reported for the pristine bilayer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='14 Thus, upon cooling down, a negative MR start developing below 200 K due to the onset of short-range interactions within the layers, then MR reaches a broad plateau at ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' 150 K, near Tc, and below 100 K it increases again (Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' However, some differences with the pristine bilayer are observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' First, in the pristine bilayer a minimum in MR, instead of a plateau, is observed at 150 K, followed at 100 K by a decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' Second, a hysteretic behavior is observed from temperatures below TN (second to fourth panels in Figure 2), increasing the coercive field and ΔMR upon cooling down, while no hysteresis is observed in the pristine bilayer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' Similar trends are observed for fields applied along different directions (Supplementary Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' b C d △MR (%) 5 500 △R (%) 5 500 AR 0 0 AMR 0 0 500 5 500 B+B 2 B +B 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0 0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='8 0 R (10°2) (%) MR (%) 2 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='8 4 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='6 MR 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4 4 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='6 8 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='2 6 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4 3 2 1 0 1 2 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4 B (T) B (T) Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='- Field and temperature dependence of the MR in orthogonally-twisted bilayer CrSBr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' a) Temperature dependence of the MR at saturated fields (B = 3 T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' b, c) Field and temperature dependence of the MR while sweeping from negative (positive) to positive (negative) fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' d) Field and temperature dependence of ΔMR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' MR is defined as MR (%) = 100·[R(B) – R(0)]/R(0), being R(0) the resistance obtained at zero field and ΔMR= MR+B→-B – MR-B→+B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' θ = φ = 0º.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' To further explore the irreversibility of the observed hysteresis in Figure 1, we perform a series of first-order reversal curves (FORC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' The FORC analysis lies behind the Preisach model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='27,28 We increment sequentially the maximum applied magnetic field (Bmax) in steps of 20 mT, after an initial saturation at negative fields (sequence: -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='6 T → +Bmax → -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='6 T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' Selected curves are shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='a (see the Supplementary Video for the whole data set).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' For sweeping fields below ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='1 T (first panel in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='a), the resistance increases/decreases upon increasing/decreasing B following the behavior already observed in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='d when sweeping from negative fields (limiting branch with positive slope).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' No hysteresis is observed for this loop, being the MR curve symmetric (ΔMR = 0 at ZF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' A more interesting scenario is offered when this field threshold is overcome (second panel in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' In this case, the resistance increases (red curve) upon increasing B, as before, until a sharp drop occurs at ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='1 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' Then, upon decreasing B (blue curve), the resistance decreases but with a smaller slope until a second drop is observed at ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='1 T, when it returns to the initial path (limiting branch with positive slope).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' This behavior results in the emergence of an asymmetric hysteresis (ΔMR ≠ 0 at ZF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' Similar asymmetric curves with successive drops in the resistance, giving rise to steps and plateaus at well-defined magnetic fields, are observed upon increasing the maximum sweeping magnetic field value (third panel in Figure 3, Supplementary Video and Supplementary Figure 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' Interestingly, each step observed for positive fields is characterized by a different slope while returning to ZF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' This slope decreases until a saturation field is reached (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='32 T in the present case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' For B > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='32 T, the limiting branch with negative slope is reached and the hysteresis loop becomes fully symmetric with respect to the R axis (fourth panel in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' Interestingly, when coming from positive saturated fields (sequence: +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='6 T → -Bmax → +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='6 T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='b and Supplementary Video), the same phenomenology is observed but reversing the modulus of the switching fields (mirror image with respect to the R axis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' Therefore, this magnetic system is formed by two ground states that are degenerated at ZF but that evolve with opposite MR slopes in presence of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' Thus, an initial saturation at negative fields leads to a state defined by the MR branch with positive slope (Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' This state is sketched as a set of blue circles in the Figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' Conversely, when coming from positive fields, a different state is obtained (set of red circles in Figure 3) leading to the MR branch with negative slope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' For B values within the range ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='32 T the system evolves hysteretically and selectively towards one of these two ground states and only for higher |B| values a change of ground state is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' This allows to select at will the ground state of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' Furthermore, in the hysteretic region such evolution takes place through successive steps at specific fields that may be associated to intermediate states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' This multistep phenomenology can be related with the Preisach model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' Thus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' starting from one of the two MR branches,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' each one of the resistance drops observed in the a b MR(%) c MR(%) p AMR% 25-20-15-10-50 25-20-15-10-50 4 0 4 0 300 300 300 B=3T B 3+B +B→-B 5 250 250 250 10 200- 200 200- 150 150 R 15 > 100 100 100 20 50 50 50 +0 0- 0- 150300 3 2 101 2 3 3 2 2-101 2 3 3-2-10 1 2 3 T (K) B (T) B (T) B (T)hysteresis curves is associated to the switch of an individual hysteron,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' leading to each one of the intermediate states postulated above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' In applied terms, every one of these switches could be potentially employed as a bit of information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' This is schematically sketched in Figure 3 by sweeping red/blue bytes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' Importantly, there is also magnetic memory at ZF since we can select between hysteretic and non-hysteretic MR scenarios depending on the initial ground state of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' In the Supplementary Figure 7, we consider different magnetic field sweep protocols and, for example, in the sequence ZF → + Bmax → ZF we observe hysteresis only after an initial saturation in negative magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' Therefore, the magnetic history allows us to control the appearance or not of hysteresis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' Regarding the physical origin of this multistep spin switching, we can tentatively attribute it to the stabilization of different domain configurations or spin textures (as the different skyrmions or vortices phases observed in metallic magnetic multilayered thin film heterostructures, like Pt/Co),6,29,30 thus motivating future magnetic imaging experiments in these CrSBr twisted bilayers to identify the nature of these states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' To manifest the robustness of these results, we present in Supplementary Figure 8 the study for other orthogonally-twisted CrSBr bilayers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' Overall, similar trends, although at different switching fields, are observed under different in-plane field orientations (Supplementary Figure 9) and temperatures (Supplementary Figure 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='- Multistep spin switching with magnetic memory in orthogonally-twisted bilayer CrSBr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' First-order reversal curves considering the sequence a) -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='6 T → +Bmax → -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='6 T and b) +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='6 T → -Bmax → +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='6 T at 10 K and θ = φ = 0 °.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' Bmax is incremented sequentially in steps of 20 mT and selected curves are shown (see Supplementary Video for the whole dataset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' The saturated state at negative (positive) magnetic fields is schematically sketched as a set of blue (red) circles configuration, being every spin switch related to the change of one individual hysteron (squared hysteresis operator characterized by a coercive field and a field shift from zero) within the Preisach model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' MR is defined as MR (%) = 100·[R(B) – R(0)]/R(0), being R(0) the resistance obtained at zero field in the symmetric case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' In conclusion, we have shown that twisting engineering of magnetic 2D materials is a fruitful platform for the emergence of new correlated phases in synthetic metamagnets, as exemplified here by the appearance of multistep spin switching accompanied by hysteretic MR effects in orthogonally-twisted bilayer CrSBr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' These field-induced features can be controlled by playing with the modulus and direction of the applied magnetic field, being absent in pristine CrSBr mono- and bi-layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' Overall, our results pinpoint twisted bilayer CrSBr as a versatile and rich platform for controlling and addressing the magnetic information on 2D magnets —of special a 606 :IBmaxl = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='06 T 主 :IBmaxl = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='16 T :IBmaxl = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='18 T IBmaxl = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='50 T 400 0 400 2 T+ B 8 N 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0 0 8 R 8 2 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='8 (10 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='6 a 6 8 8 8 8 8 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4 +-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4 B (T) B (T) B (T) B (T) b g09 IBmaxl = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='06 T 主 :IBmaxl = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='16 T :IBmaxl = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='18 T ± [Bmaxl = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='50 T 400 0 400 2 + B M E8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0 0 R 88 2 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='8 (10° .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='6 b 6 8 8 8 8 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4 B (T) B (T) B (T) B (T)relevance in areas such as spintronics or magnonics31—, as well as for motivating a new playground for fundamental studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' In particular, this orthogonally-twisted bilayer CrSBr, may offer a promising route for the creation and manipulation of non-colinear magnetic textures, like vortices or topologically protected skyrmions and merons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='18,19,32,33 On the other hand, the controlled stacking of 2D magnetic monolayers under defined angles opens new avenues to artificially increase the magnetic symmetry in the plane, thereby reducing the anisotropy energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' Of special interest is to reach the crossover from easy-axis to easy-plane anisotropy, since easy- plane (XY) systems34 are predicted to host dissipationless spin transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' 35,36 Methods Crystal growth: CrSBr crystals were synthesized by chemical vapor transport and characterized by powder and crystal X-Ray diffraction, energy dispersive X-Ray analysis (EDX), high-resolution TEM, SQUID magnetometry and temperature-dependent single crystal, as reported in our previous work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='14 van der Waals heterostructure fabrication: 2D layers were obtained by mechanical exfoliation from their bulk counterparts under strict inert conditions (argon glovebox) since CrSBr monolayers degrade in air.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='15,24 The obtained flakes were examined by optical microscopy (NIKON Eclipse LV-100 optical microscope under normal incidence) as a fast tool for identifying the number of layers and compared with our previously calibrated values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='14 Typical CrSBr flakes exhibit a ribbon shape, being the long (short) direction associated with the a (b) axis and being the c axis the out-of-plane direction, as previously verified by optical contrast, Raman spectroscopy and selected area electron diffraction patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='14 The van der Waals heterostructures were fabricated by assembling the different layers by the deterministic assembly of the flakes using polycarbonate, as reported by Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=',37 using a micromanipulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' Thus, the twisted-monolayers were placed between top and bottom few-layers metallic NbSe2 or few-layers graphene, where several insulating h-BN layers were inserted both for avoiding possible short-cuts and protecting the whole heterostructure from degradation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' The stack of 2D materials was placed on top of pre-lithographed electrodes (5 nm Ti/50 nm Au on 285 nm SiO2/Si from NOVA Electronic Materials, LCC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' The whole process was performed under inert atmosphere conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' A total of three orthogonally-twisted CrSBr bilayers were fabricated (device 1 -data shown in the main text- is based on metallic NbSe2 thin-layers while device 2 and 3 -data shown in the Supplementary Information- are based on few-layers graphene), observing a consistent phenomenology between all of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' Note that, in the case of using few-layers graphene, the intrinsic MR arising from the few-layers graphene is observed as well (in special, for out-of-plane applied magnetic fields), yielding to a finite positive value of the MR even at room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' Nonetheless, the magnetic fingerprints of the twisted-CrSBr are well noticeable, clearly developing below TN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' In particular, device 1 is formed by a top (bottom) CrSBr monolayer of 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='2 µm2 (53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='3 µm2), with an overlap area of 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='3 µm2 and a twisted angle of 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='5º.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' Device 2 is formed by a top (bottom) CrSBr monolayer of 190.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='1 µm2 (117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='3 µm2), with an overlap area of 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='9 µm2 and a twisted angle of 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='3º.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' Device 3 is formed by a top (bottom) CrSBr monolayer of 206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='6 µm2 (121.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='1 µm2), with an overlap area of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='9 µm2 and a twisted angle of 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0º.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' Magneto-transport measurements: Electrical measurements were performed in a Quantum Design PPMS-9 cryostat with a 4-probe geometry, where a DC current was passed by the outer leads and the DC voltage drop was measured in the inner ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' DC voltages and DC currents were measured (MFLI from Zurich Instruments) using an external resistance of 1 MΩ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=', a resistance much larger than the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' Temperature sweeps were performed at 1 K·min−1, field sweeps at 200 Oe/s, rotation sweeps at 3 °/s and the current bias was 1 µA, unless otherwise explicitly specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' Magneto-resistance (MR) is defined as: MR = 100[R(B) – R(0)]/R(0), where B is the external magnetic field and R(0) is the resistance at zero field in the symmetric case (see text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' Supporting Information Supporting Information is available from the authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' Acknowledgements The authors acknowledge the financial support from the European Union (ERC AdG Mol-2D 788222, FET OPEN SINFONIA 964396), the Spanish MCIN (2D-HETEROS PID2020- 117152RB-100, co-financed by FEDER, and Excellence Unit “María de Maeztu” CEX2019- 000919-M) and the Generalitat Valenciana (PROMETEO Program, PO FEDER Program IDIFEDER/2018/061, a Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='D fellowship to C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=', and the postdoctoral fellow APOSTD- CIAPOS2021/215 to S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='-V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' This study forms part of the Advanced Materials program and was supported by MCIN with funding from European Union NextGenerationEU (PRTR-C17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='I1) and by Generalitat Valenciana.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' We thank Á.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' López-Muñoz for his constant technical support and fundamental insights, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' Bedoya-Pinto for helpful discussions as well as C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' Supplementary Information for Multistep spin switching in orthogonally-twisted ferromagnetic monolayers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' Carla Boix-Constant, Samuel Mañas-Valero*, Eugenio Coronado.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' * Instituto de Ciencia Molecular (ICMol) - Universitat de València, Catedrático José Beltrán 2, Paterna 46980, Spain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' e-mail: samuel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='manas@uv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='es, eugenio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='coronado@uv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='es Supplementary Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='- Magnetic field dependence of the magneto-resistance (MR) in orthogonally-twisted bilayer CrSBr for different devices and field sweep rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' a) Device based on NbSe2 vertical van der Waals heterostructure (T = 10 K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' b-c) Devices based on few-layers graphene vertical van der Waals heterostructures (T = 2 K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' The field is applied in-plane (φ = 0 º) along the easy-axis of the CrSBr monolayer with smaller area, corresponding to θ = 0º (for a and b) and θ = 90º (for c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' MR is defined as MR (%) = 100·[R(B) – R(0)]/R(0), being R(0) the resistance obtained at zero field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' Supplementary Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='- In-plane magnetic field dependence of the magneto-resistance (MR) in orthogonally- twisted bilayer CrSBr (device 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' a) 2D plot of ΔMR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' b) Selected MR/resistance hysteresis loops (bottom panel) and its increment (top panel) at selected angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' Measurements corresponds to an orthogonally-twisted CrSBr bilayer based on metallic NbSe2 thin-layers vertical van der Waals heterostructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' MR is defined as MR (%) = 100·[R(B) – R(0)]/R(0), being R(0) the resistance obtained at zero field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' a b c (%) 6 Device 1 400 202 Device 2 200> 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='5- Device 3 100 AR R 0 AMR 0 R △MR B AMR 400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' 100 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='5 TTTTTTT 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='5 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='3 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='70 (%) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='8 (%) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='2 (10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='5 - 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='6 1 R MR MR 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='60 4 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='1 p 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0 - M 2 10:0e/s Oe/s 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='5 - 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0e/s 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='2 50:0e/s 50 8e/s 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='50 0e/s 3 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0 1000e/s ： 8 100:0e/s 100Qe/s 200:0e/s 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0 - 200:0e/s 200 0e/s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='6 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='6 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='6 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='6 B (T) B (T) B (T)a △MR (%) b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='8 6 4 2 0 4 6 8 AMR (%) 5 T = 10 K 400 $=0% 200 360 T = 10 K 0 $ = 0° 200F 315 5 400 270 B++B 0=00 2 0 = 45 ° 4 = 90 0 225 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0 0 180 2 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='8刀 (10) 135 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='6P 90 6 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4 45 8 0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='6 B (T) B (T) Supplementary Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='- Out-of-plane magnetic field dependence of the magneto-resistance (MR) in orthogonally-twisted bilayer CrSBr (device 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' a,b) Field-dependence of the resistance and MR (bottom panel) as well as its increment (top panel), defined as ΔX = X+B→-B – X-B→+B, where X states either for the resistance or the MR (T = 10 K, θ = 0º and φ = 90º).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' Sweeping up (down) trace is depicted in red (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' b 1 100 (%) 100 △R △R 0 0 0 100 100 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='5 - 2 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='8 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='90 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0 - R (%) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='85 R 4 (10° MR MR 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='80 a 6 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='5 一 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='75 8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='70 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='2 B++ E 3 2 1 0 1 2 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='8 B (T) B (T) Supplementary Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='- Magnetic field dependence of the magneto-resistance (MR) in orthogonally-twisted bilayer CrSBr based on few-layers graphene van der Waals heterostructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' Panels a-d (e-h) correspond to device 2 (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' a, b, e, f) Field-dependence of the resistance and MR (bottom panel) as well as its increment (top panel), defined as ΔX = X+B→-B – X-B→+B, where X states either for the resistance or the MR for in-plane (a, e panels) and out-of-plane (b, f panel) fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' Sweeping up (down) trace is depicted in red (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' Red/blue arrows indicate the sweeping direction of the magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' MR is defined as MR (%) = 100·[R(B) – R(0)]/R(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' c, g) 2D plot of ΔMR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' d, h) Selected MR and resistance hysteresis loops (bottom panel) and its increment (top panel) at selected angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' Note that the intrinsic MR arising from the few-layers graphene is observed as well (in special, for out-of-plane applied magnetic fields), yielding to a finite positive value of the MR even at room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' Nonetheless, the magnetic fingerprints of the twisted-CrSBr are well noticeable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' Device 2 b [%] 2 - 50 = 90 ° AR (Q) AMR ( 0 0 AMR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0 0 2 1 = 2 K L 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='5 - T = 35 K 50 + B=±+B 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='36 1- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4 F0 T 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='34 R (103 (%) R MR ( E 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='5 2 } 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4 1 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='8- F 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='3 LLLLL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0 B (T) B (T) C AMR (%) d 3 2 1 2 200 △R (Q) 200 270 m 8=46: 225 R(103 33 45- 4} E9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4 B (T) B (T) Device 3 200 △MR (%) △R (Q2) 0=0° E 50 T =2K 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0 - E0 卡-200 T= 35K 12- 8+ B=+B 10 E 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0} 卡11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='70 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='65 2 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0- MR (%) E11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='552 E8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='6 E 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='50 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0→ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4 L 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='6-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='5 B (T) B (T) AMR (%) h 3 1 1 2 (%) 2 J 360FT=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='K △R (Q) AMR 315 2 - 200 270 TLLL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='53 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='8 225- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='5↓ 8：11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='7 R (103 MR 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='53 0) E 06 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0 月 45于 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='5 3 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='5 0 LLLLLLLLI 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='6 B (T) B (T) Supplementary Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='- Temperature and magnetic field dependence in orthogonally-twisted bilayer CrSBr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' a-c) In-plane field orientations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' d) Out-of-plane orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' First panel: Temperature dependence of the MR in the saturated state (B = 3 T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' Second (third) panel: field and temperature dependence of the MR while sweeping from negative (positive) to positive (negative) fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' Fourth panel: field and temperature dependence of ΔMR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' Magneto- resistance (MR) is defined as MR (%) = 100·[R(B) – R(0)]/R(0), being R(0) the resistance obtained at zero field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' a 0=00 MR (% △MR (%) 25-20-15-10°-5 25-20-15-10° 5 A Φ=00 300- 300 +B→-B 300- B→+B 5# 250 250 250 10 200 200 200 150 Z150 100 100- 100 20- 50 50 50 01 01 0+ 150300 3 2 10 1 3 3 2 10 1 2 3 3 21 0 1 2 3 T (K) B (T) B (T) B (T) b 0=450 MR (%) MR (%) AMR (%) 20-15-10-5 05 Φ=0。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' 20 10 0 8 8 300- 300- 300- B=3T B→+B +B→B 5 250 250 250 10 200 200 200 150 R 15 100 100 100 20- 50 50 50 01 01 0 150300 3 2-1 0 1 2 3 3 2 10 1 2 3 3 2 1 0 1 2 3 T (K) B (T) B (T) B (T) c 0=90° MR (%) MR (%) △MR (%) Φ=0° 20-15-10-5 0 20-15-10 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' 4 2 300 300 300- B→+B +B→-B 5 250 250 250 10 200 200 200 150 100手 100 100 20 50 50 50 0- 0 +0 0 150300 3-2-1012 3 3-2-1012 3 3 -2 -1 0 1 2 3 T (K) B (T) B (T) B (T) d 0=0° MR (%) MR (%) △MR (%) 20-15 10 5 20-15-10° Φ=900 0 2 1 0 2 B=3T 300- 300 300- B→+B +B→-B 5 250 250 250 0-101 200 200- 200 150 150 100 100 100 20- 50 50 50 01 01 150300 3 -2 -1 0 1 2 3 3 -2 -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' 0 2 3 3-2-10 1 2 3 T (K) B (T) B (T) B (T) Supplementary Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='- Hysteresis opening in orthogonally-twisted bilayer CrSBr (device 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' a-e) Field- dependence of the resistance and MR (bottom panel) as well as its increment (top panel), defined as ΔX = X+B→-B – X- B→+B, where X states either for the resistance or the MR after sweeping up to different selected magnetic fields at 10 K and θ = φ = 0°, being the magnetic field applied in plane along the easy (intermediate) magnetic axis of the top (bottom) CrSBr monolayer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' f) ΔR 2D plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' The magnetic sweep protocol is as follows: after a first saturation at negative fields, we perform the sequence ZF → Bmax →-Bmax→ ZF, increasing in every cycle the maximum field in 20 mT step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' Sweeping up (down) trace is depicted in red (blue) in a-d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' Red/blue arrows indicate the sweeping direction of the magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' MR is defined as MR (%) = 100·[R(B) – R(0)]/R(0), being R(0) the resistance obtained at zero field in the symmetric case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' a b C % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='1 (%) Bmaxl = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='12 T 100 AR 200 4 AR AR 0 A 0 AMR 0 0 AMR 0 0 (2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='1 4 1 2 100 200 2 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='1 2 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0元 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0元 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0刀 (%) (10 (10 (10° MR 0 MR 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='9 MR 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='9 b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='1 2 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='8 21 2 + B F7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='8 B B 3 B 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='1 B (T) B (T) B (T) e f AMR (%) 400 5 400 △R (2) 4 :|Bmaxl = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='30 AR R 400 0 400 0 0 AMR 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='6 4 b 400 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='5 2 2- 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0 E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4 0 0 R (%) R 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='3 2 MR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='2 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='6 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='6 Bmax 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='087 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='12 2+ B 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='30T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='40 T 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4 B (T) B (T) B (T) Supplementary Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='- Multistep spin switching with magnetic memory in orthogonally-twisted CrSBr under different magnetic field sweep protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' T = 10 K and θ = φ = 0 ° (device 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' a, b) Sequence ZF → +Bmax → ZF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' c, d) Sequence ZF → -Bmax → ZF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' e, f) Sequence ZF → +Bmax → ZF → -Bmax → ZF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' g, h) Sequence ZF → -Bmax → ZF → +Bmax → ZF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' i) Sequence +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='6 T → -Bmax → + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='6 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' j) Sequence -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='6 T → +Bmax → - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='6 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' Panels a, c, e, g and i (b, d, f, h and j) correspond to an initial saturation at positive (negative) magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' In every field sweep, Bmax is incremented in steps of 20 mT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' a Sequence: ZF → Bmax → ZF b Sequence: ZF → Bmax → ZF Initial saturation at positive fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' Initial saturation at negative fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0 7800- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='1 8000 } 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='1 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='3菱 7800 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='2 F0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4 3 R 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='5 R 7600- 7400- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='5 7400- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='6 7200 7200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='6 B (T) B (T) c Sequence:ZF → ZF d Sequence: ZF → ZF Initial saturation at positive fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' Initial saturation at negative fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' 8000 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='1 7800- 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='6 7400 7400- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='6-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='6 B (T) B (T) e f Sequence: ZF → + Bmax Initial saturation at positive fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' Initial saturation at negative fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' 8000- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0 8000 } 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='2 7800 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='3菱 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4 3 Eto R 7600 R 7600 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='5 7400- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='6 7400 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='6 B (T) B(T) g Sequence: ZF → ZF h Sequence: ZF →ZF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' → ZF Initial saturation at positive fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' Initial saturation at negative fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' 8200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0 8000- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0 8000- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='2 7800 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='3菱 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='3菱 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4 3 R 7600 - R 7600 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='5 7400- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='6 7400- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='6 7200 7200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='6 B (T) B(T) Sequence: + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='6 T → - B, + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='6 T Sequence: - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='6 T → + Br 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='6T Initial saturation at positive fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' Initial saturation at negative fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' 8000 } 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0 8000 } 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='1 7800- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='2 7800- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='3菱 (0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='3 显 R 7600 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4 3 R 7600 - 一 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='5 7400 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='6 7400 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='6 7200 7200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='6 B (T) B (T) Supplementary Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='- Hysteresis opening in orthogonally-twisted bilayer CrSBr based on few-layers graphene vertical van der Waals heterostructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' a-e) Field-dependence of the resistance and MR (bottom panel) as well as its increment (top panel), defined as ΔX = X+B→-B – X-B→+B, where X states either for the resistance or the MR after sweeping up to different selected magnetic fields at 2 K and θ = φ = 0 ° (device 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' f-g) ΔR 2D plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' The magnetic sweep protocol is as follows: for panels a-e, after a first saturation at negative fields, we perform the sequence ZF → Bmax →-Bmax→ ZF, increasing in every cycle the maximum field in 20 mT step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' In panel f-g, the sequence is +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='6 T → -Bmax →+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='6 and increasing in every cycle the maximum negative field in 20 mT step, for device 2 (g) and device 3 (f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' Sweeping up (down) trace is depicted in red (blue) in a-d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' Arrows indicate the sweeping direction of the magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' Magneto-resistance (MR) is defined as MR (%) = 100·[R(B) – R(0)]/R(0), being R(0) the resistance obtained at zero field in the symmetric case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' a Sweep up to |Bm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='10 T b Sweep up to IBmaxl = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='16 T e 10 (%) △R 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='5- Sweepupto: 0 AMR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='10 200 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='161 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='30 100 Z 1 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='60 R 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0 0 F9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='7 Device 2 Device 2 Device 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='7 R R 200 (%) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='5- (103 (103 MR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0- MR 0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='6 a 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='70 →+BL9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='5 B B 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='5 # 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='1 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='65 B (T) B (T) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0- 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='60 c Sweep up to IB, maxl=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='30T d Sweep up to IBm maxl = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0 N0= =30g 8000 8000- 8200 - 1大 8000- R 7600 7600 7400- 7400 7400- 7400- 7200 LL TELLEE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4_0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4 _0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0 e= 60°1 8200 = 90° e=105° 8200 - 8000 - 8000 8000 - 8000 - 7800 a 7800 7800 7600 7600 R 7600 7600- 7400- 7400- 7400- 7400 LLLL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0 8200e = 120% e=135R1 =150° 8200 - = 165 8200 - 8200 - 8000 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='8000 - 8000 - 7800 R R 7600 7600 7600 7400- 7400- 7400- 7400 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4 0.' metadata={'source': 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+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0 8200 - e = 180° 8200 = 210° 1e=195°1 8200 - 16 = 225 8000 0008 8000 8000 7800 g7800 7800 7800 R 7600 R 7600 7600 7600 7400 - 7400 7400 7400- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4 0.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='2 e= 240° 8200 - [e = 255° 8200=270 Ve = 285° 8200 - E 0008 8000 - 8000 - 8000 F 0082 R R 7600 P 7600 7600 一 一 7400 7400- 7400- 7400 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4 0.' metadata={'source': 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+page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0 8200 →= 300° : e=315 @ = 330° 8200 - 1=345 8200 - 8200 - 8000 8000 - 8000 - E 0082 R 7600 7600 7600 R 0092 7400 7400 - 7400- 7400- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='- First-order reversal curves for in-plane (θ = φ = 0 °) magnetic fields at different temperatures (device 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' We consider the sequence +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='6 T → -Bmax → + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='6 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content=' In every field sweep, Bmax is incremented in steps of 20 mT.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} +page_content='6 B (T)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfiQ-H/content/2301.05647v1.pdf'} diff --git a/ldE5T4oBgHgl3EQfHQ4M/content/tmp_files/2301.05437v1.pdf.txt b/ldE5T4oBgHgl3EQfHQ4M/content/tmp_files/2301.05437v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..70d07e15ab5fc877f8b1aa768f024fdca79d4cfd --- /dev/null +++ b/ldE5T4oBgHgl3EQfHQ4M/content/tmp_files/2301.05437v1.pdf.txt @@ -0,0 +1,1767 @@ +Multiqubit entanglement due to quantum gravity +Shaomin Liu,1 Lin Chen,2, 3, ∗ and Mengfan Liang2, † +1School of Mathematics, Physics and Finance, Anhui Polytechnic University, Wuhu 241000, China +2LMIB(Beihang University), Ministry of Education, +and School of Mathematical Sciences, Beihang University, Beijing 100191, China +3International Research Institute for Multidisciplinary Science, Beihang University, Beijing 100191, China +(Dated: January 16, 2023) +Quantum gravity between masses can produce entangled states in thought experiments. +We +extend the experiments to tripartite case and construct states equivalent to Greenberger- Horne- +Zeilinger states and W states under stochastic local operations and classical communication. The +entanglement relates to the evolution phases induced by gravitational interaction. When we involve +more masses in the experiments, multipartite entangled states can be constructed in a similar way. +We measure the degree of multipartite entanglement by calculating the geometric measure. We +describe the relationship between geometric measure and the evolution phases. It helps in searching +out the states with robust entanglement. +PACS numbers: 03.65.Ud, 04.60.Ds, 03.67.Mn +I. +INTRODUCTION +Due to the extreme weakness of gravity, its quantum effects are hard to detect. Recently, authors +reported observing gravitational Aharonov-Bohm effect [1, 2]. This experiment measured the gravi- +tational phase shift induced by a kilogram-scale source mass close to the wave packets. It convinced +us about the quantum feature of gravity. Bose et al. [3] and Marletto et al. [4] suggested two similar +thought experiments to probe quantized gravity. Two neutral masses are separable initially, split +into superposition of spatially localized states in an inhomogeneous magnetic field. The mutual grav- +itational interaction between components of superposition will evolute relative phases and transform +initial separable state into bipartite entangled state. These quantum phases correlate with their +interaction time and can be detected by entanglement witnesses. Even though the dominant contri- +bution of interaction is Newtonian at the low energy limits, the entanglement between two masses +can verify quantum signatures of gravity. Because local operations and classical communication +(LOCC) does not create entanglement, the entanglement can be generated only by non-classical me- +diator. There are also some discussions on Post-Newtonian order corrections in [5, 6]. The feasibility +of the hypothetical experiments have been discussed in the original articles [3, 4]. +There are some factors which may pollute the entanglement in Bose’s experiment, such as Casimir- +Polder forces, van der Waals forces or other electro-magnetic interactions. By adjusting the param- +eters or setups of the experiments, some pollution can be reduced. +A thought experiment was +presented to study interaction mediated only by gravity between two hypothetical neutrino-like par- +ticles [7]. +A modified model of original experiment, the symmetric setup, enhanced the gravity +interaction to against the noisy dynamics, such as stochastic fluctuations of the parameters or de- +coherence induced by environmental interaction [8]. However, these studies all focused on bipartite +entanglement, as far as we know, multipartite entanglement is little understood. +In this paper, we extend the quantum gravity inducing entanglement to multiqubit case. Multi- +qubit entanglement plays an important role in quantum information, computation and communica- +∗linchen@buaa.edu.cn (corresponding author) +†lmf2021@buaa.edu.cn (corresponding author) +arXiv:2301.05437v1 [quant-ph] 13 Jan 2023 + +2 +tion. There are various platforms generating multipartite entanglement with photons or ions [9–13]. +Photonic experiments entangled 14 photons to realize Greenberger-Horne-Zeilinger (GHZ) states by +interleave single-photon emissions with atomic rotations [9]. In a linear Paul trip, GHZ states were +produced with up to 24 ions, mediated by the Mølmer-Sørensen gate [13]. These attempts went a +step further in quantum computation. In the future, the major problems still are how to increase the +efficiency of generating entanglement and protect systems against decoherence. This work suggests +a theoretical path to produce entangled states induced by mutual gravitational interaction of neutral +masses. +In multipartite system, the Hamiltonian leads to relative evolution phases with special spatial +symmetry. The interaction is similar to the bipartite case in [8]. In this protocol, the quantum +gravity can generate GHZ states, but not W states. We can classify equivalent entangled states +under stochastic LOCC (SLOCC). They contain the same kind of entanglement and are suited to +implement the same tasks of quantum information theory [14]. We denote GHZ-type states as those +states equivalent to GHZ states under SLOCC, and similarly for W-type states. In Theorem 1, we +show that the gravitation can generate N-qubit GHZ-type states. However, considering the weakness +of gravity, transform N masses into entangled states is difficult. We suggest a way to generate N- +qubit GHZ-type states by getting (N − 2)-qubit GHZ-type states and Bell states entangled. The +(N − 2)-qubit GHZ-type states can be produced in the gravitational entanglement apparatus too. +It provides an approach to extend existing multipartite entanglement platforms by involving more +qubits and makes the experiment more feasible. +To address the degree of the gravity induced +entanglement from a geometric viewpoint [15, 16], we calculate the geometric measure (GM) of +entanglement for the tripartite case. GM quantifies the entanglement by measuring the distance +between the entangled state and the nearest product state. So we can seek out the robust entangled +final states induced by gravity with the results in Theorem 2. It is helpful in multipartite platforms +design. +The rest of this paper is constructed as follows. In Sec. II, we introduce Bose’s experiment in +detail. In Sec. III, more masses are led in to construct multiqubit entanglement and GHZ-type +states in the symmetric setup. In Sec. IV, we calculate the GM of entanglement for the three-qubit +case. Sec. V makes conclusion and outlook. +II. +PRELIMINARIES +Bose et al. proposed a thought experiment in [3]. Two neutral masses m1 and m2, in an inhomo- +geneous magnetic field, both split into a superposition of two spatially separated states |L⟩ and |R⟩ +for a time τ. As we can see in Fig. 1(i), l is the distance between components of superposition, and +d is the distance between the centres of two masses. +Initially, two masses A1 and A2 are separated, each one splits into two superpositions. +|ψ(0)⟩A1A2 = 1 +√ +2 +� +|L⟩A1 + |R⟩A1 +� 1 +√ +2 +� +|L⟩A2 + |R⟩A2 +� +. +(1) +The Schr¨odinger equation reveals the time evolution of the state, +iℏ∂tψ(r, t) = +� +− ℏ2 +2m∇2 + V (r) +� +ψ(r, t). +(2) +The Hamiltonian H = − ℏ2 +2m∇2 + V = H0 + V can be separated into two parts, H0 relates to the free +particle’s behaviour and V is the mutual gravity potential. + +3 +FIG. 1: Two masses at distance d from each other, both split into superpositions of spatially localized states at +distance l. (i)The original setup proposed in [3], superpositions are parallel to the initial separations. (ii) The +symmetric setup proposed in [8], superpositions are orthogonal to the initial separations. (iii) The symmetric setup +for three-qubit. +During the interaction time, these components keep stable distances from each other, so they have +time independent gravity interaction energies. By calculating the propagator and the scattering +process, the mutual gravitation potential can be described as [17–19], +V (r) = −Gm1m2 +r +(1 + 3G(m1 + m2) +rc2 ++ +41Gℏ +10πr2c3). +(3) +The general relativistic correction and quantum correction to Newtonian potential are extremely +small. Even if we take the Newtonian approximation of potential, the entanglement can reflect the +quantum nature of gravity too. Actually in this setting, gravity acts as a mediator in quantum +mechanics, the entanglement will be detected only through non-classic dynamics. So in this paper, +we just take the Newtonian approximation of potential for simplicity. +Since the potential is independent of time, the Schr¨odinger equation for the state ψ(r, t) can be +solved by separating variables, ψ(r, t) = e +−iEt +ℏ ψ(r), the evolution phase is related to the potential +eiφ ≡ e +−iV t +ℏ . +Because the states |L⟩ and |R⟩ can be separated by different distances (including d, d − l, d + l), +the mutual gravity interaction can induce different rates of phase evolution in Stern-Gerlach(SG) +apparatus, for simplicity, written as, +|ψ(τ)⟩A1A2 = 1 +2 +� +|L(τ)⟩A1 +� +eiφ1|L(τ)⟩A2 +eiφ2|R(τ)⟩A2 +� ++|R(τ)⟩A1 +� +eiφ3|L(τ)⟩A2 +eiφ1|R(τ)⟩A2 +�� +. (4) +If we take Newtonian approximation of the potential in Eq. (3), the evolution phases are [3] +φ1 ∼ Gm1m2τ +ℏd +, +φ2 ∼ Gm1m2τ +ℏ(d + l) , +φ3 ∼ Gm1m2τ +ℏ(d − l) . +(5) +We denote the relative phases ∆φ2 = φ2 − φ1 and ∆φ3 = φ3 − φ1. The entanglement depends +on the relative phases and is irrelative to common phase φ1. There is an exceptional case, when +∆φ2 + ∆φ3 = 2nπ, n ∈ Z (Z is the integer set), |ψ(τ)⟩A1A2 is separable. +The parameters are chosen as m1, m2 ∼ 10−14kg, d ∼ 450µm, l ∼ 250µm, τ ∼ 2.5s, the en- +tanglement witness is W = X ⊗ Z + Y ⊗ Y [3]. The expectation value ⟨W⟩ > 1 is the signal of + +d +R +A2 +A2 +H3 +KI +L) + +A2 +[11 +(i) +(ii) +(ili)4 +entanglement. This witness has a suboptimal detection areas and may be ineffective for small entan- +glement which corresponds to short interaction time τ. Nevertheless, a long interaction time is also +infeasible, because when we consider the decoherence effect from earth’s gravity, it is difficult to keep +superposition states free falling in such a long time. So a much broader class of witnesses [20, 21] was +suggested to detect the greatest volume of entangled states and make the setup of experiment more +feasible. Authors constructed optimal fidelity witnesses by maximally entangled states to shorten the +required interaction time, such as W = I ⊗I −X ⊗X −Z ⊗Y −Y ⊗Z. These witnesses are sensitive +to very small entanglement, so they are valid at the beginning of free-fall. They also broaden the +detection area in the space of phases. With these instruments, the positive results will announce the +quantum gravity. An entanglement witness was suggested to detect spinless entanglement between +microspheres with massive spatial qubits too [22]. However, there are also some doubts about the +effectiveness of these witnesses of quantized gravity. General configuration-ensemble models and +mean-field semiclassical gravity models were supposed to explain the entanglement with classical +gravity [23, 24]. +A modified symmetric setup was suggested in [8], each mass splits into two superposition states, +which are in the direction orthogonal to initial separation, see Fig. 1(ii). Compared with the parallel +split mode in Fig. 1(i), the symmetric setup permits a reduced distance between masses, that is +useful in keeping the distance constant and enhancing gravity interaction. In this case, there is only +one relative phase ∆φ = Gm1m2τ +ℏ +( 1 +d − +1 +√ +l2+d2), the final state in Eq. (4) becomes +|ψ(τ)⟩A1A2 = eiφ1 +2 +� +|L(τ)⟩A1 +� +|L(τ)⟩A2 + ei∆φ|R(τ)⟩A2 +� ++ |R(τ)⟩A1 +� +ei∆φ|L(τ)⟩A2 + |R(τ)⟩A2 +�� +. +(6) +The distances of qubits in this setup display some symmetry which constraints the number of +independent phases. The next sections will consider entanglement in this setup. +III. +MULTIQUBIT ENTANGLEMENT +In this section, we extend gravity interaction induced entanglement to multipartite case. GHZ +states and W states are the maximum entangled states under geometric entanglement metric. They +play important roles in multipartite entanglement. Besides, since the equivalence class of GHZ or W +states under SLOCC contain the same kind of entanglement, we also focus on GHZ-type states and +W-type states. Two states are equivalent under SLOCC when they are related by a local invertible +operator [14]. For example, if |ψ⟩ = (M1 ⊗ ... ⊗ Mn)|GHZ⟩ , where each Mi is an invertible matrix, +then |ψ⟩ is in the equivalence class of GHZ states. We will construct GHZ-type states under the +gravitational symmetric setup. Nevertheless, if we want to construct W-type states, more freedom +in parameters of the setup should be involved in. We will respectively construct three-qubit and +four-qubit GHZ-type states in Sec. III A and III B, and then extend to N-qubit case in Sec. III C +and III D. +A. +The three-qubit case +First we introduce one more mass in Eq. (1), as described in Fig. 1(iii). +|ψ(0)⟩A1A2A3 = 1 +√ +2 +� +|L⟩A1 + |R⟩A1 +� 1 +√ +2 +� +|L⟩A2 + |R⟩A2 +� 1 +√ +2 +� +|L⟩A3 + |R⟩A3 +� +. +(7) +The time evolution of this state is similar to the bipartite case in Eq. (2). The gravitation potential +in Hamiltonian is related to the distances between three components of superpositions. We consider + +5 +the symmetric setup, and there are three different evolution phases in the final state. The gravitation +potential among the left components |L⟩A1, |L⟩A2 and |L⟩A3 in Fig. 1(iii) is V = −Gm1m2( 1 +d+ 1 +d+ 1 +2d), +so does the potential among the right components. The potential among |L⟩A1, |R⟩A2 and |L⟩A3 +is V = −Gm1m2( 1 +2d + +1 +√ +d2+l2 + +1 +√ +d2+l2), so does the potential among |R⟩A1, |L⟩A2 and |R⟩A3. The +potential among |L⟩A1, |L⟩A2 and |R⟩A3 is V = −Gm1m2( 1 +d + +1 +√ +4d2+l2 + +1 +√ +d2+l2), so do rest parts in +the state. The final state can be written as +|ψ(τ)⟩A1A2A3 = +1 +2 +√ +2 +� +eiϕ1� +|L(τ)⟩A1|L(τ)⟩A2|L(τ)⟩A3 + |R(τ)⟩A1|R(τ)⟩A2|R(τ)⟩A3 +� ++ eiϕ2� +|L(τ)⟩A1|L(τ)⟩A2|R(τ)⟩A3 + |L(τ)⟩A1|R(τ)⟩A2|R(τ)⟩A3 ++ |R(τ)⟩A1|L(τ)⟩A2|L(τ)⟩A3 + |R(τ)⟩A1|R(τ)⟩A2|L(τ)⟩A3 +� ++ eiϕ3� +|L(τ)⟩A1|R(τ)⟩A2|L(τ)⟩A3 + |R(τ)⟩A1|L(τ)⟩A2|R(τ)⟩A3 +�� +. +(8) +The evolution phases are +ϕ1 ∼ 5Gm1m2τ +2ℏd +, +ϕ2 ∼ Gm1m2τ +ℏ +(1 +d + +1 +√ +4d2 + l2 + +1 +√ +d2 + l2), +ϕ3 ∼ Gm1m2τ +ℏ +( 1 +2d + +2 +√ +d2 + l2). +(9) +For simplicity, we denote |L(τ)⟩ as |0⟩, |R(τ)⟩ as |1⟩, ∆ϕ2 = ϕ2 − ϕ1, ∆ϕ3 = ϕ3 − ϕ1. Eq. (8) can +be written as +|ψ⟩ = eiϕ1 +2 +√ +2 +� +|000⟩ + |111⟩ + ei∆ϕ2� +|001⟩ + |011⟩ + |100⟩ + |110⟩ +� ++ ei∆ϕ3� +|010⟩ + |101⟩ +�� +. +(10) +The entanglement of Eq. (10) comes from the relative evolution phases ∆ϕ2 and ∆ϕ3. In some +certain situation, the state in Eq. (10) may become the separable state, GHZ state or GHZ-type +state. In the following, we will discuss the three cases (i), (ii) and (iii). +(i) The state in Eq. (10) is a separable state. We write it as +|ψ⟩ = eiϕ1 +2 +√ +2 +� +|0⟩A1 +� +|00⟩A2A3 + ei∆ϕ2� +|01⟩A2A3 + |11⟩A2A3 +� ++ ei∆ϕ3|10⟩A2A3 +� ++ |1⟩A1 +� +|11⟩A2A3 + ei∆ϕ2� +|00⟩A2A3 + |10⟩A2A3 +� ++ ei∆ϕ3|01⟩A2A3 +�� +. +(11) +So in the range space of the reduced density operator of the second and third qubits, there are two +vectors, +|a⟩ = 1 +2 +� +|00⟩A2A3 + ei∆ϕ2� +|01⟩A2A3 + |11⟩A2A3 +� ++ ei∆ϕ3|10⟩A2A3 +� +, +|b⟩ = 1 +2 +� +|11⟩A2A3 + ei∆ϕ2� +|00⟩A2A3 + |10⟩A2A3 +� ++ ei∆ϕ3|01⟩A2A3 +� +. +(12) +If the vectors |a⟩ and |b⟩ are linearly dependent, the final state |ψ⟩ is a separable state. For example, +by choosing ∆ϕ3 = 2nπ, n ∈ Z, the state in Eq. (10) is separable, +|ψ1⟩ = eiϕ1 +2 +√ +2[|0⟩A1 ⊗ (|0⟩ + |1⟩)A2 ⊗ (|0⟩ + ei∆ϕ2|1⟩)A3 + |1⟩A1 ⊗ (|0⟩ + |1⟩)A2 ⊗ (ei∆ϕ2|0⟩ + |1⟩)A3]. +(13) + +6 +The state in Eq. (13) will be fully separable states for ∆ϕ2 = nπ. +|ψ2⟩ = eiϕ1(|0⟩ ± |1⟩ +√ +2 +)A1 ⊗ (|0⟩ ± |1⟩ +√ +2 +)A2 ⊗ (|0⟩ ± |1⟩ +√ +2 +)A3. +(14) +(ii) The state in Eq. (10) is a GHZ state. +When ei∆ϕ3 ̸= 1, the state in Eq. (10) becomes a genuinely entangled state. The reduced density +operator of the first, second and third qubit respectively is +ρA1 = ρA3 = 1 +8 +� +4(|0⟩⟨0| + |1⟩⟨1|) + (ei∆ϕ2 + e−i∆ϕ2 + ei(∆ϕ2−∆ϕ3) + ei(∆ϕ3−∆ϕ2))(|0⟩⟨1| + |1⟩⟨0|) +� +, +ρA2 = 1 +8 +� +4(|0⟩⟨0| + |1⟩⟨1|) + (ei∆ϕ3 + e−i∆ϕ3 + 2)(|0⟩⟨1| + |1⟩⟨0|) +� +. +(15) +When ei∆ϕ2 + e−i∆ϕ2 + ei(∆ϕ2−∆ϕ3) + ei(∆ϕ3−∆ϕ2) = 0 and ei∆ϕ3 + e−i∆ϕ3 + 2 = 0, the state in Eq. +(10) is a GHZ state. That means ∆ϕ3 = (2n+1)π. For example, by choosing ei∆ϕ2 = i, ei∆ϕ3 = −1. +Eq. (10) becomes +|ψ3⟩ = eiϕ1 +2 +√ +2[(|0⟩ + i|1⟩)A1 ⊗ |0⟩A2 ⊗ (|0⟩ + i|1⟩)A3 + (i|0⟩ + |1⟩)A1 ⊗ |1⟩A2 ⊗ (i|0⟩ + |1⟩)A3]. +(16) +The state in Eq. (16) is a GHZ state, because |ψ⟩ = ( σz+σy +2 +)A1 ⊗ IA2 ⊗ ( σz+σy +2 +)A3(|000⟩ + |111⟩). +When we choose ei∆ϕ2 = 1 and ei∆ϕ3 = −1, the state in Eq. (10) is LU equivalent to a GHZ state, +|ψ4⟩ = eiϕ1 +2 +√ +2[(|00⟩ + |11⟩)A1A2 ⊗ (|0⟩ + |1⟩)A3 + (|10⟩ − |01⟩)A1A2 ⊗ (|0⟩ − |1⟩)A3]. +(17) +(iii) The state in Eq. (10) is a GHZ-type state. +The state in Eq. (10) becomes a GHZ-type state when there exist two linearly independent product +vectors in the range space of the reduced density operator. That means if the vectors |a⟩ and |b⟩ +construct a product state |ψp⟩ = |a⟩ + x|b⟩, there exist two roots of x which satisfy the equation +(1 + ei∆ϕ2x)(ei∆ϕ2 + x) = (ei∆ϕ2 + ei∆ϕ3x)(ei∆ϕ3 + ei∆ϕ2x). +(18) +For example, when we choose ei∆ϕ2 = 1 and ei∆ϕ3 = i, the state in Eq. (10) becomes GHZ-type +states, +|ψ5⟩ = eiϕ1 +2 +√ +2[(|00⟩ + |11⟩)A1A2 ⊗ (|0⟩ + |1⟩)A3 + |10⟩A1A2 ⊗ (|0⟩ + i|1⟩)A3 + |01⟩A1A2 ⊗ (i|0⟩ + |1⟩)A3]. +(19) +We can see that, there are greater chances for Eq. (10) be a GHZ-type state than be a GHZ state. +So we will consider GHZ-type states in next sections only. +On the other hand if there exists a multiple root for Eq. (18), the state in Eq. (10) becomes a +W-type state. For W-type states, the phases should satisfy (1 + ei∆ϕ3)2 = 4e2i∆ϕ2. That means +ei∆ϕ3 = 1, though in this case |ψ1⟩ is separable, actually there is no solution for the equation. So +under this setup, we cannot produce W-type states. + +7 +B. +The four-qubit case +In this part, we extend the modes of Fig. 1(iii) to the four-qubit case. The final state can be +written as +|ψ⟩ =1 +4 +� +eiϕ′ +1� +|0000⟩ + |1111⟩ +� ++ eiϕ′ +2� +|0001⟩ + |0111⟩ + |1000⟩ + |1110⟩ +� ++ eiϕ′ +3� +|0010⟩ + |1101⟩ + |0100⟩ + |1011⟩ +� ++ eiϕ′ +4� +|1100⟩ + |0011⟩ +� ++ eiϕ′ +5� +|1010⟩ + |0101⟩ +� ++ eiϕ′ +6� +|1001⟩ + |0110⟩ +�� +. +(20) +The evolution phases are similar to Eq. (9), +ϕ′ +1 ∼13Gm1m2τ +3ℏd +, +ϕ′ +2 ∼Gm1m2τ +ℏ +( 5 +2d + +1 +√ +9d2 + l2 + +1 +√ +4d2 + l2 + +1 +√ +d2 + l2), +ϕ′ +3 ∼Gm1m2τ +ℏ +(11 +6d + +1 +√ +4d2 + l2 + +2 +√ +d2 + l2), +ϕ′ +4 ∼Gm1m2τ +ℏ +(2 +d + +1 +√ +9d2 + l2 + +2 +√ +4d2 + l2 + +1 +√ +d2 + l2), +ϕ′ +5 ∼Gm1m2τ +ℏ +(1 +d + +1 +√ +9d2 + l2 + +3 +√ +d2 + l2), +ϕ′ +6 ∼Gm1m2τ +ℏ +( 4 +3d + +2 +√ +4d2 + l2 + +2 +√ +d2 + l2). +(21) +When we extract eiϕ′ +1, the relative phases are ∆ϕ′ +i = ϕ′ +i − ϕ′ +1. If the relative phases are chosen as +ei∆ϕ′ +2 = ei∆ϕ′ +3 and ei∆ϕ′ +4 = ei∆ϕ′ +5 = ei∆ϕ′ +6 = 1, the state will be GHZ-type. For example, if we choose +ei∆ϕ′ +2 = ei∆ϕ′ +3 = i, then Eq. (20) becomes +|ψ⟩ =1 +4eiϕ′ +1 +� +(|00⟩ + |11⟩)A1A2 +� +|0⟩A3(|0⟩ + i|1⟩)A4 + |1⟩A3(i|0⟩ + |1⟩)A4 +� ++ (|10⟩ + |01⟩)A1A2 +� +|0⟩A3(i|0⟩ + |1⟩)A4 + |1⟩A3(|0⟩ + i|1⟩)A4 +�� +. +(22) +Looking through the three-qubit and the four-qubit cases, we can see that the relative phases are +related to the distances of qubits. As in Fig. 1(iii), the distances are constrained by some spatial +symmetry. We will present the details in App. A. +When we extend the results to N-qubit case, we have Theorem 1. +Theorem 1 When masses are split into superpositions in symmetric setup as Fig. 1(iii) described, +the gravity interaction between superpositions can produce N-qubit GHZ-type entangled states. +Since there is some difference between the (2N + 1)-qubit case and the 2N-qubit case, we will +show Theorem 1 from two aspects in Sec. III C and III D . First, let us consider the (2N + 1)-qubit +case. +C. +The (2N + 1)-qubit case +We define a one-qubit basis, +|m+⟩ = |0⟩ + |1⟩ +√ +2 +, +|m−⟩ = |0⟩ − |1⟩ +√ +2 +. +(23) + +8 +They are linearly independent, then we can construct a three-qubit state like this, +|ψ3⟩ = |00⟩ + |11⟩ +2 +|m+⟩ + |10⟩ − |01⟩ +2 +|m−⟩ += +1 +2 +√ +2 +� +|0⟩ +� +|00⟩ + |11⟩ + |01⟩ − |10⟩ +� ++ |1⟩ +� +|00⟩ + |11⟩ + |10⟩ − |01⟩ +�� +. +(24) +Obviously |c⟩ = +1 +√ +2[|00⟩ + |11⟩ + |01⟩ − |10⟩] and |d⟩ = +1 +√ +2[|00⟩ + |11⟩ + |10⟩ − |01⟩] are linearly +independent, and they can construct two separable states +|c⟩ + i|d⟩ = 1 + i +√ +2 (|0⟩ + i|1⟩)(|0⟩ − i|1⟩), +|c⟩ − i|d⟩ = 1 − i +√ +2 (|0⟩ − i|1⟩)(|0⟩ + i|1⟩). +(25) +They span bipartite Hilbert space, so the state in Eq. (24) is a three-qubit GHZ-type state, just be +the same as Eq. (17). +Next, we introduce a back-up basis +|ψ′3⟩ = |00⟩ + |11⟩ +2 +|m−⟩ − |10⟩ − |01⟩ +2 +|m+⟩. +(26) +Since |ψ′3⟩ = (σx)A3(−σz)A3|ψ3⟩, it is also a GHZ-type state, and is linearly independent with |ψ3⟩. +The states in |ψ3⟩ and |ψ′3⟩ construct a three-qubit basis. We should notice that the state in |ψ′3⟩ is +not that kind final state produced in our apparatus, because it doesn’t satisfy the spatial symmetry +described in Theorem 3. +Now we can construct a five-qubit GHZ-type state by three-qubit basis, +|ψ5⟩ = |00⟩ + |11⟩ +2 +|ψ3⟩ + |10⟩ − |01⟩ +2 +|ψ′3⟩. +(27) +It can be written as +|ψ5⟩ =1 +4 +��� +|00⟩ + |11⟩ +� ++ i +� +|01⟩ − |10⟩ +��� +|ψ3⟩ + i|ψ′3⟩ +� ++ +�� +|00⟩ + |11⟩ +� +− i +� +|01⟩ − |10⟩ +��� +|ψ3⟩ − i|ψ′3⟩ +�� +. +(28) +So it is a GHZ-type state. The five-qubit back-up basis is +|ψ′5⟩ = |00⟩ + |11⟩ +2 +|ψ′3⟩ − |10⟩ − |01⟩ +2 +|ψ3⟩. +(29) +It has the same feature as |ψ′3⟩. Then we can construct a seven-qubit GHZ-type state in the same +way. So generally, a (2N + 1)-qubit GHZ-type state can be presented as +|ψ2N+1⟩ = |00⟩ + |11⟩ +2 +|ψ2N−1⟩ + |10⟩ − |01⟩ +2 +|ψ′2N−1⟩ += 1 +4 +��� +|00⟩ + |11⟩ +� ++ i +� +|01⟩ − |10⟩ +��� +|ψ2N−1⟩ + i|ψ′2N−1⟩ +� ++ +�� +|00⟩ + |11⟩ +� +− i +� +|01⟩ − |10⟩ +��� +|ψ2N−1⟩ − i|ψ′2N−1⟩ +�� +. +(30) +As the criterion proposed in Theorem 3 shown, the (2N + 1)-qubit GHZ-type state |ψ2N+1⟩ can +be produced in the symmetric setup, but the back-up basis |ψ′2N+1⟩ can not be generated in the +apparatus. + +9 +D. +The 2N-qubit case +The 2N-qubit case is similar as the (2N + 1)-qubit case, the two-qubit basis is +|k+⟩ = 1 +2 +�� +|00⟩ + |11⟩ +� ++ i +� +|01⟩ + |10⟩ +�� +, +|k−⟩ = 1 +2 +� +i +� +|00⟩ + |11⟩ +� ++ +� +|01⟩ + |10⟩ +�� +. +(31) +The four-qubit GHZ-type state is +|ψ4⟩ = |00⟩ + |11⟩ +2 +|k+⟩ + |10⟩ + |01⟩ +2 +|k−⟩, +(32) +just the same as Eq. (22). +Next, we introduce the four-qubit back-up basis +|ψ′4⟩ = +� +|00⟩ + |11⟩ +� +|k−⟩ + +� +|10⟩ + |01⟩ +� +|k+⟩ = (σx)4|ψ4⟩. +(33) +It is GHZ-type and linearly independent with |ψ4⟩. Different from |ψ′3⟩, the state in |ψ′4⟩ can be +produced in the symmetric setup. +Then we can construct a six-qubit GHZ-type state by four-qubit basis, +|ψ6⟩ =|00⟩ + |11⟩ +2 +|ψ4⟩ + |10⟩ + |01⟩ +2 +|ψ′4⟩ +=1 +4 +��� +|00⟩ + |11⟩ +� +− +� +|10⟩ + |01⟩ +��� +|ψ4⟩ − |ψ′4⟩ +� ++ +�� +|00⟩ + |11⟩ +� ++ +� +|10⟩ + |01⟩ +��� +|ψ4⟩ + |ψ′4⟩ +�� +. +(34) +The six-qubit back-up basis is +|ψ′6⟩ = |00⟩ + |11⟩ +2 +|ψ′4⟩ + |10⟩ + |01⟩ +2 +|ψ4⟩. +(35) +Generally, 2N-qubit GHZ-type states can be presented as +|ψ2N⟩ = |00⟩ + |11⟩ +2 +|ψ2N−2⟩ + |10⟩ + |01⟩ +2 +|ψ′2N−2⟩ += 1 +4 +��� +|00⟩ + |11⟩ +� +− +� +|10⟩ + |01⟩ +��� +|ψ2N−2⟩ − |ψ′2N−2⟩ +� ++ +�� +|00⟩ + |11⟩ +� ++ +� +|10⟩ + |01⟩ +��� +|ψ2N−2⟩ + |ψ′2N−2⟩ +�� +. +(36) +Different from the (2N +1)-qubit case, the states in |ψ2N⟩ and |ψ′2N⟩ both can be produced in the +symmetric setup (details in App. A 2). It provides a new approach to generate multiqubit entangled +states. By getting (2N −2)-qubit basis and Bell states entangled, we can obtain 2N-qubit GHZ-type +state |ψ2N⟩. That means if we have produced (2N −2)-qubit GHZ-type states by gravity interaction, +entangled states with more qubits are available. This approach can be applied in improving existing +multipartite entangled platforms. If we get rid of the restrict from spatial symmetry, the approach +is also feasible for the (2N + 1)-qubit case. +In this section, we extend the gravitational entanglement to multiqubit. If we choose appropriate +parameters in the apparatus, the masses can be transformed to GHZ-type entangled states. Since the +number of independent phases are constrained in symmetric setup, the apparatus can not produce + +10 +W-type states. +W-type states need more freedom of phases, that means less symmetry in the +apparatus. For example, if we change the distances d between masses in Fig. 1(i), such as d1 and +d2, the phases of each item are independent, +|ψ6⟩ = 1 +2 +√ +2 +� +eiφ′ +1|000⟩ + eiφ′ +2|001⟩ + eiφ′ +3|010⟩ + eiφ′ +4|100⟩ ++ eiφ′ +5|110⟩ + eiφ′ +6|101⟩ + eiφ′ +7|011⟩ + eiφ′ +8|111⟩ +� +. +(37) +If we choose the phases as eiφ′ +1 = eiφ′ +2 = eiφ′ +3 = eiφ′ +4 = eiφ′ +8 = 1, eiφ′ +5 = −i, eiφ′ +6 = −1, eiφ′ +7 = i, the +modified apparatus will produce W-type states, +|ψ6⟩ = 1 +2 +√ +2 +� +|000⟩ + |001⟩ + |010⟩ + |100⟩ − i|110⟩ − |101⟩ + i|011⟩ + |111⟩ +� += 1 +2 +√ +2 +�� +|0⟩ + |1⟩ +� +A1 ⊗ |00⟩A2A3 + +� +|0⟩ − |1⟩ +� +A1 ⊗ |01⟩A2A3 ++ +� +|0⟩ − i|1⟩ +� +A1 ⊗ |1⟩A2 ⊗ +� +|0⟩ + i|1⟩ +� +A3 +� +. +(38) +The state in |ψ6⟩ can be transformed to W states under SLOCC. +IV. +MEASURE OF ENTANGLEMENT +In this section, we analyse and measure the gravitational entanglement by the GM of entanglement. +GM is the closest distance between an entangled state and the set of separable states [15, 16], +Λ2(ρ) = +max +|ϕ⟩∈PRO⟨ϕ|ρ|ϕ⟩, +G(ρ) = −2 log Λ(ρ). +(39) +Here the logarithm has base two. +We just calculate GM of tripartite system for simplicity. Suppose the three-qubit product states +be +|φ⟩ = (cos α|0⟩ + eiθ sin α|1⟩)A1 ⊗ (cos β|0⟩ + eiη sin β|1⟩)A2 ⊗ (cos γ|0⟩ + eiω sin γ|1⟩)A3. +(40) +Since there are too many parameters in the product states, we will consider the symmetric form +of the entangled state for simplicity in Sec. IV A, and study the general form in Sec. IV B. The +conclusions are presented in Theorem. 2. +Theorem 2 When we use GM to measure the entanglement, the gravity induced entanglement can +produce stable and robust entanglement for tripartite system. The robust entanglement appears gen- +erally at ∆ϕ3 ∈ [ 11π +16 , 21π +16 ] and the corresponding G(ρ) is close to one. +We will show it in next subsections. +A. +Symmetric situation +First we consider the entangled states in Eq. (10) be in symmetric form, that means ei∆ϕ2 = ei∆ϕ3. +The states can be written as +|ψ⟩ = eiϕ1 +2 +√ +2 +� +|000⟩ + |111⟩ + ei∆ϕ3� +|001⟩ + |011⟩ + |100⟩ + |110⟩ + |010⟩ + |101⟩ +�� +. +(41) + +11 +TABLE I: The data of Λ2 and the parameters of the nearest product states for some certain relative evolution +phases ∆ϕ3. +∆ϕ2 +π +8 +π +4 +3π +8 +π +2 +5π +8 +3π +4 +7π +8 +π +9π +8 +5π +4 +11π +8 +3π +2 +13π +8 +7π +4 +15π +8 +Λ2 +0.97 +0.89 +0.77 +0.625 +0.52 +0.50 +0.50 +0.50 +0.50 +0.50 +0.52 +0.625 +0.77 +0.89 +0.97 +α +π +4 +π +4 +π +4 +π +4 +π +32 +π +32 +7π +32 +π +4 +7π +32 +5π +16 +5π +16 +π +4 +π +4 +π +4 +π +4 +θ +0 +0 +0 +0 +27π +16 +25π +16 +π +2 +π +2 +3π +2 +7π +16 +5π +16 +0 +0 +0 +0 +As been showed in Proposition 4 of [16], the closest product state to any symmetric state is +necessarily symmetric. So the state in Eq. (40) should be symmetric too (which means α = β = γ, +θ = ω = η), can be written as |φ⟩ = (cos α|0⟩ + eiθ sin α|1⟩)⊗3. +The inner product of the entangled state and the product state is +⟨φ|ψ⟩ = eiϕ1 +2 +√ +2 +� +cos3 α + sin3 αe−3iθ + 3ei∆ϕ3 cos α sin αe−iθ(cos α + sin αe−iθ) +� +. +(42) +Its modular square is +|⟨φ|ψ⟩|2 = 1 +8 +� +cos6 α + cos5 α sin α +� +6 cos(∆ϕ3 − θ) +� ++ cos4 α sin2 α +� +6 cos(∆ϕ3 − 2θ) + 9 +� ++ cos3 α sin3 α(18 cos θ + 2 cos 3θ) + cos2 α sin4 α +� +6 cos(∆ϕ3 + 2θ) + 9 +� ++ cos α sin5 α +� +6 cos(∆ϕ3 + θ) +� ++ sin6 α +� +. +(43) +(i) +0 +1 +2 +3 +4 +5 +6 +7 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +∆φ3 +Λ2 +(ii) +FIG. 2: (i) The colour stands for |⟨φ|ψ⟩|2, as the function of α, θ and ∆ϕ3 in one period, the colour bar gives out +approximate value of |⟨φ|ψ⟩|2. (ii) The blue curve represents the Λ2 as the function of ∆ϕ3, and the red curve +represents the data fitting function in Eq. (44). +Eq. (43) is a function of α, θ and ∆ϕ3, which is presented in Fig. 2 (i). As the colour bar shows, +for ∆ϕ3 close to zero, Λ2 get close to one, and the nearest product state appears around α = π/4, +θ = 0. The criterion proposed in Sec. III A shows that the state in Eq. (41) becomes a separable +state for ∆ϕ3 = 0, so GM is vanishing and Λ2 = 1 (G(ρ)=0). At the beginning of the gravitational +interaction, the relative evolution phase ∆ϕ3 is small, and the entangled state in Eq. (41) is near the +product state, so Λ2 is big. With the interaction going on, the entanglement of Eq. (41) increases, +the distance between entangled state and the nearest product state increases, and Λ2 falls down. +With numerical computation, the blue curve in Fig. 2 (ii) describes Λ2 as the function of ∆ϕ3 +only, the nearest product states for each ∆ϕ3 are different, the corresponding α and θ were partly +listed in Table I. Both ends of the curve (∆ϕ3 < π +4 or ∆ϕ3 > 7π +4 ) correlate to weak entanglement + +0.3 +0 +8 +0.2 +6 +2 +1.5 +0.10.5 +0 +αt.0.9 +0.80.7 +0.6 +0.5 +0.412 +with Λ2 close to one. The nearest product states have stable phases α = π +4 and θ = 0. In the rapid +change ranges (∆ϕ3 ∈ [ π +4, π +2] and [ 3π +2 , 7π +4 ]) of the curve, Λ2 changes rapidly. The nearest product +states also are |φ⟩ = ( |0⟩+|1⟩ +√ +2 )⊗3. The middle range of the curve (∆ϕ3 ∈ ( π +2, 3π +2 )) represents the +final states entangled strongly, with small Λ2 around 0.5. However, the nearest product states are +oscillating. In this range, entanglement is great and stable, sensitive to witnesses. It is an important +region for studying the multipartite entanglement. GM in Eq. (39) becomes G(ρ) = 1 in this region. +Nevertheless, the apparatus can only generate small relative evolution phase ∆ϕ3 in [3], since the +interaction can not last for a long time because of decoherence. If we want to detect robust entangled +state, we can consider heavier masses as in [4]. When we detect some entangled states with certain +phases ∆ϕ3, corresponding α and θ are listed in Table I, or we can use the fitting function of the +curve to compute. +Λ2 = 0.164[arctan(5.71∆ϕ2 − 28.68) + arctan(−3.79∆ϕ2 + 4.83)] + 0.98. +(44) +The red curve in Fig. 2 (ii) presents the theoretical result. +B. +General Situation +In this subsection, we consider the product states be chosen as Eq. (40), the inner product becomes +⟨φ|ψ⟩ = eiϕ1 +2 +√ +2 +� +cos α cos β cos γ + sin α sin β sin γe−i(θ+η+ω) ++ ei∆ϕ2� +cos α cos β sin γe−iω + cos α sin β sin γe−i(η+ω) + sin α cos β cos γe−iθ ++ sin α sin β cos γe−i(θ+η)� ++ ei∆ϕ3� +cos α sin β cos γe−iη + sin α cos β sin γe−i(θ+ω)�� +. +(45) +Since there are many parameters in Eq. (45), we study it by numerical analysis. We just present +how Λ2 trends with ∆ϕ2 and ∆ϕ3 in Fig. 3, because it is hard to give the exact function form. The +states in Eq. (10) become product states when ei∆ϕ2 = ±1 and ei∆ϕ3 = 1, GM vanishes and the +entanglement is weak in adjacent regions (the red parts in Fig. 3). The nearest product states are +|φ⟩ = +1 +2 +√ +2(|0⟩ ± |1⟩)A1 ⊗ (|0⟩ + |1⟩)A2 ⊗ (|0⟩ ± |1⟩)A3, with α = β = γ = π/4 and θ = ω = η = 0 +(or θ = η = π, ω = 0). There are also some regions presenting robust and stable entanglement in +Fig. 3 (the blue parts). For different ∆ϕ2, the ranges of ∆ϕ3 (corresponding to blue parts) are a +little different, we choose the intersection ∆ϕ3 ∈ [ 11π +16 , 21π +16 ], it is valid for arbitrary ∆ϕ2. In these +regions, Λ2 is a little smaller than 0.5, the maximum G(ρ) is about 1.14. The corresponding nearest +product states vary with relative evolution phases ∆ϕ2 and ∆ϕ3. For example, when ∆ϕ2 = 2π and +∆ϕ3 = π, the nearest product states is |φ⟩ = +1 +2 +√ +2(|0⟩ + i|1⟩)A1 ⊗ (|0⟩ − i|1⟩)A2 ⊗ (|0⟩ + i|1⟩)A3, with +α = β = γ = π/4 and θ = η = π/2, ω = 3π/2. +In this section, we calculated GM of the gravitational entanglement. It depends on the relative +evolution phases ∆ϕ2 and ∆ϕ3. With numerical computation, we constructed the phase map of GM, +it marks the degree of entanglement of the final states for each couple of relative evolution phases. +We found that the robust entanglement appears generally at ∆ϕ3 ∈ [ 11π +16 , 21π +16 ]. +V. +CONCLUSIONS +In this article, we studied the multiqubit entanglement caused by gravity of neutral masses. The +mutual gravitation interaction could transform the separable states into GHZ-type states. It sug- + +13 +0 +2 +4 +6 +8 +0 +1 +2 +3 +4 +5 +6 +7 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +∆φ3 +∆φ2 +Λ2 +FIG. 3: The maximum of modular square Λ2 as the function of ∆ϕ2 and ∆ϕ3 in one period. +gested a way to construct N-qubit GHZ-type states by getting Bell states and (N − 2)-qubit GHZ- +type states entangled. This approach is not only feasible in gravity induced entanglement, but also +can be applied in ions trap, because Coulomb force between ions acts in the same way as gravity +except for a minus sign. Since electromagnetic interaction is much stronger than gravity, the ions’ +entanglement is more feasible in experiment. As described in [13], ions in a Paul trap can form a +linear string. The positions of them satisfy the spatial symmetry in this work. For ions’ string, +the evolution phases of superpositions are caused by oscillating electric field and mutual coulomb +force between ions. With appropriate parameters, electric field induces common phase, and relative +phases are only related to mutual interaction. So the results in this article can be applied in ions +trap to involve more ions in the entangled states. That may be a feasible path to realize robust +multiqubit entanglement and will play a part in quantum computation. +We also calculated GM of three-qubit gravity induced entangled states. This value measures the +degree of entanglement for a certain final state. We constructed functions to describe the relationship +of GM and the relative evolution phases. The phase map of GM enables us to pick out the ranges +with robust entanglement. We will pay attention on these regions in the future study of multipartite +entanglement. It is helpful in experiment design. +Appendix A: The spatial symmetry +As we can see in Sec. II, the masses split into two superpositions, components |L⟩ and |R⟩ keep +stable distances from each other in the apparatus of symmetric setup (as Fig. 1(ii, iii) described). +The mutual gravity interaction can induce different rates of phase evolution in the final state. +In Eq. (9) and Eq. (21), these phases are decided by the distances between |L⟩ and |R⟩. For +example, for the three-qubit case in Eq. (10), the evolution phase of |001⟩ is ϕ2 in Eq. (9). It relates +to the distances, including d between |0⟩A1 and |0⟩A2, +√ +d2 + l2 between |0⟩A2 and |1⟩A3, +√ +4d2 + l2 +between |0⟩A1 and |1⟩A3, as Fig. 4(i) described. We check Fig. 4 (ii, iii, iv), and find that |110⟩, |100⟩ +and |011⟩ should have the same phase ϕ2, because the sum of distances in these cases are equivalent +to Fig. 4(i). +Other parts in Eq. (10) reveal the same feature, |000⟩ and |111⟩ share the same phase ϕ1, |010⟩ and +|101⟩ share the same phase ϕ3. However, ϕ1, ϕ2, ϕ3 are independent. That is the spatial symmetry +we talked about in Sec. III. +For the four-qubit case in Eq. +(20), +we classify the state into six groups with in- +dependent phases, +� +|0000⟩, |1111⟩ +� +, +� +|0001⟩, |0111⟩, |1000⟩, |1110⟩ +� +, +� +|0010⟩, |1101⟩, |0100⟩, |1011⟩ +� +, + +14 +(i) |001⟩ +(ii) |110⟩ +(iii) |100⟩ +(iv) |011⟩ +FIG. 4: Vertical view of Fig. 1(iii). The dots stand for the superpositions of the first (A1), second (A2) and third +(A3) masses from left to right. The lines refer to the distances between components of the masses. +(i) |1001⟩ +(ii) |0110⟩ +FIG. 5: In the four-qubit case, |1100⟩ and |0011⟩ have the same phase since they share the symmetric distances. +� +|1100⟩, |0011⟩ +� +, +� +|1010⟩, |0101⟩ +� +, +� +|1001⟩, |0110⟩ +� +, each group of states in one bracket have the same +phase. The states in the last bracket are presented in Fig. 5 for example. We find that the states in +one bracket are symmetric if we exchange |0⟩ and |1⟩ or turn over the qubit. +We define some notations to help understanding. +First, we denote � +|ψ⟩ as the invert state of +|ψ⟩, that means � +|0⟩ = |1⟩, � +|1⟩ = |0⟩ for each qubit, for example if |ψ⟩ = |001⟩ then � +|ψ⟩ = |110⟩. +Next, if we turn over the qubit in the state |ψ⟩, it becomes the turn over state |ψ⟩To, such as +(|0⟩A|0⟩B|1⟩C)To = |1⟩A|0⟩B|0⟩C. For example if |ψ⟩ = |101001⟩ + |110001⟩, the turn over state +|ψ⟩To = |100101⟩ + |100011⟩. +We describe the spatial symmetry in Theorem 3. +Theorem 3 In the symmetric setup described by Fig. 1 (ii, iii), the multipartite entangled state |ψ⟩ +should obey spatial symmetry that each symmetric groups in the final state should share the same +evolution phase. It equals to the statements � +|ψ⟩ = |ψ⟩ and |ψ⟩To = |ψ⟩. +The states we have constructed in Eq. (30) and Eq. (36) under symmetric setup should follow +Theorem 3. We will check whether they satisfy above statements from Eq. (23) to Eq. (36). +1. +The (2N + 1)-qubit case +Obviously, the basis in Eq. (23) satisfy, +� +|m+⟩ = |1⟩ + |0⟩ +√ +2 += |m+⟩, +|m+⟩To = |m+⟩, +� +|m−⟩ = |1⟩ − |0⟩ +√ +2 += −|m−⟩, +|m−⟩To = |m−⟩. +(A1) + +1 +1 +1 +1 +0 +0 +0 +00 +0 +01 +0 +0 +01 +1 +1 +0 +01 +1 +1 +0 +0 +01 +1 +0 +0 +0 +015 +For three-qubit basis in Eq. (24) and Eq. (26), the invert states are +� +|ψ3⟩ = |11⟩ + |00⟩ +2 +� +|m+⟩ + |01⟩ − |10⟩ +2 +� +|m−⟩ = |ψ3⟩, +� +|ψ′3⟩ = |11⟩ + |00⟩ +2 +� +|m−⟩ − |01⟩ − |10⟩ +2 +� +|m+⟩ = −|ψ′3⟩. +(A2) +The turn over state of |ψ3⟩ is +|ψ3⟩To = |m+⟩To|00⟩ + |11⟩ +2 ++ |m−⟩To|01⟩ − |10⟩ +2 += +1 +2 +√ +2 +�� +|0⟩ + |1⟩ +�� +|00⟩ + |11⟩ +� ++ +� +|0⟩ − |1⟩ +�� +|01⟩ − |10⟩ +�� += +1 +2 +√ +2 +� +|0⟩ +� +|00⟩ + |01⟩ +� ++ |1⟩ +� +|11⟩ + |10⟩ +� ++ |0⟩ +� +|11⟩ − |10⟩ +� ++ |1⟩ +� +|00⟩ − |01⟩ +�� += +1 +2 +√ +2 +� +|00⟩ +� +|0⟩ + |1⟩ +� ++ |11⟩ +� +|1⟩ + |0⟩ +� ++ |01⟩ +� +|1⟩ − |0⟩ +� ++ |10⟩ +� +|0⟩ − |1⟩ +�� += +1 +2 +√ +2 +�� +|00⟩ + |11⟩ +�� +|0⟩ + |1⟩ +� ++ +� +|10⟩ − |01⟩ +�� +|0⟩ − |1⟩ +�� += |ψ3⟩. +(A3) +Similar, we have +|ψ′3⟩To = |m−⟩To|00⟩ + |11⟩ +2 +− |m+⟩To|01⟩ − |10⟩ +2 += |ψ′3⟩. +(A4) +For five-qubit basis in Eq. (27) and Eq. (29), the invert states are +� +|ψ5⟩ = |11⟩ + |00⟩ +2 +� +|ψ3⟩ + |01⟩ − |10⟩ +2 +� +|ψ′3⟩ = |ψ5⟩, +� +|ψ′5⟩ = |11⟩ + |00⟩ +2 +� +|ψ′3⟩ − |01⟩ − |10⟩ +2 +� +|ψ3⟩ = −|ψ′5⟩. +(A5) +The turn over state of Eq. (27) is +|ψ5⟩To = |ψ3⟩To|00⟩ + |11⟩ +2 ++ |ψ′3⟩To|01⟩ − |10⟩ +2 += +1 +4 +√ +2 +��� +|00⟩ + |11⟩ +�� +|0⟩ + |1⟩ +� ++ +� +|10⟩ − |01⟩ +�� +|0⟩ − |1⟩ +��� +|00⟩ + |11⟩ +� ++ +�� +|00⟩ + |11⟩ +�� +|0⟩ − |1⟩ +� +− +� +|10⟩ − |01⟩ +�� +|0⟩ + |1⟩ +��� +|01⟩ − |10⟩ +�� += +1 +4 +√ +2 +�� +|00⟩ + |11⟩ +��� +|0⟩ + |1⟩ +�� +|00⟩ + |11⟩ +� ++ +� +|0⟩ − |1⟩ +�� +|01⟩ − |10⟩ +�� ++ +� +|10⟩ − |01⟩ +��� +|0⟩ − |1⟩ +�� +|00⟩ + |11⟩ +� +− +� +|0⟩ + |1⟩ +�� +|01⟩ − |10⟩ +��� += |00⟩ + |11⟩ +2 +|ψ3⟩To + |10⟩ − |01⟩ +2 +|ψ′3⟩To = |00⟩ + |11⟩ +2 +|ψ3⟩ + |10⟩ − |01⟩ +2 +|ψ′3⟩ = |ψ5⟩. +(A6) +We can show |ψ′5⟩To = |ψ′5⟩ in the same way. +With mathematical induction, we can deduce |ψ7⟩ and |ψ′7⟩ satisfy the same statements, and so +do (2N + 1)-qubit. So the (2N + 1)-qubit final states we constructed in Sec. III C obey Theorem +3. We should notice that, the back-up basis |ψ′2N+1⟩ does not obey Theorem 3, so it can not be +generated in the symmetric setup. + +16 +2. +The 2N-qubit case +Now we consider the two-qubit basis in Eq. (31), +� +|k+⟩ = 1 +2 +�� +|11⟩ + |00⟩ +� ++ i(|10⟩ + |01⟩ +�� += |k+⟩, +|k+⟩To = 1 +2 +�� +|00⟩ + |11⟩ +� ++ i(|10⟩ + |01⟩ +�� += |k+⟩, +� +|k−⟩ = 1 +2 +� +i +� +|11⟩ + |00⟩ +� ++ (|10⟩ + |01⟩ +�� += |k−⟩, +|k−⟩To = 1 +2 +� +i +� +|00⟩ + |11⟩ +� ++ (|10⟩ + |01⟩ +�� += |k−⟩. +(A7) +The four-qubit basis in Eq. (32) and Eq. (33) satisfy, +� +|ψ4⟩ = |11⟩ + |00⟩ +2 +� +|k+⟩ + |10⟩ + |01⟩ +2 +� +|k−⟩ = |ψ4⟩, +� +|ψ′4⟩ = |11⟩ + |00⟩ +2 +� +|k−⟩ + |10⟩ + |01⟩ +2 +� +|k+⟩ = |ψ′4⟩. +(A8) +|ψ4⟩To = |k+⟩To|00⟩ + |11⟩ +2 ++ |k−⟩To|10⟩ + |01⟩ +2 += 1 +4 +��� +|00⟩ + |11⟩ +� ++ i +� +|01⟩ + |10⟩ +��� +|00⟩ + |11⟩ +� ++ +� +i +� +|00⟩ + |11⟩ +� ++ +� +|01⟩ + |10⟩ +��� +|10⟩ + |01⟩ +�� += 1 +4 +�� +|00⟩ + |11⟩ +��� +|00⟩ + |11⟩ +� ++ i +� +|10⟩ + |01⟩ +�� ++ +� +|01⟩ + |10⟩ +�� +i +� +|00⟩ + |11⟩ +� ++ +� +|10⟩ + |01⟩ +��� += |00⟩ + |11⟩ +2 +|k+⟩ + |01⟩ + |10⟩ +2 +|k−⟩ = |ψ4⟩. +(A9) +We can show |ψ′4⟩To = |ψ′4⟩ similarly. +Inducing in the same way, 2N-qubit basis also satisfy Theorem 3. So 2N-qubit final states con- +structed in Sec. III D can be produced in the symmetric setup. In this case, all back-up bases |ψ′2N⟩ +obey Theorem 3 too. +Acknowledgements +Authors thank the interesting discussion with Professor Qilin Zhang and Mingguo Sun. MFL and +LC were supported by the NNSF of China (Grant No. 11871089), and the Fundamental Research +Funds for the Central Universities(Grant No. ZG216S2005). +[1] Overstreet C, Asenbaum P, Curti J, Kim M, Kasevich MA. Observation of a gravitational Aharonov-Bohm effect. Science. +2022 Jan 14;375(6577):226-229. doi: 10.1126/science.abl7152. Epub 2022 Jan 13. PMID: 35025635. + +17 +[2] Hohensee MA, Estey B, Hamilton P, Zeilinger A, M¨uller H. Force-free gravitational redshift: proposed gravitational +Aharonov-Bohm experiment. Phys Rev Lett. 2012 Jun 8;108(23):230404. doi: 10.1103/PhysRevLett.108.230404. Epub +2012 Jun 7. PMID: 23003927. +[3] S. Bose, A. Mazumdar, G.W. Morley, H. Ulbricht, M. Toroˇs, M. Paternostro, A.A. Geraci, P.F. Barker, M.S. Kim, G. +Milburn. Spin engtanglement witness for quantum gravity. Phys. Rev. Lett. 119, 240401 (2017). +[4] C. Marletto, V. Vedral. 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Is gravitational entanglement evidence for the quantization of spacetime? +arXiv: +2205.00939v1 (2022). + diff --git a/ldE5T4oBgHgl3EQfHQ4M/content/tmp_files/load_file.txt b/ldE5T4oBgHgl3EQfHQ4M/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..04339a0bea340daa1b03ccd41789f63c9ddc1bb6 --- /dev/null +++ b/ldE5T4oBgHgl3EQfHQ4M/content/tmp_files/load_file.txt @@ -0,0 +1,803 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf,len=802 +page_content='Multiqubit entanglement due to quantum gravity Shaomin Liu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='1 Lin Chen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' ∗ and Mengfan Liang2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' † 1School of Mathematics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' Physics and Finance,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' Anhui Polytechnic University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' Wuhu 241000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' China 2LMIB(Beihang University),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' Ministry of Education,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' and School of Mathematical Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' Beihang University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' Beijing 100191,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' China 3International Research Institute for Multidisciplinary Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' Beihang University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' Beijing 100191,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' China (Dated: January 16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' 2023) Quantum gravity between masses can produce entangled states in thought experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' We extend the experiments to tripartite case and construct states equivalent to Greenberger- Horne- Zeilinger states and W states under stochastic local operations and classical communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' The entanglement relates to the evolution phases induced by gravitational interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' When we involve more masses in the experiments, multipartite entangled states can be constructed in a similar way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' We measure the degree of multipartite entanglement by calculating the geometric measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' We describe the relationship between geometric measure and the evolution phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' It helps in searching out the states with robust entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' PACS numbers: 03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='Ud, 04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='Ds, 03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='Mn I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' INTRODUCTION Due to the extreme weakness of gravity, its quantum effects are hard to detect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' Recently, authors reported observing gravitational Aharonov-Bohm effect [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' This experiment measured the gravi- tational phase shift induced by a kilogram-scale source mass close to the wave packets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' It convinced us about the quantum feature of gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' Bose et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' [3] and Marletto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' [4] suggested two similar thought experiments to probe quantized gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' Two neutral masses are separable initially, split into superposition of spatially localized states in an inhomogeneous magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' The mutual grav- itational interaction between components of superposition will evolute relative phases and transform initial separable state into bipartite entangled state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' These quantum phases correlate with their interaction time and can be detected by entanglement witnesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' Even though the dominant contri- bution of interaction is Newtonian at the low energy limits, the entanglement between two masses can verify quantum signatures of gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' Because local operations and classical communication (LOCC) does not create entanglement, the entanglement can be generated only by non-classical me- diator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' There are also some discussions on Post-Newtonian order corrections in [5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' The feasibility of the hypothetical experiments have been discussed in the original articles [3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' There are some factors which may pollute the entanglement in Bose’s experiment, such as Casimir- Polder forces, van der Waals forces or other electro-magnetic interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' By adjusting the param- eters or setups of the experiments, some pollution can be reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' A thought experiment was presented to study interaction mediated only by gravity between two hypothetical neutrino-like par- ticles [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' A modified model of original experiment, the symmetric setup, enhanced the gravity interaction to against the noisy dynamics, such as stochastic fluctuations of the parameters or de- coherence induced by environmental interaction [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' However, these studies all focused on bipartite entanglement, as far as we know, multipartite entanglement is little understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' In this paper, we extend the quantum gravity inducing entanglement to multiqubit case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' Multi- qubit entanglement plays an important role in quantum information, computation and communica- ∗linchen@buaa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='cn (corresponding author) †lmf2021@buaa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='cn (corresponding author) arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='05437v1 [quant-ph] 13 Jan 2023 2 tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' There are various platforms generating multipartite entanglement with photons or ions [9–13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' Photonic experiments entangled 14 photons to realize Greenberger-Horne-Zeilinger (GHZ) states by interleave single-photon emissions with atomic rotations [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' In a linear Paul trip, GHZ states were produced with up to 24 ions, mediated by the Mølmer-Sørensen gate [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' These attempts went a step further in quantum computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' In the future, the major problems still are how to increase the efficiency of generating entanglement and protect systems against decoherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' This work suggests a theoretical path to produce entangled states induced by mutual gravitational interaction of neutral masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' In multipartite system, the Hamiltonian leads to relative evolution phases with special spatial symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' The interaction is similar to the bipartite case in [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' In this protocol, the quantum gravity can generate GHZ states, but not W states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' We can classify equivalent entangled states under stochastic LOCC (SLOCC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' They contain the same kind of entanglement and are suited to implement the same tasks of quantum information theory [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' We denote GHZ-type states as those states equivalent to GHZ states under SLOCC, and similarly for W-type states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' In Theorem 1, we show that the gravitation can generate N-qubit GHZ-type states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' However, considering the weakness of gravity, transform N masses into entangled states is difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' We suggest a way to generate N- qubit GHZ-type states by getting (N − 2)-qubit GHZ-type states and Bell states entangled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' The (N − 2)-qubit GHZ-type states can be produced in the gravitational entanglement apparatus too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' It provides an approach to extend existing multipartite entanglement platforms by involving more qubits and makes the experiment more feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' To address the degree of the gravity induced entanglement from a geometric viewpoint [15, 16], we calculate the geometric measure (GM) of entanglement for the tripartite case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' GM quantifies the entanglement by measuring the distance between the entangled state and the nearest product state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' So we can seek out the robust entangled final states induced by gravity with the results in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' It is helpful in multipartite platforms design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' The rest of this paper is constructed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' II, we introduce Bose’s experiment in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' III, more masses are led in to construct multiqubit entanglement and GHZ-type states in the symmetric setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' IV, we calculate the GM of entanglement for the three-qubit case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' V makes conclusion and outlook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' PRELIMINARIES Bose et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' proposed a thought experiment in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' Two neutral masses m1 and m2, in an inhomo- geneous magnetic field, both split into a superposition of two spatially separated states |L⟩ and |R⟩ for a time τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' As we can see in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' 1(i), l is the distance between components of superposition, and d is the distance between the centres of two masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' Initially, two masses A1 and A2 are separated, each one splits into two superpositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' |ψ(0)⟩A1A2 = 1 √ 2 � |L⟩A1 + |R⟩A1 � 1 √ 2 � |L⟩A2 + |R⟩A2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (1) The Schr¨odinger equation reveals the time evolution of the state, iℏ∂tψ(r, t) = � − ℏ2 2m∇2 + V (r) � ψ(r, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (2) The Hamiltonian H = − ℏ2 2m∇2 + V = H0 + V can be separated into two parts, H0 relates to the free particle’s behaviour and V is the mutual gravity potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' 3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' 1: Two masses at distance d from each other, both split into superpositions of spatially localized states at distance l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (i)The original setup proposed in [3], superpositions are parallel to the initial separations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (ii) The symmetric setup proposed in [8], superpositions are orthogonal to the initial separations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (iii) The symmetric setup for three-qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' During the interaction time, these components keep stable distances from each other, so they have time independent gravity interaction energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' By calculating the propagator and the scattering process, the mutual gravitation potential can be described as [17–19], V (r) = −Gm1m2 r (1 + 3G(m1 + m2) rc2 + 41Gℏ 10πr2c3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (3) The general relativistic correction and quantum correction to Newtonian potential are extremely small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' Even if we take the Newtonian approximation of potential, the entanglement can reflect the quantum nature of gravity too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' Actually in this setting, gravity acts as a mediator in quantum mechanics, the entanglement will be detected only through non-classic dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' So in this paper, we just take the Newtonian approximation of potential for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' Since the potential is independent of time, the Schr¨odinger equation for the state ψ(r, t) can be solved by separating variables, ψ(r, t) = e −iEt ℏ ψ(r), the evolution phase is related to the potential eiφ ≡ e −iV t ℏ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' Because the states |L⟩ and |R⟩ can be separated by different distances (including d, d − l, d + l), the mutual gravity interaction can induce different rates of phase evolution in Stern-Gerlach(SG) apparatus, for simplicity, written as, |ψ(τ)⟩A1A2 = 1 2 � |L(τ)⟩A1 � eiφ1|L(τ)⟩A2 +eiφ2|R(τ)⟩A2 � +|R(τ)⟩A1 � eiφ3|L(τ)⟩A2 +eiφ1|R(τ)⟩A2 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (4) If we take Newtonian approximation of the potential in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (3), the evolution phases are [3] φ1 ∼ Gm1m2τ ℏd , φ2 ∼ Gm1m2τ ℏ(d + l) , φ3 ∼ Gm1m2τ ℏ(d − l) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (5) We denote the relative phases ∆φ2 = φ2 − φ1 and ∆φ3 = φ3 − φ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' The entanglement depends on the relative phases and is irrelative to common phase φ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' There is an exceptional case, when ∆φ2 + ∆φ3 = 2nπ, n ∈ Z (Z is the integer set), |ψ(τ)⟩A1A2 is separable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' The parameters are chosen as m1, m2 ∼ 10−14kg, d ∼ 450µm, l ∼ 250µm, τ ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='5s, the en- tanglement witness is W = X ⊗ Z + Y ⊗ Y [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' The expectation value ⟨W⟩ > 1 is the signal of d R A2 A2 H3 KI L) A2 [11 (i) (ii) (ili)4 entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' This witness has a suboptimal detection areas and may be ineffective for small entan- glement which corresponds to short interaction time τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' Nevertheless, a long interaction time is also infeasible, because when we consider the decoherence effect from earth’s gravity, it is difficult to keep superposition states free falling in such a long time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' So a much broader class of witnesses [20, 21] was suggested to detect the greatest volume of entangled states and make the setup of experiment more feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' Authors constructed optimal fidelity witnesses by maximally entangled states to shorten the required interaction time, such as W = I ⊗I −X ⊗X −Z ⊗Y −Y ⊗Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' These witnesses are sensitive to very small entanglement, so they are valid at the beginning of free-fall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' They also broaden the detection area in the space of phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' With these instruments, the positive results will announce the quantum gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' An entanglement witness was suggested to detect spinless entanglement between microspheres with massive spatial qubits too [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' However, there are also some doubts about the effectiveness of these witnesses of quantized gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' General configuration-ensemble models and mean-field semiclassical gravity models were supposed to explain the entanglement with classical gravity [23, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' A modified symmetric setup was suggested in [8], each mass splits into two superposition states, which are in the direction orthogonal to initial separation, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' 1(ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' Compared with the parallel split mode in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' 1(i), the symmetric setup permits a reduced distance between masses, that is useful in keeping the distance constant and enhancing gravity interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' In this case, there is only one relative phase ∆φ = Gm1m2τ ℏ ( 1 d − 1 √ l2+d2), the final state in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (4) becomes |ψ(τ)⟩A1A2 = eiφ1 2 � |L(τ)⟩A1 � |L(τ)⟩A2 + ei∆φ|R(τ)⟩A2 � + |R(τ)⟩A1 � ei∆φ|L(τ)⟩A2 + |R(τ)⟩A2 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (6) The distances of qubits in this setup display some symmetry which constraints the number of independent phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' The next sections will consider entanglement in this setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' MULTIQUBIT ENTANGLEMENT In this section, we extend gravity interaction induced entanglement to multipartite case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' GHZ states and W states are the maximum entangled states under geometric entanglement metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' They play important roles in multipartite entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' Besides, since the equivalence class of GHZ or W states under SLOCC contain the same kind of entanglement, we also focus on GHZ-type states and W-type states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' Two states are equivalent under SLOCC when they are related by a local invertible operator [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' For example, if |ψ⟩ = (M1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' ⊗ Mn)|GHZ⟩ , where each Mi is an invertible matrix, then |ψ⟩ is in the equivalence class of GHZ states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' We will construct GHZ-type states under the gravitational symmetric setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' Nevertheless, if we want to construct W-type states, more freedom in parameters of the setup should be involved in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' We will respectively construct three-qubit and four-qubit GHZ-type states in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' III A and III B, and then extend to N-qubit case in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' III C and III D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' The three-qubit case First we introduce one more mass in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (1), as described in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' 1(iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' |ψ(0)⟩A1A2A3 = 1 √ 2 � |L⟩A1 + |R⟩A1 � 1 √ 2 � |L⟩A2 + |R⟩A2 � 1 √ 2 � |L⟩A3 + |R⟩A3 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (7) The time evolution of this state is similar to the bipartite case in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' The gravitation potential in Hamiltonian is related to the distances between three components of superpositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' We consider 5 the symmetric setup, and there are three different evolution phases in the final state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' The gravitation potential among the left components |L⟩A1, |L⟩A2 and |L⟩A3 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' 1(iii) is V = −Gm1m2( 1 d+ 1 d+ 1 2d), so does the potential among the right components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' The potential among |L⟩A1, |R⟩A2 and |L⟩A3 is V = −Gm1m2( 1 2d + 1 √ d2+l2 + 1 √ d2+l2), so does the potential among |R⟩A1, |L⟩A2 and |R⟩A3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' The potential among |L⟩A1, |L⟩A2 and |R⟩A3 is V = −Gm1m2( 1 d + 1 √ 4d2+l2 + 1 √ d2+l2), so do rest parts in the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' The final state can be written as |ψ(τ)⟩A1A2A3 = 1 2 √ 2 � eiϕ1� |L(τ)⟩A1|L(τ)⟩A2|L(τ)⟩A3 + |R(τ)⟩A1|R(τ)⟩A2|R(τ)⟩A3 � + eiϕ2� |L(τ)⟩A1|L(τ)⟩A2|R(τ)⟩A3 + |L(τ)⟩A1|R(τ)⟩A2|R(τ)⟩A3 + |R(τ)⟩A1|L(τ)⟩A2|L(τ)⟩A3 + |R(τ)⟩A1|R(τ)⟩A2|L(τ)⟩A3 � + eiϕ3� |L(τ)⟩A1|R(τ)⟩A2|L(τ)⟩A3 + |R(τ)⟩A1|L(τ)⟩A2|R(τ)⟩A3 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (8) The evolution phases are ϕ1 ∼ 5Gm1m2τ 2ℏd , ϕ2 ∼ Gm1m2τ ℏ (1 d + 1 √ 4d2 + l2 + 1 √ d2 + l2), ϕ3 ∼ Gm1m2τ ℏ ( 1 2d + 2 √ d2 + l2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (9) For simplicity, we denote |L(τ)⟩ as |0⟩, |R(τ)⟩ as |1⟩, ∆ϕ2 = ϕ2 − ϕ1, ∆ϕ3 = ϕ3 − ϕ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (8) can be written as |ψ⟩ = eiϕ1 2 √ 2 � |000⟩ + |111⟩ + ei∆ϕ2� |001⟩ + |011⟩ + |100⟩ + |110⟩ � + ei∆ϕ3� |010⟩ + |101⟩ �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (10) The entanglement of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (10) comes from the relative evolution phases ∆ϕ2 and ∆ϕ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' In some certain situation, the state in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (10) may become the separable state, GHZ state or GHZ-type state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' In the following, we will discuss the three cases (i), (ii) and (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (i) The state in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (10) is a separable state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' We write it as |ψ⟩ = eiϕ1 2 √ 2 � |0⟩A1 � |00⟩A2A3 + ei∆ϕ2� |01⟩A2A3 + |11⟩A2A3 � + ei∆ϕ3|10⟩A2A3 � + |1⟩A1 � |11⟩A2A3 + ei∆ϕ2� |00⟩A2A3 + |10⟩A2A3 � + ei∆ϕ3|01⟩A2A3 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (11) So in the range space of the reduced density operator of the second and third qubits, there are two vectors, |a⟩ = 1 2 � |00⟩A2A3 + ei∆ϕ2� |01⟩A2A3 + |11⟩A2A3 � + ei∆ϕ3|10⟩A2A3 � , |b⟩ = 1 2 � |11⟩A2A3 + ei∆ϕ2� |00⟩A2A3 + |10⟩A2A3 � + ei∆ϕ3|01⟩A2A3 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (12) If the vectors |a⟩ and |b⟩ are linearly dependent, the final state |ψ⟩ is a separable state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' For example, by choosing ∆ϕ3 = 2nπ, n ∈ Z, the state in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (10) is separable, |ψ1⟩ = eiϕ1 2 √ 2[|0⟩A1 ⊗ (|0⟩ + |1⟩)A2 ⊗ (|0⟩ + ei∆ϕ2|1⟩)A3 + |1⟩A1 ⊗ (|0⟩ + |1⟩)A2 ⊗ (ei∆ϕ2|0⟩ + |1⟩)A3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (13) 6 The state in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (13) will be fully separable states for ∆ϕ2 = nπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' |ψ2⟩ = eiϕ1(|0⟩ ± |1⟩ √ 2 )A1 ⊗ (|0⟩ ± |1⟩ √ 2 )A2 ⊗ (|0⟩ ± |1⟩ √ 2 )A3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (14) (ii) The state in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (10) is a GHZ state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' When ei∆ϕ3 ̸= 1, the state in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (10) becomes a genuinely entangled state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' The reduced density operator of the first, second and third qubit respectively is ρA1 = ρA3 = 1 8 � 4(|0⟩⟨0| + |1⟩⟨1|) + (ei∆ϕ2 + e−i∆ϕ2 + ei(∆ϕ2−∆ϕ3) + ei(∆ϕ3−∆ϕ2))(|0⟩⟨1| + |1⟩⟨0|) � , ρA2 = 1 8 � 4(|0⟩⟨0| + |1⟩⟨1|) + (ei∆ϕ3 + e−i∆ϕ3 + 2)(|0⟩⟨1| + |1⟩⟨0|) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (15) When ei∆ϕ2 + e−i∆ϕ2 + ei(∆ϕ2−∆ϕ3) + ei(∆ϕ3−∆ϕ2) = 0 and ei∆ϕ3 + e−i∆ϕ3 + 2 = 0, the state in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (10) is a GHZ state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' That means ∆ϕ3 = (2n+1)π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' For example, by choosing ei∆ϕ2 = i, ei∆ϕ3 = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (10) becomes |ψ3⟩ = eiϕ1 2 √ 2[(|0⟩ + i|1⟩)A1 ⊗ |0⟩A2 ⊗ (|0⟩ + i|1⟩)A3 + (i|0⟩ + |1⟩)A1 ⊗ |1⟩A2 ⊗ (i|0⟩ + |1⟩)A3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (16) The state in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (16) is a GHZ state, because |ψ⟩ = ( σz+σy 2 )A1 ⊗ IA2 ⊗ ( σz+σy 2 )A3(|000⟩ + |111⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' When we choose ei∆ϕ2 = 1 and ei∆ϕ3 = −1, the state in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (10) is LU equivalent to a GHZ state, |ψ4⟩ = eiϕ1 2 √ 2[(|00⟩ + |11⟩)A1A2 ⊗ (|0⟩ + |1⟩)A3 + (|10⟩ − |01⟩)A1A2 ⊗ (|0⟩ − |1⟩)A3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (17) (iii) The state in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (10) is a GHZ-type state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' The state in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (10) becomes a GHZ-type state when there exist two linearly independent product vectors in the range space of the reduced density operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' That means if the vectors |a⟩ and |b⟩ construct a product state |ψp⟩ = |a⟩ + x|b⟩, there exist two roots of x which satisfy the equation (1 + ei∆ϕ2x)(ei∆ϕ2 + x) = (ei∆ϕ2 + ei∆ϕ3x)(ei∆ϕ3 + ei∆ϕ2x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (18) For example, when we choose ei∆ϕ2 = 1 and ei∆ϕ3 = i, the state in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (10) becomes GHZ-type states, |ψ5⟩ = eiϕ1 2 √ 2[(|00⟩ + |11⟩)A1A2 ⊗ (|0⟩ + |1⟩)A3 + |10⟩A1A2 ⊗ (|0⟩ + i|1⟩)A3 + |01⟩A1A2 ⊗ (i|0⟩ + |1⟩)A3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (19) We can see that, there are greater chances for Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (10) be a GHZ-type state than be a GHZ state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' So we will consider GHZ-type states in next sections only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' On the other hand if there exists a multiple root for Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (18), the state in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (10) becomes a W-type state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' For W-type states, the phases should satisfy (1 + ei∆ϕ3)2 = 4e2i∆ϕ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' That means ei∆ϕ3 = 1, though in this case |ψ1⟩ is separable, actually there is no solution for the equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' So under this setup, we cannot produce W-type states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' 7 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' The four-qubit case In this part, we extend the modes of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' 1(iii) to the four-qubit case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' The final state can be written as |ψ⟩ =1 4 � eiϕ′ 1� |0000⟩ + |1111⟩ � + eiϕ′ 2� |0001⟩ + |0111⟩ + |1000⟩ + |1110⟩ � + eiϕ′ 3� |0010⟩ + |1101⟩ + |0100⟩ + |1011⟩ � + eiϕ′ 4� |1100⟩ + |0011⟩ � + eiϕ′ 5� |1010⟩ + |0101⟩ � + eiϕ′ 6� |1001⟩ + |0110⟩ �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (20) The evolution phases are similar to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (9), ϕ′ 1 ∼13Gm1m2τ 3ℏd , ϕ′ 2 ∼Gm1m2τ ℏ ( 5 2d + 1 √ 9d2 + l2 + 1 √ 4d2 + l2 + 1 √ d2 + l2), ϕ′ 3 ∼Gm1m2τ ℏ (11 6d + 1 √ 4d2 + l2 + 2 √ d2 + l2), ϕ′ 4 ∼Gm1m2τ ℏ (2 d + 1 √ 9d2 + l2 + 2 √ 4d2 + l2 + 1 √ d2 + l2), ϕ′ 5 ∼Gm1m2τ ℏ (1 d + 1 √ 9d2 + l2 + 3 √ d2 + l2), ϕ′ 6 ∼Gm1m2τ ℏ ( 4 3d + 2 √ 4d2 + l2 + 2 √ d2 + l2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (21) When we extract eiϕ′ 1, the relative phases are ∆ϕ′ i = ϕ′ i − ϕ′ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' If the relative phases are chosen as ei∆ϕ′ 2 = ei∆ϕ′ 3 and ei∆ϕ′ 4 = ei∆ϕ′ 5 = ei∆ϕ′ 6 = 1, the state will be GHZ-type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' For example, if we choose ei∆ϕ′ 2 = ei∆ϕ′ 3 = i, then Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (20) becomes |ψ⟩ =1 4eiϕ′ 1 � (|00⟩ + |11⟩)A1A2 � |0⟩A3(|0⟩ + i|1⟩)A4 + |1⟩A3(i|0⟩ + |1⟩)A4 � + (|10⟩ + |01⟩)A1A2 � |0⟩A3(i|0⟩ + |1⟩)A4 + |1⟩A3(|0⟩ + i|1⟩)A4 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (22) Looking through the three-qubit and the four-qubit cases, we can see that the relative phases are related to the distances of qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' As in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' 1(iii), the distances are constrained by some spatial symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' We will present the details in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' When we extend the results to N-qubit case, we have Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' Theorem 1 When masses are split into superpositions in symmetric setup as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' 1(iii) described, the gravity interaction between superpositions can produce N-qubit GHZ-type entangled states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' Since there is some difference between the (2N + 1)-qubit case and the 2N-qubit case, we will show Theorem 1 from two aspects in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' III C and III D .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' First, let us consider the (2N + 1)-qubit case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' The (2N + 1)-qubit case We define a one-qubit basis, |m+⟩ = |0⟩ + |1⟩ √ 2 , |m−⟩ = |0⟩ − |1⟩ √ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (23) 8 They are linearly independent, then we can construct a three-qubit state like this, |ψ3⟩ = |00⟩ + |11⟩ 2 |m+⟩ + |10⟩ − |01⟩ 2 |m−⟩ = 1 2 √ 2 � |0⟩ � |00⟩ + |11⟩ + |01⟩ − |10⟩ � + |1⟩ � |00⟩ + |11⟩ + |10⟩ − |01⟩ �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (24) Obviously |c⟩ = 1 √ 2[|00⟩ + |11⟩ + |01⟩ − |10⟩] and |d⟩ = 1 √ 2[|00⟩ + |11⟩ + |10⟩ − |01⟩] are linearly independent, and they can construct two separable states |c⟩ + i|d⟩ = 1 + i √ 2 (|0⟩ + i|1⟩)(|0⟩ − i|1⟩), |c⟩ − i|d⟩ = 1 − i √ 2 (|0⟩ − i|1⟩)(|0⟩ + i|1⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (25) They span bipartite Hilbert space, so the state in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (24) is a three-qubit GHZ-type state, just be the same as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' Next, we introduce a back-up basis |ψ′3⟩ = |00⟩ + |11⟩ 2 |m−⟩ − |10⟩ − |01⟩ 2 |m+⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (26) Since |ψ′3⟩ = (σx)A3(−σz)A3|ψ3⟩, it is also a GHZ-type state, and is linearly independent with |ψ3⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' The states in |ψ3⟩ and |ψ′3⟩ construct a three-qubit basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' We should notice that the state in |ψ′3⟩ is not that kind final state produced in our apparatus, because it doesn’t satisfy the spatial symmetry described in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' Now we can construct a five-qubit GHZ-type state by three-qubit basis, |ψ5⟩ = |00⟩ + |11⟩ 2 |ψ3⟩ + |10⟩ − |01⟩ 2 |ψ′3⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (27) It can be written as |ψ5⟩ =1 4 ��� |00⟩ + |11⟩ � + i � |01⟩ − |10⟩ ��� |ψ3⟩ + i|ψ′3⟩ � + �� |00⟩ + |11⟩ � − i � |01⟩ − |10⟩ ��� |ψ3⟩ − i|ψ′3⟩ �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (28) So it is a GHZ-type state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' The five-qubit back-up basis is |ψ′5⟩ = |00⟩ + |11⟩ 2 |ψ′3⟩ − |10⟩ − |01⟩ 2 |ψ3⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (29) It has the same feature as |ψ′3⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' Then we can construct a seven-qubit GHZ-type state in the same way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' So generally, a (2N + 1)-qubit GHZ-type state can be presented as |ψ2N+1⟩ = |00⟩ + |11⟩ 2 |ψ2N−1⟩ + |10⟩ − |01⟩ 2 |ψ′2N−1⟩ = 1 4 ��� |00⟩ + |11⟩ � + i � |01⟩ − |10⟩ ��� |ψ2N−1⟩ + i|ψ′2N−1⟩ � + �� |00⟩ + |11⟩ � − i � |01⟩ − |10⟩ ��� |ψ2N−1⟩ − i|ψ′2N−1⟩ �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (30) As the criterion proposed in Theorem 3 shown, the (2N + 1)-qubit GHZ-type state |ψ2N+1⟩ can be produced in the symmetric setup, but the back-up basis |ψ′2N+1⟩ can not be generated in the apparatus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' 9 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' The 2N-qubit case The 2N-qubit case is similar as the (2N + 1)-qubit case, the two-qubit basis is |k+⟩ = 1 2 �� |00⟩ + |11⟩ � + i � |01⟩ + |10⟩ �� , |k−⟩ = 1 2 � i � |00⟩ + |11⟩ � + � |01⟩ + |10⟩ �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (31) The four-qubit GHZ-type state is |ψ4⟩ = |00⟩ + |11⟩ 2 |k+⟩ + |10⟩ + |01⟩ 2 |k−⟩, (32) just the same as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' Next, we introduce the four-qubit back-up basis |ψ′4⟩ = � |00⟩ + |11⟩ � |k−⟩ + � |10⟩ + |01⟩ � |k+⟩ = (σx)4|ψ4⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (33) It is GHZ-type and linearly independent with |ψ4⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' Different from |ψ′3⟩, the state in |ψ′4⟩ can be produced in the symmetric setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' Then we can construct a six-qubit GHZ-type state by four-qubit basis, |ψ6⟩ =|00⟩ + |11⟩ 2 |ψ4⟩ + |10⟩ + |01⟩ 2 |ψ′4⟩ =1 4 ��� |00⟩ + |11⟩ � − � |10⟩ + |01⟩ ��� |ψ4⟩ − |ψ′4⟩ � + �� |00⟩ + |11⟩ � + � |10⟩ + |01⟩ ��� |ψ4⟩ + |ψ′4⟩ �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (34) The six-qubit back-up basis is |ψ′6⟩ = |00⟩ + |11⟩ 2 |ψ′4⟩ + |10⟩ + |01⟩ 2 |ψ4⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (35) Generally, 2N-qubit GHZ-type states can be presented as |ψ2N⟩ = |00⟩ + |11⟩ 2 |ψ2N−2⟩ + |10⟩ + |01⟩ 2 |ψ′2N−2⟩ = 1 4 ��� |00⟩ + |11⟩ � − � |10⟩ + |01⟩ ��� |ψ2N−2⟩ − |ψ′2N−2⟩ � + �� |00⟩ + |11⟩ � + � |10⟩ + |01⟩ ��� |ψ2N−2⟩ + |ψ′2N−2⟩ �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (36) Different from the (2N +1)-qubit case, the states in |ψ2N⟩ and |ψ′2N⟩ both can be produced in the symmetric setup (details in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' A 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' It provides a new approach to generate multiqubit entangled states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' By getting (2N −2)-qubit basis and Bell states entangled, we can obtain 2N-qubit GHZ-type state |ψ2N⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' That means if we have produced (2N −2)-qubit GHZ-type states by gravity interaction, entangled states with more qubits are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' This approach can be applied in improving existing multipartite entangled platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' If we get rid of the restrict from spatial symmetry, the approach is also feasible for the (2N + 1)-qubit case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' In this section, we extend the gravitational entanglement to multiqubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' If we choose appropriate parameters in the apparatus, the masses can be transformed to GHZ-type entangled states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' Since the number of independent phases are constrained in symmetric setup, the apparatus can not produce 10 W-type states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' W-type states need more freedom of phases, that means less symmetry in the apparatus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' For example, if we change the distances d between masses in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' 1(i), such as d1 and d2, the phases of each item are independent, |ψ6⟩ = 1 2 √ 2 � eiφ′ 1|000⟩ + eiφ′ 2|001⟩ + eiφ′ 3|010⟩ + eiφ′ 4|100⟩ + eiφ′ 5|110⟩ + eiφ′ 6|101⟩ + eiφ′ 7|011⟩ + eiφ′ 8|111⟩ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (37) If we choose the phases as eiφ′ 1 = eiφ′ 2 = eiφ′ 3 = eiφ′ 4 = eiφ′ 8 = 1, eiφ′ 5 = −i, eiφ′ 6 = −1, eiφ′ 7 = i, the modified apparatus will produce W-type states, |ψ6⟩ = 1 2 √ 2 � |000⟩ + |001⟩ + |010⟩ + |100⟩ − i|110⟩ − |101⟩ + i|011⟩ + |111⟩ � = 1 2 √ 2 �� |0⟩ + |1⟩ � A1 ⊗ |00⟩A2A3 + � |0⟩ − |1⟩ � A1 ⊗ |01⟩A2A3 + � |0⟩ − i|1⟩ � A1 ⊗ |1⟩A2 ⊗ � |0⟩ + i|1⟩ � A3 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (38) The state in |ψ6⟩ can be transformed to W states under SLOCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' MEASURE OF ENTANGLEMENT In this section, we analyse and measure the gravitational entanglement by the GM of entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' GM is the closest distance between an entangled state and the set of separable states [15, 16], Λ2(ρ) = max |ϕ⟩∈PRO⟨ϕ|ρ|ϕ⟩, G(ρ) = −2 log Λ(ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (39) Here the logarithm has base two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' We just calculate GM of tripartite system for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' Suppose the three-qubit product states be |φ⟩ = (cos α|0⟩ + eiθ sin α|1⟩)A1 ⊗ (cos β|0⟩ + eiη sin β|1⟩)A2 ⊗ (cos γ|0⟩ + eiω sin γ|1⟩)A3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (40) Since there are too many parameters in the product states, we will consider the symmetric form of the entangled state for simplicity in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' IV A, and study the general form in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' IV B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' The conclusions are presented in Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' Theorem 2 When we use GM to measure the entanglement, the gravity induced entanglement can produce stable and robust entanglement for tripartite system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' The robust entanglement appears gen- erally at ∆ϕ3 ∈ [ 11π 16 , 21π 16 ] and the corresponding G(ρ) is close to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' We will show it in next subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' Symmetric situation First we consider the entangled states in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (10) be in symmetric form, that means ei∆ϕ2 = ei∆ϕ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' The states can be written as |ψ⟩ = eiϕ1 2 √ 2 � |000⟩ + |111⟩ + ei∆ϕ3� |001⟩ + |011⟩ + |100⟩ + |110⟩ + |010⟩ + |101⟩ �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (41) 11 TABLE I: The data of Λ2 and the parameters of the nearest product states for some certain relative evolution phases ∆ϕ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' ∆ϕ2 π 8 π 4 3π 8 π 2 5π 8 3π 4 7π 8 π 9π 8 5π 4 11π 8 3π 2 13π 8 7π 4 15π 8 Λ2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='89 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='77 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='625 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='52 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='52 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='625 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='77 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='89 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='97 α π 4 π 4 π 4 π 4 π 32 π 32 7π 32 π 4 7π 32 5π 16 5π 16 π 4 π 4 π 4 π 4 θ 0 0 0 0 27π 16 25π 16 π 2 π 2 3π 2 7π 16 5π 16 0 0 0 0 As been showed in Proposition 4 of [16], the closest product state to any symmetric state is necessarily symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' So the state in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (40) should be symmetric too (which means α = β = γ, θ = ω = η), can be written as |φ⟩ = (cos α|0⟩ + eiθ sin α|1⟩)⊗3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' The inner product of the entangled state and the product state is ⟨φ|ψ⟩ = eiϕ1 2 √ 2 � cos3 α + sin3 αe−3iθ + 3ei∆ϕ3 cos α sin αe−iθ(cos α + sin αe−iθ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (42) Its modular square is |⟨φ|ψ⟩|2 = 1 8 � cos6 α + cos5 α sin α � 6 cos(∆ϕ3 − θ) � + cos4 α sin2 α � 6 cos(∆ϕ3 − 2θ) + 9 � + cos3 α sin3 α(18 cos θ + 2 cos 3θ) + cos2 α sin4 α � 6 cos(∆ϕ3 + 2θ) + 9 � + cos α sin5 α � 6 cos(∆ϕ3 + θ) � + sin6 α � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (43) (i) 0 1 2 3 4 5 6 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='9 1 ∆φ3 Λ2 (ii) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' 2: (i) The colour stands for |⟨φ|ψ⟩|2, as the function of α, θ and ∆ϕ3 in one period, the colour bar gives out approximate value of |⟨φ|ψ⟩|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (ii) The blue curve represents the Λ2 as the function of ∆ϕ3, and the red curve represents the data fitting function in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (44).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (43) is a function of α, θ and ∆ϕ3, which is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' 2 (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' As the colour bar shows, for ∆ϕ3 close to zero, Λ2 get close to one, and the nearest product state appears around α = π/4, θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' The criterion proposed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' III A shows that the state in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (41) becomes a separable state for ∆ϕ3 = 0, so GM is vanishing and Λ2 = 1 (G(ρ)=0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' At the beginning of the gravitational interaction, the relative evolution phase ∆ϕ3 is small, and the entangled state in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (41) is near the product state, so Λ2 is big.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' With the interaction going on, the entanglement of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (41) increases, the distance between entangled state and the nearest product state increases, and Λ2 falls down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' With numerical computation, the blue curve in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' 2 (ii) describes Λ2 as the function of ∆ϕ3 only, the nearest product states for each ∆ϕ3 are different, the corresponding α and θ were partly listed in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' Both ends of the curve (∆ϕ3 < π 4 or ∆ϕ3 > 7π 4 ) correlate to weak entanglement 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='3 0 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='2 6 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='5 0 αt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='412 with Λ2 close to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' The nearest product states have stable phases α = π 4 and θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' In the rapid change ranges (∆ϕ3 ∈ [ π 4, π 2] and [ 3π 2 , 7π 4 ]) of the curve, Λ2 changes rapidly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' The nearest product states also are |φ⟩ = ( |0⟩+|1⟩ √ 2 )⊗3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' The middle range of the curve (∆ϕ3 ∈ ( π 2, 3π 2 )) represents the final states entangled strongly, with small Λ2 around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' However, the nearest product states are oscillating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' In this range, entanglement is great and stable, sensitive to witnesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' It is an important region for studying the multipartite entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' GM in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (39) becomes G(ρ) = 1 in this region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' Nevertheless, the apparatus can only generate small relative evolution phase ∆ϕ3 in [3], since the interaction can not last for a long time because of decoherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' If we want to detect robust entangled state, we can consider heavier masses as in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' When we detect some entangled states with certain phases ∆ϕ3, corresponding α and θ are listed in Table I, or we can use the fitting function of the curve to compute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' Λ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='164[arctan(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='71∆ϕ2 − 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='68) + arctan(−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='79∆ϕ2 + 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='83)] + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (44) The red curve in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' 2 (ii) presents the theoretical result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' General Situation In this subsection, we consider the product states be chosen as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (40), the inner product becomes ⟨φ|ψ⟩ = eiϕ1 2 √ 2 � cos α cos β cos γ + sin α sin β sin γe−i(θ+η+ω) + ei∆ϕ2� cos α cos β sin γe−iω + cos α sin β sin γe−i(η+ω) + sin α cos β cos γe−iθ + sin α sin β cos γe−i(θ+η)� + ei∆ϕ3� cos α sin β cos γe−iη + sin α cos β sin γe−i(θ+ω)�� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (45) Since there are many parameters in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (45), we study it by numerical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' We just present how Λ2 trends with ∆ϕ2 and ∆ϕ3 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' 3, because it is hard to give the exact function form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' The states in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (10) become product states when ei∆ϕ2 = ±1 and ei∆ϕ3 = 1, GM vanishes and the entanglement is weak in adjacent regions (the red parts in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' The nearest product states are |φ⟩ = 1 2 √ 2(|0⟩ ± |1⟩)A1 ⊗ (|0⟩ + |1⟩)A2 ⊗ (|0⟩ ± |1⟩)A3, with α = β = γ = π/4 and θ = ω = η = 0 (or θ = η = π, ω = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' There are also some regions presenting robust and stable entanglement in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' 3 (the blue parts).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' For different ∆ϕ2, the ranges of ∆ϕ3 (corresponding to blue parts) are a little different, we choose the intersection ∆ϕ3 ∈ [ 11π 16 , 21π 16 ], it is valid for arbitrary ∆ϕ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' In these regions, Λ2 is a little smaller than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='5, the maximum G(ρ) is about 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' The corresponding nearest product states vary with relative evolution phases ∆ϕ2 and ∆ϕ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' For example, when ∆ϕ2 = 2π and ∆ϕ3 = π, the nearest product states is |φ⟩ = 1 2 √ 2(|0⟩ + i|1⟩)A1 ⊗ (|0⟩ − i|1⟩)A2 ⊗ (|0⟩ + i|1⟩)A3, with α = β = γ = π/4 and θ = η = π/2, ω = 3π/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' In this section, we calculated GM of the gravitational entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' It depends on the relative evolution phases ∆ϕ2 and ∆ϕ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' With numerical computation, we constructed the phase map of GM, it marks the degree of entanglement of the final states for each couple of relative evolution phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' We found that the robust entanglement appears generally at ∆ϕ3 ∈ [ 11π 16 , 21π 16 ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' CONCLUSIONS In this article, we studied the multiqubit entanglement caused by gravity of neutral masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' The mutual gravitation interaction could transform the separable states into GHZ-type states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' It sug- 13 0 2 4 6 8 0 1 2 3 4 5 6 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='9 1 ∆φ3 ∆φ2 Λ2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' 3: The maximum of modular square Λ2 as the function of ∆ϕ2 and ∆ϕ3 in one period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' gested a way to construct N-qubit GHZ-type states by getting Bell states and (N − 2)-qubit GHZ- type states entangled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' This approach is not only feasible in gravity induced entanglement, but also can be applied in ions trap, because Coulomb force between ions acts in the same way as gravity except for a minus sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' Since electromagnetic interaction is much stronger than gravity, the ions’ entanglement is more feasible in experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' As described in [13], ions in a Paul trap can form a linear string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' The positions of them satisfy the spatial symmetry in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' For ions’ string, the evolution phases of superpositions are caused by oscillating electric field and mutual coulomb force between ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' With appropriate parameters, electric field induces common phase, and relative phases are only related to mutual interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' So the results in this article can be applied in ions trap to involve more ions in the entangled states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' That may be a feasible path to realize robust multiqubit entanglement and will play a part in quantum computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' We also calculated GM of three-qubit gravity induced entangled states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' This value measures the degree of entanglement for a certain final state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' We constructed functions to describe the relationship of GM and the relative evolution phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' The phase map of GM enables us to pick out the ranges with robust entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' We will pay attention on these regions in the future study of multipartite entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' It is helpful in experiment design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' Appendix A: The spatial symmetry As we can see in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' II, the masses split into two superpositions, components |L⟩ and |R⟩ keep stable distances from each other in the apparatus of symmetric setup (as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' 1(ii, iii) described).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' The mutual gravity interaction can induce different rates of phase evolution in the final state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (9) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (21), these phases are decided by the distances between |L⟩ and |R⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' For example, for the three-qubit case in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (10), the evolution phase of |001⟩ is ϕ2 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' It relates to the distances, including d between |0⟩A1 and |0⟩A2, √ d2 + l2 between |0⟩A2 and |1⟩A3, √ 4d2 + l2 between |0⟩A1 and |1⟩A3, as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' 4(i) described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' We check Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' 4 (ii, iii, iv), and find that |110⟩, |100⟩ and |011⟩ should have the same phase ϕ2, because the sum of distances in these cases are equivalent to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' 4(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' Other parts in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (10) reveal the same feature, |000⟩ and |111⟩ share the same phase ϕ1, |010⟩ and |101⟩ share the same phase ϕ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' However, ϕ1, ϕ2, ϕ3 are independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' That is the spatial symmetry we talked about in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' For the four-qubit case in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (20), we classify the state into six groups with in- dependent phases, � |0000⟩, |1111⟩ � , � |0001⟩, |0111⟩, |1000⟩, |1110⟩ � , � |0010⟩, |1101⟩, |0100⟩, |1011⟩ � , 14 (i) |001⟩ (ii) |110⟩ (iii) |100⟩ (iv) |011⟩ FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' 4: Vertical view of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' 1(iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' The dots stand for the superpositions of the first (A1), second (A2) and third (A3) masses from left to right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' The lines refer to the distances between components of the masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (i) |1001⟩ (ii) |0110⟩ FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' 5: In the four-qubit case, |1100⟩ and |0011⟩ have the same phase since they share the symmetric distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' � |1100⟩, |0011⟩ � , � |1010⟩, |0101⟩ � , � |1001⟩, |0110⟩ � , each group of states in one bracket have the same phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' The states in the last bracket are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' 5 for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' We find that the states in one bracket are symmetric if we exchange |0⟩ and |1⟩ or turn over the qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' We define some notations to help understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' First, we denote � |ψ⟩ as the invert state of |ψ⟩, that means � |0⟩ = |1⟩, � |1⟩ = |0⟩ for each qubit, for example if |ψ⟩ = |001⟩ then � |ψ⟩ = |110⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' Next, if we turn over the qubit in the state |ψ⟩, it becomes the turn over state |ψ⟩To, such as (|0⟩A|0⟩B|1⟩C)To = |1⟩A|0⟩B|0⟩C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' For example if |ψ⟩ = |101001⟩ + |110001⟩, the turn over state |ψ⟩To = |100101⟩ + |100011⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' We describe the spatial symmetry in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' Theorem 3 In the symmetric setup described by Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' 1 (ii, iii), the multipartite entangled state |ψ⟩ should obey spatial symmetry that each symmetric groups in the final state should share the same evolution phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' It equals to the statements � |ψ⟩ = |ψ⟩ and |ψ⟩To = |ψ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' The states we have constructed in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (30) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (36) under symmetric setup should follow Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' We will check whether they satisfy above statements from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (23) to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' The (2N + 1)-qubit case Obviously, the basis in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (23) satisfy, � |m+⟩ = |1⟩ + |0⟩ √ 2 = |m+⟩, |m+⟩To = |m+⟩, � |m−⟩ = |1⟩ − |0⟩ √ 2 = −|m−⟩, |m−⟩To = |m−⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (A1) 1 1 1 1 0 0 0 00 0 01 0 0 01 1 1 0 01 1 1 0 0 01 1 0 0 0 015 For three-qubit basis in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (24) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (26), the invert states are � |ψ3⟩ = |11⟩ + |00⟩ 2 � |m+⟩ + |01⟩ − |10⟩ 2 � |m−⟩ = |ψ3⟩, � |ψ′3⟩ = |11⟩ + |00⟩ 2 � |m−⟩ − |01⟩ − |10⟩ 2 � |m+⟩ = −|ψ′3⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='(A2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='The turn over state of |ψ3⟩ is ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='|ψ3⟩To = |m+⟩To|00⟩ + |11⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='+ |m−⟩To|01⟩ − |10⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='|0⟩ + |1⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='|00⟩ + |11⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='|0⟩ − |1⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='|01⟩ − |10⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='�� ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='|00⟩ + |01⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='+ |1⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='|11⟩ + |10⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='+ |0⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='|11⟩ − |10⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='+ |1⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='|00⟩ − |01⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='|00⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='|0⟩ + |1⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='+ |11⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='|1⟩ + |0⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='+ |01⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='|1⟩ − |0⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='+ |10⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='|0⟩ − |1⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='|00⟩ + |11⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='|0⟩ + |1⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='|10⟩ − |01⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='|0⟩ − |1⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='= |ψ3⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (A3) Similar, we have |ψ′3⟩To = |m−⟩To|00⟩ + |11⟩ 2 − |m+⟩To|01⟩ − |10⟩ 2 = |ψ′3⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (A4) For five-qubit basis in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (27) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (29), the invert states are � |ψ5⟩ = |11⟩ + |00⟩ 2 � |ψ3⟩ + |01⟩ − |10⟩ 2 � |ψ′3⟩ = |ψ5⟩, � |ψ′5⟩ = |11⟩ + |00⟩ 2 � |ψ′3⟩ − |01⟩ − |10⟩ 2 � |ψ3⟩ = −|ψ′5⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (A5) The turn over state of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (27) is ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='|ψ5⟩To = |ψ3⟩To|00⟩ + |11⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='+ |ψ′3⟩To|01⟩ − |10⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='|00⟩ + |11⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='|0⟩ + |1⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='� ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='|0⟩ + |1⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='|01⟩ − |10⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='= |00⟩ + |11⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='|ψ3⟩To + |10⟩ − |01⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='|ψ′3⟩To = |00⟩ + |11⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='|ψ3⟩ + |10⟩ − |01⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='|ψ′3⟩ = |ψ5⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (A6) We can show |ψ′5⟩To = |ψ′5⟩ in the same way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' With mathematical induction, we can deduce |ψ7⟩ and |ψ′7⟩ satisfy the same statements, and so do (2N + 1)-qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' So the (2N + 1)-qubit final states we constructed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' III C obey Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' We should notice that, the back-up basis |ψ′2N+1⟩ does not obey Theorem 3, so it can not be generated in the symmetric setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' 16 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' The 2N-qubit case Now we consider the two-qubit basis in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (31), � |k+⟩ = 1 2 �� |11⟩ + |00⟩ � + i(|10⟩ + |01⟩ �� = |k+⟩, |k+⟩To = 1 2 �� |00⟩ + |11⟩ � + i(|10⟩ + |01⟩ �� = |k+⟩, � |k−⟩ = 1 2 � i � |11⟩ + |00⟩ � + (|10⟩ + |01⟩ �� = |k−⟩, |k−⟩To = 1 2 � i � |00⟩ + |11⟩ � + (|10⟩ + |01⟩ �� = |k−⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (A7) The four-qubit basis in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (32) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (33) satisfy, � |ψ4⟩ = |11⟩ + |00⟩ 2 � |k+⟩ + |10⟩ + |01⟩ 2 � |k−⟩ = |ψ4⟩, � |ψ′4⟩ = |11⟩ + |00⟩ 2 � |k−⟩ + |10⟩ + |01⟩ 2 � |k+⟩ = |ψ′4⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (A8) |ψ4⟩To = |k+⟩To|00⟩ + |11⟩ 2 + |k−⟩To|10⟩ + |01⟩ 2 = 1 4 ��� |00⟩ + |11⟩ � + i � |01⟩ + |10⟩ ��� |00⟩ + |11⟩ � + � i � |00⟩ + |11⟩ � + � |01⟩ + |10⟩ ��� |10⟩ + |01⟩ �� = 1 4 �� |00⟩ + |11⟩ ��� |00⟩ + |11⟩ � + i � |10⟩ + |01⟩ �� + � |01⟩ + |10⟩ �� i � |00⟩ + |11⟩ � + � |10⟩ + |01⟩ ��� = |00⟩ + |11⟩ 2 |k+⟩ + |01⟩ + |10⟩ 2 |k−⟩ = |ψ4⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' (A9) We can show |ψ′4⟩To = |ψ′4⟩ similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' Inducing in the same way, 2N-qubit basis also satisfy Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' So 2N-qubit final states con- structed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' III D can be produced in the symmetric setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' In this case, all back-up bases |ψ′2N⟩ obey Theorem 3 too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' Acknowledgements Authors thank the interesting discussion with Professor Qilin Zhang and Mingguo Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' MFL and LC were supported by the NNSF of China (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' 11871089), and the Fundamental Research Funds for the Central Universities(Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' ZG216S2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' [1] Overstreet C, Asenbaum P, Curti J, Kim M, Kasevich MA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' Observation of a gravitational Aharonov-Bohm effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' Science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' 2022 Jan 14;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='375(6577):226-229.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='1126/science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='abl7152.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' On two recent proposals for witnessing nonclassical gravity[J].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' Journal of Physics A Mathematical and Theoretical, (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' [24] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' Kemal D¨oner, Andr´e Groβardt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' Is gravitational entanglement evidence for the quantization of spacetime?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content=' arXiv: 2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} +page_content='00939v1 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfHQ4M/content/2301.05437v1.pdf'} diff --git a/mNE5T4oBgHgl3EQfiw-y/vector_store/index.faiss b/mNE5T4oBgHgl3EQfiw-y/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..d476944da0703fcd23c8eebaadff62b4d01d4e53 --- /dev/null +++ b/mNE5T4oBgHgl3EQfiw-y/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:d3d27155d78dd606d7c618426fc5b079abe15310567a73826000b832ee402bd2 +size 4194349 diff --git a/oNAyT4oBgHgl3EQflfjb/content/tmp_files/2301.00455v1.pdf.txt b/oNAyT4oBgHgl3EQflfjb/content/tmp_files/2301.00455v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..6cb928559a99d6e92ed21cdc2d13ff45a50931cd --- /dev/null +++ b/oNAyT4oBgHgl3EQflfjb/content/tmp_files/2301.00455v1.pdf.txt @@ -0,0 +1,971 @@ +Refraction beats attenuation in breast CT +MICHAŁ RAWLIK1,2,*, ALEXANDRE PEREIRA1,2, SIMON SPINDLER1,2, ZHENTIAN WANG1,3, +LUCIA ROMANO1,2, KONSTANTINS JEFIMOVS2, ZHITIAN SHI1,2, MAXIM POLIKARPOV1,2, JINQIU +XU1,2, MARIE-CHRISTINE ZDORA1,2, STEFANO VAN GOGH1,2, MARTIN STAUBER4, EDUARDO +YUKIHARA2, JEPPE B. CHRISTENSEN2, RAHEL A. KUBIK-HUCH5, TILO NIEMANN5, CORNELIA +LEO6, ZSUZSANNA VARGA7, ANDREAS BOSS8, AND MARCO STAMPANONI1,2 +1Institute for Biomedical Engineering, ETH Z¨urich and University of Z¨urich, Switzerland +2Paul Scherrer Institute, Villigen, Switzerland +3Department of Engineering Physics, Tsinghua University, Haidian District, 100080 Beijing, China +4GratXray AG, Villigen, Switzerland +5Department of Radiology, Kantonsspital Baden, Switzerland +6Interdisciplinary Breast Center, Kantonsspital Baden, Switzerland +7Department of Pathology and Molecular Pathology, University Hospital Z¨urich, Switzerland +8Institute for Diagnostic and Interventional Radiology, University Hospital Z¨urich, Switzerland +*corresponding author, email: mrawlik@ethz.ch +January 3, 2023 +For a century, clinical X-ray imaging has visualised only the attenuation properties of tissue, which fundamentally limits +the contrast, particularly in soft tissues like the breast. Imaging based on refraction can overcome this limitation, but so +far has been constrained to high-dose ex-vivo applications or required highly coherent X-ray sources, like synchrotrons. +It has been predicted that grating interferometry (GI) could eventually allow computed tomography (CT) to be more +dose-efcient. However, the benefit of refraction in clinical CT has not been demonstrated so far. Here we show that +GI-CT is more dose-efcient in imaging of breast tissue than conventional CT. Our system, based on a 70 kVp X-ray tube +source and commercially available gratings, demonstrated superior quality, in terms of adipose-to-glandular tissue +contrast-to-noise ratio (CNR), of refraction-contrast compared to the attenuation images. The fusion of the two modes +of contrast outperformed conventional CT for spatial resolutions better than 263 µm and an average dose to the breast +of 16 mGy, which is in the clinical breast CT range. Our results show that grating interferometry can significantly reduce +the dose, while maintaining the image quality, in diagnostic breast CT. Unlike conventional absorption-based CT, the +sensitivity of refraction-based imaging is far from being fully exploited, and further progress will lead to significant +improvements of clinical X-ray CT. +In 2020, breast cancer was the most commonly diagnosed +cancer overall, with over two million cases. +Among +women it makes up 24.5 % of the cancer cases and 15.5 % +of the cancer-related deaths1. The prevalence of breast +cancer has prompted most developed countries to estab- +lish mammography screening programmes, which have +been shown to reduce mortality2,3. However, the effec- +tiveness of mammography is disputed. A retrospective +study found that only 46 % of screen-detected cancers are +true positives, while 22 % are missed4. +The reason is that mammography images are difficult +to read. Not only does the soft tissue in the breast provide +limited X-ray contrast, but also the complicated morphol- +ogy of the breast is ambiguous when rendered in a two- +dimensional projection. Even among experienced read- +ers, the agreement in identifying masses is far from per- +fect (κ = 0.67)5. This is despite the painful measure of +compressing the breast to spread it out on the image and +make it thinner, so that the contrast can be improved by +using lower-energy X-rays. +The shortcomings of other breast-imaging modali- +ties have so far hindered their widespread use. +Dig- +ital Breast Tomosynthesis partially removes the tissue- +overlap-related ambiguity but it has been shown to pro- +vide only an incremental improvement over mammog- +raphy6. +Breast ultrasound plays mainly a supportive +role and, while MRI provides excellent contrast, its res- +olution is lower than mammography, it cannot visualise +microcalcifications and the modality is expensive, time- +consuming and uncomfortable. +Dedicated breast CT has recently been introduced in +clinical practice with promising results, summarised in a +recent review7. With the volumetric data, the tissue over- +lap is completely alleviated, and the breast does not need +to be compressed. In the review, its authors identify the +major shortcoming of the method to be the near identi- +cal attenuation contrast between breast tumours without +microcalcifications and glandular parenchyma. +Like visible light, X-rays are not only attenuated but +also refracted when traversing matter. In the last years, +several methods to detect the refraction of X-rays, the +phase contrast (PC), have been developed. In the con- +text of breast imaging, three are noteworthy. +One is +propagation-based PC, which does not need additional op- +1 +arXiv:2301.00455v1 [physics.med-ph] 1 Jan 2023 + +Fig. 1. The principle of GI-CT. a An illustration of the imaging setup. The formation of the Talbot interference pattern behind +G1 and its distortion by the breast are shown schematically. b The intensity of a monochromatic wavefront in a GI-CT system +in a horizontal plane. The G0 absorption grating acts as an array of vertical slits, each providing enough spatial coherence in +the transverse direction so that an interference pattern is created behind the periodically-phase-shifting, by π, G1 grating. At +the position of G2 the interference produces a pattern of parallel bright-and-dark lines. The period of G0 is chosen such, that +the line patterns from its slits add constructively (but incoherently). The period of the line pattern is smaller than the pixel size +of the detector, so an absorption grating G2, with the same period as the line pattern, is used to analyse it. The refraction of +X-rays in the sample distorts the line pattern, which results in an intensity change behind the G2 grating. The gratings are shown +schematically, and their lamellae are vertical, i.e. perpendicular to the plane of the image. For illustration purposes the image is +stretched and shows only an approximately 600 µm-wide part of the system. c Change in the intensity incident on pixels as G0 is +shifted: without interaction with the object (blue), with pure attenuation interaction (orange) and with refraction (green). d A slice +of a phantom reconstructed from the attenuation signal in a simulated GI-CT measurement. The yellow arrow points to a small +feature, not discernible in the attenuation image. e The phase reconstruction from the same simulated measurement. The small +feature can be recognised, but low-spatial-frequency noise is prominent. f GI-CT fusion combines the low spatial frequencies of +the attenuation reconstruction, where it has lower Noise Power Spectrum (NPS), with the high ones of the phase reconstruction +– the regime where the phase has lower NPS. g The fused GI-CT image. The feature pointed out by the arrow is visible and the +low-spatial-frequency noise is suppressed. +2 + +phase +attenuation +0100mm +fused +G1 +G2 +GO +△Itical elements, but requires the high spatial coherence of a +synchrotron or a specialised laboratory source. The other +two, edge illumination8 and grating interferometry (GI)9, +use optical elements, and work with standard X-ray tube +sources with large spot sizes. We discuss GI further, and +for a broad overview of PC imaging we direct the reader +to a recent review10. +The principle of GI is illustrated in Fig. 1a and b. Let us +assume for the moment that the G0 element is a narrow +slit that provides spatial coherence for the X-ray beam. +G1 is a grating that introduces a periodic π-shift in the +beam, which results downstream in an interference pat- +tern of parallel bright and dark lines. Refraction on large- +scale structures in the sample, the phase contrast, causes +the pattern to shift laterally. Diffusion by refraction on +small-scale structures beyond the resolving power of the +detector, the dark-field (DF) signal, blurs the line pattern. +For X-rays, the refraction angles are in the microradian +range requiring the period of the pattern (and G1) to be +in the order of a few micrometres. A pattern this small +cannot be resolved directly with standard large-scale de- +tectors, so a periodically opaque analyser G2 with the pe- +riod matching the one of the pattern is used. The final +observation is that G0 can, in fact, also be a periodically- +opaque grating: an array of slits with the spacing chosen +so that the line patterns created by each add construc- +tively (but incoherently) in the plane of the analyser9. +The gratings are decisive when it comes to the sensi- +tivity of GI to the refraction of X-rays. The minimal de- +tectable refraction angle is proportional to the pitch of +G2, favouring pitch sizes in the few-micrometre range, as +well as to the contrast in the interference pattern analysed +by G2, called the visibility11. To periodically block X-rays +with the energy in the clinical regime, heavy-element, +like gold, lines of 100 µm–200 µm thickness are necessary. +Fabrication of these high-aspect-ratio (line-thickness–to– +half-pitch–ratio) microstructures with sufficient quality is +challenging. Deep X-ray lithography (LIGA) can man- +ufacture gratings with thick gold lines (200 µm but the +pitch size is limited to several micrometres12,13. Silicon- +based manufacturing is now pushing the aspect ratio for +the pitch size in a micrometre14 and sub-micrometre15,16 +regimes, for both etching of silicon template and gold fill- +ing17. +The additional refraction information that GI provides +is attractive for breast imaging. PC promises higher con- +trast18 for better differentiation of tissues and DF was +shown to distinguish benign from malignant calcifica- +tions19,20. Two-dimensional GI mammography is already +close to first clinical trials21. +It is natural to pursue the extension of PC to 3D, given +the advantages it has shown in two-dimensional imag- +ing. It has been demonstrated that the benefit of PC-CT +depends on the spatial resolution or, for a fixed contrast- +to-noise ratio, equivalently, the dose. It was predicted +that with the increase of sensitivity through the progress +in the technology of grating fabrication high-resolution +breast imaging would be the first clinical area where PC- +CT will be advantageous22,23. +So far PC-CT of the breast tissue has been investi- +gated in the high-dose, high-resolution context of virtual +histopathology, demonstrating a clear benefit over atten- +uation18,24,25. +Propagation-based PC breast imaging is +pursued at the Elettra synchrotron in Trieste26 and the +Australian Synchrotron27, but the method fundamentally +relies on the high coherence of synchrotron X-ray beams +and cannot be translated to a wide-spread clinical use. +Small systems for propagation-based PC have been built, +but the low power of the required microfocal sources re- +sults in very long scan times28. Edge-illumination was +demonstrated for ex-vivo studies of larger specimens29. +A recent review can be found in30. The predicted tipping +point of the advantage of clinically-applicable phase con- +trast has not yet been reported. +We have constructed a GI-CT system with the aim to +demonstrate additional value of GI in clinical breast CT. +We have imaged a formalin-fixed human-breast speci- +men and compared the dose efficiency of the two GI-CT +contrasts, attenuation and phase, for a range of resolu- +tions, as well as the combination of the GI-CT contrast +versus conventional, attenuation-based CT. +Results +The 1.8 m-long GI-CT system consisted of a tungsten- +anode X-ray source operated at 70 kVp, +a photon- +counting detector with an active area of 195 × 19.2 mm2 +and a Talbot–Lau interferometer designed for the photon +energy of 46 keV in a symmetric geometry, with all grat- +ings being 4.2 µm-pitch. The system is depicted in Fig. 2 +and described in detail in the Methods section. +We imaged a formalin-fixed human breast from a body +donation with the average dose delivered to the speci- +men in the range of 5.5 mGy–219 mGy. The axial slices +of the reconstructed volumes for both attenuation and +phase contrast at mean absorbed dose to the breast of +11 mGy, 22 mGy and 219 mGy are shown in Fig. 3. The +visual quality of both contrasts increases with the deliv- +ered dose, which is particularly clear in the insets show- +ing an enlarged portion of the image. Moreover, even +though the phase-contrast image for the lowest dose ap- +pears visually inferior to the attenuation-contrast one, for +the higher dose it appears superior. +We analysed the images quantitatively following an +approach inspired by Raupach et al.23. +We assumed +that, in order to resolve the morphology of the breast, +a contrast-to-noise ratio (CNR) of 5 between the adi- +pose and glandular tissue is necessary (the Rose crite- +rion31). With decreasing dose, and consequently increas- +ing noise, a smoother filtering is necessary to reach this +CNR value. The additional point spread function (PSF) +of the filtering introduces limits to the imaging resolu- +tion. For each image, we determined the full-width-half- +maximum (FWHM) size of an isotropic Gaussian-blur +kernel necessary to achieve the CNR of 5. The results +depicted in Fig. 4 can be interpreted as the dose neces- +sary to resolve the morphology as a function of the reso- +lution. We observed that the dose requirement rises with +the power of 3.44 of the inverse kernel size for attenua- +tion and 1.56 for phase. The two curves, having different +3 + +Fig. 2. The GI-CT setup. From the left: the tungsten-anode X-ray source operated at 70 kVp, 3 mm Aluminium filter, the G0 atten- +uation grating, the phase-shifting G1 grating, the breast in a sample holder mounted on a rotation and vertical stages, the array +of three G2 analyser gratings, and the detector. +attenuation +11 mGy +5 mm +2 cm +22 mGy +5 mm +2 cm +219 mGy +2 cm +5 mm +−200 +−100 +0 +100 +HU +phase +11 mGy +5 mm +2 cm +22 mGy +5 mm +2 cm +219 mGy +2 cm +5 mm +−210 +−100 +0 +100 +200 +HUp +−300 +−200 +−100 +0 +100 +200 +HU / HUp +Fig. 3. The axial slices of reconstructed volumes of the human-breast specimen. +The attenuation contrast is shown in the top +row, the phase contrast in the bottom row. Three values of the average dose absorbed in the breast are depicted in the columns: +11 mGy, 22 mGy and 219 mGy. Visually, the image quality increases with the dose for both contrasts, but for the phase one it +increases faster. At the dose of 219 mGy, the image quality of phase contrast is superior, which is visible particularly well in the +enlarged region in the fourth column. In the top-left image, the regions of interest used to calculate the CNR are marked. +4 + +X-RAY +FILTER +GO +G1 +BREAST +G2 +DETECTOR +SOURCE +3mm Al +phase-shifting +attenuation i +analyser +70kVp +grating +grating +grating0600 +400 +300 +200 +150 +100 +isotropic resolution limit (µm) +5 +10 +20 +30 +40 +60 +100 +200 +dose for CNR of 5 (mGy) +phase +attenuation +grating-interferometry CT +clinical +breast CT +conventional-CT +equivalent +limit with +ideal gratings +Fig. 4. The average absorbed dose requirement of the conven- +tional CT and GI-CT imaging of the breast as a function of the +resolution. Each point represents an image (like in Fig. 3) for +which the minimal FWHM size of the isotropic Gaussian kernel to +reach the contrast-to-noise ratio (CNR) of 5 was found. The ker- +nel size constitutes a lower limit on the resolution. The best-fit +lines of the attenuation contrast (slope 3.44) and phase contrast +(slope 1.56) intersect at the kernel size of 214 µm and dose of +65 mGy. The ones of the conventional-CT equivalent (the atten- +uation signal with half the dose, as G2 would not be necessary; +slope 3.44) and fused GI-CT (slope 2.30) intersect at 263 µm and +16 mGy. The intersection can be interpreted as the point beyond +which GI-CT delivers superior image quality per unit dose. No- +tably, it is within the range of clinical breast CT. The numerically- +derived limit for fused GI-CT with ideal gratings, and otherwise +the same geometry, is also shown. The points for phase below +16 mGy are omitted for clarity. +slopes, intersect at a kernel size of 214 µm, which can be +interpreted as the resolution above which, on our system, +phase-contrast imaging is more dose-efficient than atten- +uation. +Due to the differential nature of phase contrast in +GI, the reconstructed volumes exhibit different in-plane +noise properties than attenuation contrast22. The noise +power spectrum of PC-CT, in comparison with attenua- +tion CT, is smaller for high spatial frequencies, but larger +for low frequencies (Fig. 1f). This observation, in addi- +tion to the fact that the two contrasts come from a sin- +gle acquisition, and are therefore naturally registered, +motivates fusing the images32. Because attenuation and +phase carry fundamentally different information there is +no generic way to fuse them. However, in the context +of a particular imaging task, which we define to be dif- +ferentiating glandular and adipose tissue with the maxi- +mal possible CNR, we used a simple scheme, illustrated +in Fig. 1f and g. First, we normalised the reconstructed +phase-contrast volume so that the grey levels of the adi- +pose and glandular tissues were equal to the ones in the +attenuation. Then, we added the high-pass filtered phase +volume to the low-pass filtered attenuation one, both us- +ing the same in-plane Gaussian kernel. We found the op- +timal size of the kernel, the one maximising the CNR of +the fused image, to be close to σ = 1.5 px for all measure- +ments. The quantitative CNR analysis of the fused im- +ages, depicted in Fig. 4, showed that in the investigated +range of kernel sizes they approximately follow a power +law with the exponent of 2.30 lying between the one of +the images derived from attenuation and phase contrast. +Moreover, the dose requirement is lower than the ones of +each of the single-contrast images everywhere. +Without the G2 grating, which attenuates approxi- +mately half of the photon flux downstream of the spec- +imen, it would be possible to acquire an attenuation im- +age with the same photon-counting statistics at half the +dose. We therefore consider the attenuation contrast with +half the dose to be approximately equivalent to a con- +ventional CT image. The comparison of the CNR-dose- +efficiency of the fused GI-CT and the conventional-CT +equivalent, depicted in Fig. 4, shows that the former out- +performs the latter for isotropic kernel sizes sharper than +263 µm and doses larger than 16 mGy (for CNR = 5). In +Fig. 5, we show a comparison of an enlarged fragment +of fused-GI-CT and conventional-CT-equivalent images. +While at the dose of 22 mGy the benefit of GI-CT is not +visually impressing, it increases with the dose, and it be- +comes evident at 66 mGy, in particular for small features. +We would like to point out that the low-frequency con- +trast information in the fused image comes from the at- +tenuation contrast and thus for both images the quanti- +tative unit is the one of attenuation, that is Hounsfield’s +(HU). +Discussion +Soon after the first demonstration of grating interferom- +etry with a regular, tube-based X-ray source9 it has been +predicted that the method can greatly improve clinical CT +imaging, in particular of the breast23. Despite a decade of +intensive research, the practical demonstration has been, +so far, elusive. With the aim to settle the long discussion +we have constructed a GI-CT system with a tube-based +X-ray source operated at a typical breast-CT energy of +70 kVp, a 10 cm-wide field of view, and a Talbot–Lau in- +terferometer based on commercially-available gratings. +We evaluated its performance by imaging a formalin- +fixed human breast specimen. +We could show that the additional information pro- +vided by refraction led to two breakthroughs. Firstly, the +phase-contrast images are superior, in terms of adipose- +to-glandular tissue CNR per unit dose, than attenuation +for kernels sharper than 214 µm. +Previous demonstra- +tions of refraction-based CT imaging of the breast with +a large field of view either did not show the benefit over +attenuation29 or relied on a synchrotron26, which greatly +limits the applicability. +Secondly, even though GI-CT utilises only half of +the photon flux compared to conventional CT due to +the absorbing analyser grating, the combination of the +attenuation- and phase-contrast signals provides suffi- +cient information to compensate the loss already at a +kernel size of 263 µm and dose of 16 mGy (for CNR of +5), both of which lie in their respective ranges used in +the clinics: +150 µm–300 µm and 5.8 mGy–26.1 mGy7,33. +For sharper kernels GI-CT outperforms conventional +5 + +conventional-CT equivalent +22 mGy +5 mm +66 mGy +5 mm +GI-CT +5 mm +5 mm +−200 +−150 +−100 +−50 +0 +50 +100 +HU +Fig. 5. A comparison of conventional-CT equivalent and fused +GI-CT images +At the dose of 22 mGy the additional information +coming from refraction allows GI-CT to just about overcome the +reduction in statistics coming from the G2 absorbing half of the +photon flux and the image quality is comparable to that of a +conventional-CT equivalent. At 66 mGy the image quality of GI- +CT is superior, which is visible particularly well in small features, +like the one indicated by the arrow. +attenuation-based CT with increasing benefit, requiring +only 53 % of the dose at 150 µm. +Further improvements are possible. +The limit of +the sensitivity of GI-CT to refraction is driven by the +microfabrication of the gratings. +Even the currently +commercially-available gratings performed already well +enough for GI-CT to outperform conventional CT. The +performance limit under an assumption of defect-free +gratings, depicted in Fig. 4, suggests that GI-CT could re- +quire a factor of 2 to 3 less dose than conventional CT +in the range of clinical breast CT. Improvements in the +grating-fabrication technology will take GI-CT closer to +that limit and, with smaller grating pitches, possibly be- +yond it. +Our aim was to compare the phase and attenuation +contrasts on an even ground. Here, we have therefore +deliberately kept the analysis to a minimum, refraining +from the use of iterative reconstruction algorithms, reg- +ularisation and post-processing. The use of those meth- +ods is likely to improve the performance as we quanti- +fied it in Fig. 4, and a more specific analysis will be avail- +able34. In particular, we advice caution in comparing the +values with other results obtained with elaborate analy- +sis. The particularity of these advanced analysis methods +and their differences for the phase and attenuation con- +trasts would, in our opinion, only weaken our otherwise +general conclusion: that GI-CT provides fundamentally +more information to start with. +We have demonstrated that GI-CT is a new relevant +clinical imaging modality, which can be more dose- +efficient than conventional CT. X-ray grating interferom- +etry, unlike other imaging techniques exploiting refrac- +tion, is compatible with conventional medical CT scan- +ners35 and, therefore, suitable for widespread use in hos- +pitals. The technique is immediately applicable to dedi- +cated breast CT systems, for which we have shown that +it already offers an improvement. In the future, GI could +allow dose reduction in all aspects of clinical CT. +Methods +Measurement setup +The measurement system consisted of a +Comet MXR-225HP/11 tungsten-anode X-ray source operated +at 70 kVp and 10 mA for the 22 mGy–222 mGy measurements +and 2.5 mA for the 5.5 mGy–16.5 mGy ones. The size of the fo- +cal spot was measured by the manufacturer to be 250 µm (at +30 % drop). +The X-ray beam was filtered with a 3 mm-thick +aluminium plate. The images were recorded with a photon- +counting detector with 750 µm-thick CdTe sensor and 75 µm +pixel size, which was manufactured by Dectris AG, Switzer- +land. The sensor size was 3072 × 256 pixels, but only an area +of 2600 × 256 pixels could be used. The Talbot-Lau interferom- +eter was configured in a 5th-Talbot-order symmetric geometry +with a G1 designed to introduce a phase shift of π at 46 kV. The +G0–G1 and G1–G2 distances were both 818.1 mm. The source– +G0 distance was 100 mm. G0 and G1 were a single piece each, +and for G2 three gratings were tiled together. All gratings were +bent around the vertical axis going through the X-ray source’s +focal spot. For phase-stepping, G0 was moved with a Physik– +Instrumente P-841.6B piezo actuator. The detector was 1756 mm +and the rotation centre 1003 mm away from the source. +Gratings +The 4.2 µm-pitch attenuation gratings G0 and G2 +had gold lamellae electroplated onto a graphite substrate, +and were manufactured with the LIGA process by Mi- +croworks GmbH, Germany. The gratings had a duty cycle of +0.5, and gold thickness was in the range of 150 µm–180 µm. The +polymer template was not stripped. +The π-shifting 4.2 µm-pitch phase grating G1 was manufac- +tured on a double side polished 8-inch silicon wafer by deep +reactive ion etching in a SPTS Rapier system. +A pattern in +MEGAPOSIT SPR220-3.0 positive tone photoresist was realised +by direct laser writing (Heidelberg DWL66+) (see36 for further +details). The process was optimised to ensure uniform etch- +ing depth and vertical trench sidewalls, as reported in14. The +G1 grating had a duty cycle of 0.5, and the grating lines were +59 µm thick; The thickness of the remaining silicon substrate +was 240 µm. A single tiled G1 grating was diced out from the +wafer to a size of 203 mm × 75 mm. +Specimen +The female breast specimen was a human breast tis- +sue from an adult autopsy after a body donation for research +(ethical agreement KEK-2012 554). It was without any grossly +visible pathology, and was fixed in 10 % buffered formaldehyd. +The specimen was vacuum-sealed in a plastic bag and placed in +a cylindrical PMMA container with 100 mm outer and 90 mm +inner diameter. The container was filled with water to avoid air +gaps. +6 + +Measurement protocol +During the CT scan, the specimen re- +volved continuously at 1 rpm for five full rotations while the +frames were acquired at 20 Hz. After each rotation, the G0 grat- +ing was shifted by one sixth of its period. The five-rotations pro- +tocol was repeated ten times with the 10 mA tube current, and +three with 2.5 mA, and was interleaved with a reference phase- +stepping measurements with the sample out of the beam. The +ten-repetition scan took 1.5 h of wall-time. +Dose estimation +The term dose was used to indicate the +mean absorbed dose to the breast, approximated as the ab- +sorbed dose to a 0.25–0.75 volumetric mixture of the ICRU44 +glandular and adipose tissues37, homogeneously distributed +in a PMMA cylinder with a 100 mm outer diameter and a +90 mm inner diameter. The absorbed dose was estimated by +the means of Monte Carlo simulations (GEANT438), where the +simulation geometry and source parameters were validated +through measurements using BeO optically stimulated lumi- +nescence dosimeters (OSLDs) and LiF:Mg,Ti thermolumines- +cence dosimeters (TLDs)39. The OSLDs and TLDs were cali- +brated in dose-to-water using a ISO N-60 photon field (mean +energy 47.9 keV)40 to approximate the mean energy of the X-ray +field used for imaging (mean energy 43.6 keV). The photons in +the simulation were sampled from a spectrum approximating +the X-ray tube in the experiment. The source was collimated +to match the extent of the absorption gratings G0 and G1 out- +lined in Fig. 1. In a first experiment, the OSLDs and TLDs were +placed upstream of the PMMA cylinder on the beam axis to +establish a conversion factor between the simulated dose-per- +primary to the absorbed dose to the dosimeters for a 10 min +irradiation. In a second experiment, which served to validate +the Monte Carlo model, the TLDs were placed on both exter- +nal sides of the cylinder, upstream and downstream. The mea- +sured doses to water (145 mGy upstream and 11.1 mGy down- +stream for a 10 min irradiation at 10 mA) were in agreement +with the simulated dose-to-water in volumes matching the lu- +minescence detectors (151 mGy upstream and 10.8 mGy down- +stream), thus validating the Monte Carlo model parameters. +The Monte Carlo model was then used to score the dose to the +homogeneous 0.25–0.75 volumetric mixture of the ICRU44 glan- +dular and adipose tissues placed in the PMMA cylinder. The +volumetric fractions were established with a threshold-based +segmentation of the reconstructed volumes. The mean dose to +the tissue mixture placed in the PMMA cylinder was calculated +to be 21.9(24) mGy, which corresponds to a 5 min-long CT mea- +surement series at 10 mA. +Data processing +The sinograms corresponding to the five ro- +tations with different G0 positions xj were stacked and we per- +formed a signal-retrieval with a linear least-squares fitting of a +sine to find the phase φi, visibility vi and intensity Ii in each ith +pixel: +Ij +i = Ii +2 +� +vi sin +�2π +p xj − φi +� ++ 1 +� +. +(1) +We used an overarching least-squares optimisation to find the +best-fit period of the sine p common to all pixels. The reference +measurements, acquired between the tomography scans, were +analysed in the same way to obtain the reference maps φr +i , vr +i +and Ir +i . We then constructed the attenuation pI +i and differential- +phase-contrast (DPC) pφ +i sinograms taking as the reference the +average of the two adjacent reference scans: +pI +i = − log +� Ii +Ir +i +� +, +pφ +i = φi − φr +i . +(2) +The attenuation sinogram pI was corrected for beam-hardening +effects, for which we used a separate measurement of PMMA +slabs of different thicknesses. We further applied to the attenu- +ation sinogram a ring-removal algorithm based on a combined +wavelet-FFT filter with damping of 1, 3 wavelet transform lev- +els and a db5 wavelet filter41. The attenuation volume was re- +constructed with the FDK algorithm, for the phase-contrast we +first used the Hilbert filter and then back-projection. In both +cases we used the ASTRA Toolbox GPU implementations42, +cone-beam geometry and the voxel size of 85.68 µm. The re- +constructed attenuation volume was treated with a TomoPy im- +plementation of a reconstruction-space ring-removal algorithm +(θmin = 80, threshold = 0)43. +Calibration to HU and HUp +units44 was done by setting air to −1000 and a water region +to zero. The fused images were obtained by first normalising +the phase-contrast reconstructed volume such, that the grey- +levels of the adipose and glandular tissues, measured in three +manually-selected ROIs each, corresponded to the ones in the +attenuation volume. Then, the attenuation volume was low- +pass-filtered and the phase one high-pass-filtered with an in- +plane Gaussian kernel of σ = 1.5 px. The resulting volumes +were added. +Quantitative analysis +The reconstructed volume slices were +obtained by averaging a varying number of sinograms before +the reconstruction. For the images corresponding to the doses +5.5 mGy–16.5 mGy, the average of 1–3 series with the current of +2.5 mA was used. For the 22 mGy–222 mGy ones, 1–10 series +with 10 mA were used. Three circular regions containing the +adipose and three with the glandular tissue were selected. They +are marked in Fig. 3. The CNR was estimated by the average +contrast between the tissue types and the standard deviation in +the adipose regions. For each image, we found numerically the +minimal FWHM size of an isotropic 3D Gaussian kernel nec- +essary to reach a CNR of 531. The value was chosen based on +the Rose criterion, which states that a CNR of 5 is sufficient to +detect features. We considered only the point spread function +(PSF) introduced by the filtering, which is system-independent +and sets the lower limit on the resolution. The total resolution of +the imaging system is also influenced by the PSFs of the source +and the detector. +Derivation of the ideal-gratings limit +We estimated the +limit of the performance of the system assuming ideal gratings +in a numerical Fresnel wave-propagation simulation. The focal +spot size of the X-ray source and its spectrum were considered +by propagating accordingly weighted source fields. We have +modelled gratings with ideal, defect-free lamellae, but other- +wise their geometry and material content, to the best of our +knowledge, corresponded to reality. The visibility in the model +was 17.6 %, which we interpret as the upper limit for the per- +formance of the interferometer with this geometry. The sensi- +tivity of a GI-CT system to refraction increases linearly with the +visibility and, further, the dose requirement inversely with the +square of the sensitivity. The increase of the visibility from 9.4 % +(currently achieved by our system) to 17.6 % (theoretical limit) +would then lower the dose requirement for the phase contrast +by a factor of 3.5. We assumed that with the ideal gratings the +attenuation would not change, so the intersection point of the +attenuation and phase contrast best-fit lines would be at the +resolution of 417 µm and the dose of 6.50 mGy. In Fig. 4, we +show the correspondingly shifted fused GI-CT curve. To avoid +extrapolation the limit does not extend beyond the measured +points. +7 + +References +1. +Sung, H. et al. Global Cancer Statistics 2020: GLOBOCAN Esti- +mates of Incidence and Mortality Worldwide for 36 Cancers in 185 +Countries. CA: a cancer journal for clinicians 71, 209–249 (2021). +2. +Broeders, M. et al. The impact of mammographic screening on +breast cancer mortality in Europe: a review of observational stud- +ies. Journal of medical screening 19, 14–25 (2012). +3. +Lauby-Secretan, B. et al. Breast-Cancer Screening — Viewpoint of +the IARC Working Group. New England Journal of Medicine 372, +2353–2358 (2015). +4. +Hovda, T., Tsuruda, K., Hoff, S. R., Sahlberg, K. 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D., Xiao, X. & Jacobsen, C. TomoPy: a frame- +work for the analysis of synchrotron tomographic data. Journal of +Synchrotron Radiation 21, 1188–1193 (2014). +44. +Donath, T. et al. Toward clinical X-ray phase-contrast CT: Demon- +stration of enhanced soft-tissue contrast in human specimen. In- +vestigative Radiology 45, 445–452 (2010). +8 + +Acknowledgements +The authors are grateful to Gordan Mikuljan and Philipp Zup- +piger of PSI for their fantastic technical expertise and support. +The authors acknowledge the clean room facilities of PSI and +the technical staff for the support in gratings fabrication. This +work has been funded by the SNF R’Equip grant 206021 189662 +(SiDRY), the ETH-Research Commission Grant Nr. ETH-12 20-2, +an ETH Doc.Mobility Fellowship, the Promedica Stiftung Chur, +the SNF Sinergia Grant Nr. CRSII5 183568, the PHRT-Pioneer +Project Nr. 2021-612 CLARINET as well as the Swisslos Lottery +Fund of Kanton Aargau. +Author contributions +Z.W., M.R., M. Stampanoni and M. Stauber conceptualised the +GI-CT system; M.R. designed and built the system and concep- +tualised the measurement; M.R. and A.P. analysed the data with +contributions from S.S., S.v.G., J.X., M.P. and M.-C.Z.; S.S. and +M.R. set up the control system; S.S. and A.P. implemented the +wave-propagation simulation; L.R., K.J. and Z.S. manufactured +the G1 grating; E.Y. and J.C. estimated the dose; Z.V. provided +the breast specimen; R.A.K.-H., C.L., T.N., Z.V. and A.B. pro- +vided clinical expertise; M.R. wrote the manuscript with contri- +butions of all authors. +Competing interests +M. Stauber is the CEO and a co-founder of GratXray AG, +Z. Wang is a co-founder of GratXray AG, M. Stampanoni is a +member of the BoD and a co-founder of GratXray AG, A. Boss +is a member of the BoD of GratXray AG, L. Romano is the act- +ing CSO of GratXray AG and M. Rawlik is the acting CTO of +GratXray AG. +9 + +Supplementary material +3000 +4500 +6000 +−π +0 +π +0.05 +0.09 +0.13 +3000 +4500 +6000 +intensity +−π +0 +π +phase +0.05 +0.09 +0.13 +visibility +Fig. 6. The reference phase-stepping scan without the sample in the beam. Top: The intensity profile and its histogram. Middle: +The phase profile. Bottom: The visibility profile. The average visibility is 0.094, the best regions are 0.12. +Fig. 7. SEM image of the surface of the G0 grating. The gold lamellae were electroplated in high-aspect-ratio cavities in polymer +(brighter on the image). A slight overplating defect is indicated with the arrow on the right-hand side. The arrow on the left side +indicates where the lamellae detached from the polymer forming a gap. The grating was manufactured by Microworks GmbH, +Germany. +10 + +20 μmSupplementary material +3000 +4500 +6000 +−π +0 +π +0.05 +0.09 +0.13 +3000 +4500 +6000 +intensity +−π +0 +π +phase +0.05 +0.09 +0.13 +visibility +Fig. 1: The reference phase-stepping scan without the sample in the beam. Top: The intensity profile and its histogram. Middle: +The phase profile. Bottom: The visibility profile. The average visibility is 0.094, the best regions are 0.12. +Fig. 2: SEM image of the surface of the G0 grating. The gold lamellae were electroplated in high-aspect-ratio cavities in polymer +(brighter on the image). +A slight overplating defect is indicated with the arrow on the right-hand side. +The arrow on the left +side indicates where the lamellae detached from the polymer forming a gap. The grating was manufactured by Microworks GmbH, +Germany. +1 +arXiv:2301.00455v1 [physics.med-ph] 1 Jan 2023 + +20 μm \ No newline at end of file diff --git a/oNAyT4oBgHgl3EQflfjb/content/tmp_files/load_file.txt b/oNAyT4oBgHgl3EQflfjb/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..dd21f5852106cb6d7becc205f5c15d485bb7a555 --- /dev/null +++ b/oNAyT4oBgHgl3EQflfjb/content/tmp_files/load_file.txt @@ -0,0 +1,698 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf,len=697 +page_content='Refraction beats attenuation in breast CT MICHAŁ RAWLIK1,2,*, ALEXANDRE PEREIRA1,2, SIMON SPINDLER1,2, ZHENTIAN WANG1,3, LUCIA ROMANO1,2, KONSTANTINS JEFIMOVS2, ZHITIAN SHI1,2, MAXIM POLIKARPOV1,2, JINQIU XU1,2, MARIE-CHRISTINE ZDORA1,2, STEFANO VAN GOGH1,2, MARTIN STAUBER4, EDUARDO YUKIHARA2, JEPPE B.' metadata={'source': 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+page_content=' AND MARCO STAMPANONI1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='2 1Institute for Biomedical Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' ETH Z¨urich and University of Z¨urich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' Switzerland 2Paul Scherrer Institute,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' Villigen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' Switzerland 3Department of Engineering Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' Tsinghua University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' Haidian District,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' 100080 Beijing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' China 4GratXray AG,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' Villigen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' Switzerland 5Department of Radiology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' Kantonsspital Baden,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' Switzerland 6Interdisciplinary Breast Center,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' Kantonsspital Baden,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' Switzerland 7Department of Pathology and Molecular Pathology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' University Hospital Z¨urich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' Switzerland 8Institute for Diagnostic and Interventional Radiology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' University Hospital Z¨urich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' Switzerland corresponding author,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' email: mrawlik@ethz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='ch January 3, 2023 For a century, clinical X-ray imaging has visualised only the attenuation properties of tissue, which fundamentally limits the contrast, particularly in soft tissues like the breast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' Imaging based on refraction can overcome this limitation, but so far has been constrained to high-dose ex-vivo applications or required highly coherent X-ray sources, like synchrotrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' It has been predicted that grating interferometry (GI) could eventually allow computed tomography (CT) to be more dose-efcient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' However, the benefit of refraction in clinical CT has not been demonstrated so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' Here we show that GI-CT is more dose-efcient in imaging of breast tissue than conventional CT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' Our system, based on a 70 kVp X-ray tube source and commercially available gratings, demonstrated superior quality, in terms of adipose-to-glandular tissue contrast-to-noise ratio (CNR), of refraction-contrast compared to the attenuation images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' The fusion of the two modes of contrast outperformed conventional CT for spatial resolutions better than 263 µm and an average dose to the breast of 16 mGy, which is in the clinical breast CT range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' Our results show that grating interferometry can significantly reduce the dose, while maintaining the image quality, in diagnostic breast CT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' Unlike conventional absorption-based CT, the sensitivity of refraction-based imaging is far from being fully exploited, and further progress will lead to significant improvements of clinical X-ray CT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' In 2020, breast cancer was the most commonly diagnosed cancer overall, with over two million cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' Among women it makes up 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='5 % of the cancer cases and 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='5 % of the cancer-related deaths1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' The prevalence of breast cancer has prompted most developed countries to estab- lish mammography screening programmes, which have been shown to reduce mortality2,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' However, the effec- tiveness of mammography is disputed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' A retrospective study found that only 46 % of screen-detected cancers are true positives, while 22 % are missed4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' The reason is that mammography images are difficult to read.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' Not only does the soft tissue in the breast provide limited X-ray contrast, but also the complicated morphol- ogy of the breast is ambiguous when rendered in a two- dimensional projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' Even among experienced read- ers, the agreement in identifying masses is far from per- fect (κ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='67)5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' This is despite the painful measure of compressing the breast to spread it out on the image and make it thinner, so that the contrast can be improved by using lower-energy X-rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' The shortcomings of other breast-imaging modali- ties have so far hindered their widespread use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' Dig- ital Breast Tomosynthesis partially removes the tissue- overlap-related ambiguity but it has been shown to pro- vide only an incremental improvement over mammog- raphy6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' Breast ultrasound plays mainly a supportive role and, while MRI provides excellent contrast, its res- olution is lower than mammography, it cannot visualise microcalcifications and the modality is expensive, time- consuming and uncomfortable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' Dedicated breast CT has recently been introduced in clinical practice with promising results, summarised in a recent review7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' With the volumetric data, the tissue over- lap is completely alleviated, and the breast does not need to be compressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' In the review, its authors identify the major shortcoming of the method to be the near identi- cal attenuation contrast between breast tumours without microcalcifications and glandular parenchyma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' Like visible light, X-rays are not only attenuated but also refracted when traversing matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' In the last years, several methods to detect the refraction of X-rays, the phase contrast (PC), have been developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' In the con- text of breast imaging, three are noteworthy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' One is propagation-based PC, which does not need additional op- 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='00455v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='med-ph] 1 Jan 2023 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' The principle of GI-CT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' a An illustration of the imaging setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' The formation of the Talbot interference pattern behind G1 and its distortion by the breast are shown schematically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' b The intensity of a monochromatic wavefront in a GI-CT system in a horizontal plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' The G0 absorption grating acts as an array of vertical slits, each providing enough spatial coherence in the transverse direction so that an interference pattern is created behind the periodically-phase-shifting, by π, G1 grating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' At the position of G2 the interference produces a pattern of parallel bright-and-dark lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' The period of G0 is chosen such, that the line patterns from its slits add constructively (but incoherently).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' The period of the line pattern is smaller than the pixel size of the detector, so an absorption grating G2, with the same period as the line pattern, is used to analyse it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' The refraction of X-rays in the sample distorts the line pattern, which results in an intensity change behind the G2 grating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' The gratings are shown schematically, and their lamellae are vertical, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' perpendicular to the plane of the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' For illustration purposes the image is stretched and shows only an approximately 600 µm-wide part of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' c Change in the intensity incident on pixels as G0 is shifted: without interaction with the object (blue), with pure attenuation interaction (orange) and with refraction (green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' d A slice of a phantom reconstructed from the attenuation signal in a simulated GI-CT measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' The yellow arrow points to a small feature, not discernible in the attenuation image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' e The phase reconstruction from the same simulated measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' The small feature can be recognised, but low-spatial-frequency noise is prominent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' f GI-CT fusion combines the low spatial frequencies of the attenuation reconstruction, where it has lower Noise Power Spectrum (NPS), with the high ones of the phase reconstruction – the regime where the phase has lower NPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' g The fused GI-CT image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' The feature pointed out by the arrow is visible and the low-spatial-frequency noise is suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' 2 phase attenuation 0100mm fused G1 G2 GO △Itical elements, but requires the high spatial coherence of a synchrotron or a specialised laboratory source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' The other two, edge illumination8 and grating interferometry (GI)9, use optical elements, and work with standard X-ray tube sources with large spot sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' We discuss GI further, and for a broad overview of PC imaging we direct the reader to a recent review10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' The principle of GI is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' 1a and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' Let us assume for the moment that the G0 element is a narrow slit that provides spatial coherence for the X-ray beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' G1 is a grating that introduces a periodic π-shift in the beam, which results downstream in an interference pat- tern of parallel bright and dark lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' Refraction on large- scale structures in the sample, the phase contrast, causes the pattern to shift laterally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' Diffusion by refraction on small-scale structures beyond the resolving power of the detector, the dark-field (DF) signal, blurs the line pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' For X-rays, the refraction angles are in the microradian range requiring the period of the pattern (and G1) to be in the order of a few micrometres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' A pattern this small cannot be resolved directly with standard large-scale de- tectors, so a periodically opaque analyser G2 with the pe- riod matching the one of the pattern is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' The final observation is that G0 can, in fact, also be a periodically- opaque grating: an array of slits with the spacing chosen so that the line patterns created by each add construc- tively (but incoherently) in the plane of the analyser9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' The gratings are decisive when it comes to the sensi- tivity of GI to the refraction of X-rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' The minimal de- tectable refraction angle is proportional to the pitch of G2, favouring pitch sizes in the few-micrometre range, as well as to the contrast in the interference pattern analysed by G2, called the visibility11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' To periodically block X-rays with the energy in the clinical regime, heavy-element, like gold, lines of 100 µm–200 µm thickness are necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' Fabrication of these high-aspect-ratio (line-thickness–to– half-pitch–ratio) microstructures with sufficient quality is challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' Deep X-ray lithography (LIGA) can man- ufacture gratings with thick gold lines (200 µm but the pitch size is limited to several micrometres12,13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' Silicon- based manufacturing is now pushing the aspect ratio for the pitch size in a micrometre14 and sub-micrometre15,16 regimes, for both etching of silicon template and gold fill- ing17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' The additional refraction information that GI provides is attractive for breast imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' PC promises higher con- trast18 for better differentiation of tissues and DF was shown to distinguish benign from malignant calcifica- tions19,20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' Two-dimensional GI mammography is already close to first clinical trials21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' It is natural to pursue the extension of PC to 3D, given the advantages it has shown in two-dimensional imag- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' It has been demonstrated that the benefit of PC-CT depends on the spatial resolution or, for a fixed contrast- to-noise ratio, equivalently, the dose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' It was predicted that with the increase of sensitivity through the progress in the technology of grating fabrication high-resolution breast imaging would be the first clinical area where PC- CT will be advantageous22,23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' So far PC-CT of the breast tissue has been investi- gated in the high-dose, high-resolution context of virtual histopathology, demonstrating a clear benefit over atten- uation18,24,25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' Propagation-based PC breast imaging is pursued at the Elettra synchrotron in Trieste26 and the Australian Synchrotron27, but the method fundamentally relies on the high coherence of synchrotron X-ray beams and cannot be translated to a wide-spread clinical use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' Small systems for propagation-based PC have been built, but the low power of the required microfocal sources re- sults in very long scan times28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' Edge-illumination was demonstrated for ex-vivo studies of larger specimens29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' A recent review can be found in30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' The predicted tipping point of the advantage of clinically-applicable phase con- trast has not yet been reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' We have constructed a GI-CT system with the aim to demonstrate additional value of GI in clinical breast CT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' We have imaged a formalin-fixed human-breast speci- men and compared the dose efficiency of the two GI-CT contrasts, attenuation and phase, for a range of resolu- tions, as well as the combination of the GI-CT contrast versus conventional, attenuation-based CT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' Results The 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='8 m-long GI-CT system consisted of a tungsten- anode X-ray source operated at 70 kVp, a photon- counting detector with an active area of 195 × 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='2 mm2 and a Talbot–Lau interferometer designed for the photon energy of 46 keV in a symmetric geometry, with all grat- ings being 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='2 µm-pitch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' The system is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' 2 and described in detail in the Methods section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' We imaged a formalin-fixed human breast from a body donation with the average dose delivered to the speci- men in the range of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='5 mGy–219 mGy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' The axial slices of the reconstructed volumes for both attenuation and phase contrast at mean absorbed dose to the breast of 11 mGy, 22 mGy and 219 mGy are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' The visual quality of both contrasts increases with the deliv- ered dose, which is particularly clear in the insets show- ing an enlarged portion of the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' Moreover, even though the phase-contrast image for the lowest dose ap- pears visually inferior to the attenuation-contrast one, for the higher dose it appears superior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' We analysed the images quantitatively following an approach inspired by Raupach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' We assumed that, in order to resolve the morphology of the breast, a contrast-to-noise ratio (CNR) of 5 between the adi- pose and glandular tissue is necessary (the Rose crite- rion31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' With decreasing dose, and consequently increas- ing noise, a smoother filtering is necessary to reach this CNR value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' The additional point spread function (PSF) of the filtering introduces limits to the imaging resolu- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' For each image, we determined the full-width-half- maximum (FWHM) size of an isotropic Gaussian-blur kernel necessary to achieve the CNR of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' The results depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' 4 can be interpreted as the dose neces- sary to resolve the morphology as a function of the reso- lution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' We observed that the dose requirement rises with the power of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='44 of the inverse kernel size for attenua- tion and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='56 for phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' The two curves, having different 3 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' The GI-CT setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' From the left: the tungsten-anode X-ray source operated at 70 kVp, 3 mm Aluminium filter, the G0 atten- uation grating, the phase-shifting G1 grating, the breast in a sample holder mounted on a rotation and vertical stages, the array of three G2 analyser gratings, and the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' attenuation 11 mGy 5 mm 2 cm 22 mGy 5 mm 2 cm 219 mGy 2 cm 5 mm −200 −100 0 100 HU phase 11 mGy 5 mm 2 cm 22 mGy 5 mm 2 cm 219 mGy 2 cm 5 mm −210 −100 0 100 200 HUp −300 −200 −100 0 100 200 HU / HUp Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' The axial slices of reconstructed volumes of the human-breast specimen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' The attenuation contrast is shown in the top row, the phase contrast in the bottom row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' Three values of the average dose absorbed in the breast are depicted in the columns: 11 mGy, 22 mGy and 219 mGy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' Visually, the image quality increases with the dose for both contrasts, but for the phase one it increases faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' At the dose of 219 mGy, the image quality of phase contrast is superior, which is visible particularly well in the enlarged region in the fourth column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' In the top-left image, the regions of interest used to calculate the CNR are marked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' 4 X-RAY FILTER GO G1 BREAST G2 DETECTOR SOURCE 3mm Al phase-shifting attenuation i analyser 70kVp grating grating grating0600 400 300 200 150 100 isotropic resolution limit (µm) 5 10 20 30 40 60 100 200 dose for CNR of 5 (mGy) phase attenuation grating-interferometry CT clinical breast CT conventional-CT equivalent limit with ideal gratings Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' The average absorbed dose requirement of the conven- tional CT and GI-CT imaging of the breast as a function of the resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' Each point represents an image (like in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' 3) for which the minimal FWHM size of the isotropic Gaussian kernel to reach the contrast-to-noise ratio (CNR) of 5 was found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' The ker- nel size constitutes a lower limit on the resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' The best-fit lines of the attenuation contrast (slope 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='44) and phase contrast (slope 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='56) intersect at the kernel size of 214 µm and dose of 65 mGy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' The ones of the conventional-CT equivalent (the atten- uation signal with half the dose, as G2 would not be necessary;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' slope 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='44) and fused GI-CT (slope 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='30) intersect at 263 µm and 16 mGy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' The intersection can be interpreted as the point beyond which GI-CT delivers superior image quality per unit dose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' No- tably, it is within the range of clinical breast CT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' The numerically- derived limit for fused GI-CT with ideal gratings, and otherwise the same geometry, is also shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' The points for phase below 16 mGy are omitted for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' slopes, intersect at a kernel size of 214 µm, which can be interpreted as the resolution above which, on our system, phase-contrast imaging is more dose-efficient than atten- uation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' Due to the differential nature of phase contrast in GI, the reconstructed volumes exhibit different in-plane noise properties than attenuation contrast22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' The noise power spectrum of PC-CT, in comparison with attenua- tion CT, is smaller for high spatial frequencies, but larger for low frequencies (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' 1f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' This observation, in addi- tion to the fact that the two contrasts come from a sin- gle acquisition, and are therefore naturally registered, motivates fusing the images32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' Because attenuation and phase carry fundamentally different information there is no generic way to fuse them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' However, in the context of a particular imaging task, which we define to be dif- ferentiating glandular and adipose tissue with the maxi- mal possible CNR, we used a simple scheme, illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' 1f and g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' First, we normalised the reconstructed phase-contrast volume so that the grey levels of the adi- pose and glandular tissues were equal to the ones in the attenuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' Then, we added the high-pass filtered phase volume to the low-pass filtered attenuation one, both us- ing the same in-plane Gaussian kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' We found the op- timal size of the kernel, the one maximising the CNR of the fused image, to be close to σ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='5 px for all measure- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' The quantitative CNR analysis of the fused im- ages, depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' 4, showed that in the investigated range of kernel sizes they approximately follow a power law with the exponent of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='30 lying between the one of the images derived from attenuation and phase contrast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' Moreover, the dose requirement is lower than the ones of each of the single-contrast images everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' Without the G2 grating, which attenuates approxi- mately half of the photon flux downstream of the spec- imen, it would be possible to acquire an attenuation im- age with the same photon-counting statistics at half the dose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' We therefore consider the attenuation contrast with half the dose to be approximately equivalent to a con- ventional CT image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' The comparison of the CNR-dose- efficiency of the fused GI-CT and the conventional-CT equivalent, depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' 4, shows that the former out- performs the latter for isotropic kernel sizes sharper than 263 µm and doses larger than 16 mGy (for CNR = 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' 5, we show a comparison of an enlarged fragment of fused-GI-CT and conventional-CT-equivalent images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' While at the dose of 22 mGy the benefit of GI-CT is not visually impressing, it increases with the dose, and it be- comes evident at 66 mGy, in particular for small features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' We would like to point out that the low-frequency con- trast information in the fused image comes from the at- tenuation contrast and thus for both images the quanti- tative unit is the one of attenuation, that is Hounsfield’s (HU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' Discussion Soon after the first demonstration of grating interferom- etry with a regular, tube-based X-ray source9 it has been predicted that the method can greatly improve clinical CT imaging, in particular of the breast23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' Despite a decade of intensive research, the practical demonstration has been, so far, elusive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' With the aim to settle the long discussion we have constructed a GI-CT system with a tube-based X-ray source operated at a typical breast-CT energy of 70 kVp, a 10 cm-wide field of view, and a Talbot–Lau in- terferometer based on commercially-available gratings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' We evaluated its performance by imaging a formalin- fixed human breast specimen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' We could show that the additional information pro- vided by refraction led to two breakthroughs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' Firstly, the phase-contrast images are superior, in terms of adipose- to-glandular tissue CNR per unit dose, than attenuation for kernels sharper than 214 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' Previous demonstra- tions of refraction-based CT imaging of the breast with a large field of view either did not show the benefit over attenuation29 or relied on a synchrotron26, which greatly limits the applicability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' Secondly, even though GI-CT utilises only half of the photon flux compared to conventional CT due to the absorbing analyser grating, the combination of the attenuation- and phase-contrast signals provides suffi- cient information to compensate the loss already at a kernel size of 263 µm and dose of 16 mGy (for CNR of 5), both of which lie in their respective ranges used in the clinics: 150 µm–300 µm and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='8 mGy–26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='1 mGy7,33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' For sharper kernels GI-CT outperforms conventional 5 conventional-CT equivalent 22 mGy 5 mm 66 mGy 5 mm GI-CT 5 mm 5 mm −200 −150 −100 −50 0 50 100 HU Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' A comparison of conventional-CT equivalent and fused GI-CT images At the dose of 22 mGy the additional information coming from refraction allows GI-CT to just about overcome the reduction in statistics coming from the G2 absorbing half of the photon flux and the image quality is comparable to that of a conventional-CT equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' At 66 mGy the image quality of GI- CT is superior, which is visible particularly well in small features, like the one indicated by the arrow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' attenuation-based CT with increasing benefit, requiring only 53 % of the dose at 150 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' Further improvements are possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' The limit of the sensitivity of GI-CT to refraction is driven by the microfabrication of the gratings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' Even the currently commercially-available gratings performed already well enough for GI-CT to outperform conventional CT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' The performance limit under an assumption of defect-free gratings, depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' 4, suggests that GI-CT could re- quire a factor of 2 to 3 less dose than conventional CT in the range of clinical breast CT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' Improvements in the grating-fabrication technology will take GI-CT closer to that limit and, with smaller grating pitches, possibly be- yond it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' Our aim was to compare the phase and attenuation contrasts on an even ground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' Here, we have therefore deliberately kept the analysis to a minimum, refraining from the use of iterative reconstruction algorithms, reg- ularisation and post-processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' The use of those meth- ods is likely to improve the performance as we quanti- fied it in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' 4, and a more specific analysis will be avail- able34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' In particular, we advice caution in comparing the values with other results obtained with elaborate analy- sis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' The particularity of these advanced analysis methods and their differences for the phase and attenuation con- trasts would, in our opinion, only weaken our otherwise general conclusion: that GI-CT provides fundamentally more information to start with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' We have demonstrated that GI-CT is a new relevant clinical imaging modality, which can be more dose- efficient than conventional CT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' X-ray grating interferom- etry, unlike other imaging techniques exploiting refrac- tion, is compatible with conventional medical CT scan- ners35 and, therefore, suitable for widespread use in hos- pitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' The technique is immediately applicable to dedi- cated breast CT systems, for which we have shown that it already offers an improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' In the future, GI could allow dose reduction in all aspects of clinical CT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' Methods Measurement setup The measurement system consisted of a Comet MXR-225HP/11 tungsten-anode X-ray source operated at 70 kVp and 10 mA for the 22 mGy–222 mGy measurements and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='5 mA for the 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='5 mGy–16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='5 mGy ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' The size of the fo- cal spot was measured by the manufacturer to be 250 µm (at 30 % drop).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' The X-ray beam was filtered with a 3 mm-thick aluminium plate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' The images were recorded with a photon- counting detector with 750 µm-thick CdTe sensor and 75 µm pixel size, which was manufactured by Dectris AG, Switzer- land.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' The sensor size was 3072 × 256 pixels, but only an area of 2600 × 256 pixels could be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' The Talbot-Lau interferom- eter was configured in a 5th-Talbot-order symmetric geometry with a G1 designed to introduce a phase shift of π at 46 kV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' The G0–G1 and G1–G2 distances were both 818.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='1 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' The source– G0 distance was 100 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' G0 and G1 were a single piece each, and for G2 three gratings were tiled together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' All gratings were bent around the vertical axis going through the X-ray source’s focal spot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' For phase-stepping, G0 was moved with a Physik– Instrumente P-841.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='6B piezo actuator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' The detector was 1756 mm and the rotation centre 1003 mm away from the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' Gratings The 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='2 µm-pitch attenuation gratings G0 and G2 had gold lamellae electroplated onto a graphite substrate, and were manufactured with the LIGA process by Mi- croworks GmbH, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' The gratings had a duty cycle of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='5, and gold thickness was in the range of 150 µm–180 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' The polymer template was not stripped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' The π-shifting 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='2 µm-pitch phase grating G1 was manufac- tured on a double side polished 8-inch silicon wafer by deep reactive ion etching in a SPTS Rapier system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' A pattern in MEGAPOSIT SPR220-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='0 positive tone photoresist was realised by direct laser writing (Heidelberg DWL66+) (see36 for further details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' The process was optimised to ensure uniform etch- ing depth and vertical trench sidewalls, as reported in14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' The G1 grating had a duty cycle of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='5, and the grating lines were 59 µm thick;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' The thickness of the remaining silicon substrate was 240 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' A single tiled G1 grating was diced out from the wafer to a size of 203 mm × 75 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' Specimen The female breast specimen was a human breast tis- sue from an adult autopsy after a body donation for research (ethical agreement KEK-2012 554).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' It was without any grossly visible pathology, and was fixed in 10 % buffered formaldehyd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' The specimen was vacuum-sealed in a plastic bag and placed in a cylindrical PMMA container with 100 mm outer and 90 mm inner diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' The container was filled with water to avoid air gaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' 6 Measurement protocol During the CT scan, the specimen re- volved continuously at 1 rpm for five full rotations while the frames were acquired at 20 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' After each rotation, the G0 grat- ing was shifted by one sixth of its period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' The five-rotations pro- tocol was repeated ten times with the 10 mA tube current, and three with 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='5 mA, and was interleaved with a reference phase- stepping measurements with the sample out of the beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' The ten-repetition scan took 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='5 h of wall-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' Dose estimation The term dose was used to indicate the mean absorbed dose to the breast, approximated as the ab- sorbed dose to a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='25–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='75 volumetric mixture of the ICRU44 glandular and adipose tissues37, homogeneously distributed in a PMMA cylinder with a 100 mm outer diameter and a 90 mm inner diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' The absorbed dose was estimated by the means of Monte Carlo simulations (GEANT438), where the simulation geometry and source parameters were validated through measurements using BeO optically stimulated lumi- nescence dosimeters (OSLDs) and LiF:Mg,Ti thermolumines- cence dosimeters (TLDs)39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' The OSLDs and TLDs were cali- brated in dose-to-water using a ISO N-60 photon field (mean energy 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='9 keV)40 to approximate the mean energy of the X-ray field used for imaging (mean energy 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='6 keV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' The photons in the simulation were sampled from a spectrum approximating the X-ray tube in the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' The source was collimated to match the extent of the absorption gratings G0 and G1 out- lined in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' In a first experiment, the OSLDs and TLDs were placed upstream of the PMMA cylinder on the beam axis to establish a conversion factor between the simulated dose-per- primary to the absorbed dose to the dosimeters for a 10 min irradiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' In a second experiment, which served to validate the Monte Carlo model, the TLDs were placed on both exter- nal sides of the cylinder, upstream and downstream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' The mea- sured doses to water (145 mGy upstream and 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='1 mGy down- stream for a 10 min irradiation at 10 mA) were in agreement with the simulated dose-to-water in volumes matching the lu- minescence detectors (151 mGy upstream and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='8 mGy down- stream), thus validating the Monte Carlo model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' The Monte Carlo model was then used to score the dose to the homogeneous 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='25–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='75 volumetric mixture of the ICRU44 glan- dular and adipose tissues placed in the PMMA cylinder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' The volumetric fractions were established with a threshold-based segmentation of the reconstructed volumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' The mean dose to the tissue mixture placed in the PMMA cylinder was calculated to be 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='9(24) mGy, which corresponds to a 5 min-long CT mea- surement series at 10 mA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' Data processing The sinograms corresponding to the five ro- tations with different G0 positions xj were stacked and we per- formed a signal-retrieval with a linear least-squares fitting of a sine to find the phase φi, visibility vi and intensity Ii in each ith pixel: Ij i = Ii 2 � vi sin �2π p xj − φi � + 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' (1) We used an overarching least-squares optimisation to find the best-fit period of the sine p common to all pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' The reference measurements, acquired between the tomography scans, were analysed in the same way to obtain the reference maps φr i , vr i and Ir i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' We then constructed the attenuation pI i and differential- phase-contrast (DPC) pφ i sinograms taking as the reference the average of the two adjacent reference scans: pI i = − log � Ii Ir i � , pφ i = φi − φr i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' (2) The attenuation sinogram pI was corrected for beam-hardening effects, for which we used a separate measurement of PMMA slabs of different thicknesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' We further applied to the attenu- ation sinogram a ring-removal algorithm based on a combined wavelet-FFT filter with damping of 1, 3 wavelet transform lev- els and a db5 wavelet filter41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' The attenuation volume was re- constructed with the FDK algorithm, for the phase-contrast we first used the Hilbert filter and then back-projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' In both cases we used the ASTRA Toolbox GPU implementations42, cone-beam geometry and the voxel size of 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='68 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' The re- constructed attenuation volume was treated with a TomoPy im- plementation of a reconstruction-space ring-removal algorithm (θmin = 80, threshold = 0)43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' Calibration to HU and HUp units44 was done by setting air to −1000 and a water region to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' The fused images were obtained by first normalising the phase-contrast reconstructed volume such, that the grey- levels of the adipose and glandular tissues, measured in three manually-selected ROIs each, corresponded to the ones in the attenuation volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' Then, the attenuation volume was low- pass-filtered and the phase one high-pass-filtered with an in- plane Gaussian kernel of σ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='5 px.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' The resulting volumes were added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' Quantitative analysis The reconstructed volume slices were obtained by averaging a varying number of sinograms before the reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' For the images corresponding to the doses 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='5 mGy–16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='5 mGy, the average of 1–3 series with the current of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='5 mA was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' For the 22 mGy–222 mGy ones, 1–10 series with 10 mA were used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' Three circular regions containing the adipose and three with the glandular tissue were selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' They are marked in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' The CNR was estimated by the average contrast between the tissue types and the standard deviation in the adipose regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' For each image, we found numerically the minimal FWHM size of an isotropic 3D Gaussian kernel nec- essary to reach a CNR of 531.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' The value was chosen based on the Rose criterion, which states that a CNR of 5 is sufficient to detect features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' We considered only the point spread function (PSF) introduced by the filtering, which is system-independent and sets the lower limit on the resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' The total resolution of the imaging system is also influenced by the PSFs of the source and the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' Derivation of the ideal-gratings limit We estimated the limit of the performance of the system assuming ideal gratings in a numerical Fresnel wave-propagation simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' The focal spot size of the X-ray source and its spectrum were considered by propagating accordingly weighted source fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' We have modelled gratings with ideal, defect-free lamellae, but other- wise their geometry and material content, to the best of our knowledge, corresponded to reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' The visibility in the model was 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='6 %, which we interpret as the upper limit for the per- formance of the interferometer with this geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' The sensi- tivity of a GI-CT system to refraction increases linearly with the visibility and, further, the dose requirement inversely with the square of the sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' The increase of the visibility from 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='4 % (currently achieved by our system) to 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='6 % (theoretical limit) would then lower the dose requirement for the phase contrast by a factor of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' We assumed that with the ideal gratings the attenuation would not change, so the intersection point of the attenuation and phase contrast best-fit lines would be at the resolution of 417 µm and the dose of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='50 mGy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' 4, we show the correspondingly shifted fused GI-CT curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' To avoid extrapolation the limit does not extend beyond the measured points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' 7 References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' Sung, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' Global Cancer Statistics 2020: GLOBOCAN Esti- 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+page_content=' TomoPy: a frame- work for the analysis of synchrotron tomographic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' Journal of Synchrotron Radiation 21, 1188–1193 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' Donath, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' Toward clinical X-ray phase-contrast CT: Demon- stration of enhanced soft-tissue contrast in human specimen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' In- vestigative Radiology 45, 445–452 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' 8 Acknowledgements The authors are grateful to Gordan Mikuljan and Philipp Zup- piger of PSI for their fantastic technical expertise and support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' The authors acknowledge the clean room facilities of PSI and the technical staff for the support in gratings fabrication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' This work has been funded by the SNF R’Equip grant 206021 189662 (SiDRY), the ETH-Research Commission Grant Nr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' ETH-12 20-2, an ETH Doc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='Mobility Fellowship, the Promedica Stiftung Chur, the SNF Sinergia Grant Nr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' CRSII5 183568, the PHRT-Pioneer Project Nr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' 2021-612 CLARINET as well as the Swisslos Lottery Fund of Kanton Aargau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' Author contributions Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=', M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=', M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' Stampanoni and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' Stauber conceptualised the GI-CT system;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' designed and built the system and concep- tualised the measurement;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' analysed the data with contributions from S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=', S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=', J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=', M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' set up the control system;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' implemented the wave-propagation simulation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=', K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' manufactured the G1 grating;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' estimated the dose;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' provided the breast specimen;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=', C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=', T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=', Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' pro- vided clinical expertise;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' wrote the manuscript with contri- butions of all authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' Competing interests M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' Stauber is the CEO and a co-founder of GratXray AG, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' Wang is a co-founder of GratXray AG, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' Stampanoni is a member of the BoD and a co-founder of GratXray AG, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' Boss is a member of the BoD of GratXray AG, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' Romano is the act- ing CSO of GratXray AG and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' Rawlik is the acting CTO of GratXray AG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' 9 Supplementary material 3000 4500 6000 −π 0 π 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='13 3000 4500 6000 intensity −π 0 π phase 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='13 visibility Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' The reference phase-stepping scan without the sample in the beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' Top: The intensity profile and its histogram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' Middle: The phase profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' Bottom: The visibility profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' The average visibility is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='094, the best regions are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' SEM image of the surface of the G0 grating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' The gold lamellae were electroplated in high-aspect-ratio cavities in polymer (brighter on the image).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' A slight overplating defect is indicated with the arrow on the right-hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' The arrow on the left side indicates where the lamellae detached from the polymer forming a gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' The grating was manufactured by Microworks GmbH, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' 10 20 μmSupplementary material 3000 4500 6000 −π 0 π 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='13 3000 4500 6000 intensity −π 0 π phase 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='13 visibility Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' 1: The reference phase-stepping scan without the sample in the beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' Top: The intensity profile and its histogram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' Middle: The phase profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' Bottom: The visibility profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' The average visibility is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='094, the best regions are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' 2: SEM image of the surface of the G0 grating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' The gold lamellae were electroplated in high-aspect-ratio cavities in polymer (brighter on the image).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' A slight overplating defect is indicated with the arrow on the right-hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' The arrow on the left side indicates where the lamellae detached from the polymer forming a gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' The grating was manufactured by Microworks GmbH, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='00455v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} +page_content='med-ph] 1 Jan 2023 20 μm' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQflfjb/content/2301.00455v1.pdf'} diff --git a/p9FRT4oBgHgl3EQfdzfW/vector_store/index.faiss b/p9FRT4oBgHgl3EQfdzfW/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..ad6d87c0553fe0cfca031ac415a27072bb1b2e8e --- /dev/null +++ b/p9FRT4oBgHgl3EQfdzfW/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:fefc9ef578dec9d4a49854c48854e624e453ba5422c5a72dac51f1f32e0b3578 +size 9240621 diff --git a/ptE0T4oBgHgl3EQfqwEz/content/tmp_files/2301.02556v1.pdf.txt b/ptE0T4oBgHgl3EQfqwEz/content/tmp_files/2301.02556v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..7cde705f8772b4e160e98a1930707b05697c1057 --- /dev/null +++ b/ptE0T4oBgHgl3EQfqwEz/content/tmp_files/2301.02556v1.pdf.txt @@ -0,0 +1,2041 @@ +Performance of the nanopost single-photon source: beyond the single-mode +model +Martin Arentoft Jacobsen,∗a Yujing Wang,a Luca Vannucci,a Julien Claudonb, Jean-Michel Gérard,b and Niels Gregersena +We present a detailed analysis of the physics governing the collection efficiency and the Purcell enhancement of the nanopost single- +photon source. We show that a standard single-mode Fabry-Pérot model is insufficient to describe the device performance, which +benefits significantly from scattering from the fundamental mode to radiation modes. +We show how the scattering mechanism +decouples the collection efficiency from the Purcell enhancement, such that maximum collection efficiency is obtained off-resonance. +Finally, we discuss how this scattering mechanism can be beneficial for future single-photon source designs. +1 +Introduction +The construction of scalable optical quantum technologies1,2 re- +lies on the development of sources of single indistinguishable +photons3–6 and of entangled photon pairs7. +The ideal single- +photon source (SPS) should be deterministic and feature pure +emission of single photons. The main figure of merit6 is the col- +lection efficiency ε defined as the number of photons detected in +the out-coupling channel per trigger. In a multi-photon interfer- +ence experiment8 with N photons, the success probability P scales +as P = εN, and increasing ε towards 1 is thus critical to achieve +scalable optical quantum information processing. +The sponta- +neous parametric down conversion process9 is a straight-forward +technique widely used within the quantum optics community for +production of highly indistinguishable photons, however its prob- +abilistic nature limits the efficiency of pure photon emission to a +few percent. +For this reason, the community has turned its attention to- +wards two level systems, in particular the semiconductor quan- +tum dot3,4,10 (QD), capable of deterministic emission of single +photons. For a QD in a bulk material, ε is limited to a few percent +this time due to the large index contrast at the semiconductor-air +interface. It is thus necessary to place the QD inside a photonic +nanostructure5,6 directing the light towards the collection optics. +A main strategy for controlling the light emission is to place the +QD inside a micro cavity and exploit cavity quantum electrody- +namics (CQED) in the weak coupling regime to selectively en- +hance the light emission into the optical mode of the microcavity +using the Purcell effect11. Detailed understanding of the CQED +physics governing the collection efficiency can be obtained using +a single-mode Fabry-Pérot description12–14 of the light emission. +Here, the spontaneous emission β factor describes the emission +rate ΓC of the QD into a fundamental HE11 cavity mode divided +by the total emission rate ΓT = ΓC + ΓB including a contribution +ΓB to background radiation modes. The rate ΓC into the cavity +mode normalized to the rate ΓBulk in a bulk medium is quantified +by the Purcell15 factor Fp = ΓC/ΓBulk = +3 +4π2 +Q +Vn at resonance, where +Q is the cavity quality factor and Vn is the mode volume in units +of material cubic wavelengths (λ/n)3. The spontaneous emission +a DTU Electro, Department of Electrical and Photonics Engineering, Technical University +of Denmark, DK-2800 Kongens Lyngby, Denmark. E-mail: maaja@dtu.dk +b Univ. Grenoble Alpes, CEA, Grenoble INP, IRIG, PHELIQS, “Nanophysique et Semicon- +ducteurs” Group, F-38000 Grenoble, France. +β factor can then be written in terms of the Purcell factor as +β = +ΓC +ΓC +ΓB += +Fp +Fp +ΓB/ΓBulk +. +(1) +Furthermore, we define the transmission γ as the fraction of +power in the cavity mode detected by the collection optics. Fi- +nally, we can then define a single-mode Fabry-Pérot model (SMM) +εs for the efficiency as εs = βγ. From Eq. (1), we observe that +increasing the Purcell factor Fp will improve the collection effi- +ciency, and maximum efficiency is thus expected for a QD on res- +onance with the cavity. +Indeed, this design paradigm that Purcell enhancement is ben- +eficial for achieving high collection efficiency is well-established +within SPS engineering: The most succesful SPS design strategies +today include the microcavity pillar14,16–18 and the open cavity +approach19 demonstrating up to ε ∼ 0.6 into a first lens16 and +into a fiber19, respectively, combined with highly indistinguish- +able photon emission. These narrowband approaches, for which +the single mode model εs = βγ is an excellent approximation14, +rely critically on resonant Purcell enhancement and thus on con- +trol of the spectral alignment17 to achieve high efficiency. On +the other hand, broadband approaches including the photonic +nanowire13,20–23 and the photonic crystal waveguide24–26 de- +signs exploit suppression of the background emission rate using +the dielectric screening effect20,24,25 to non-resonantly maximize +the β factor. Even so, these broadband approaches also benefit +from resonant cavity13,21 and slow-light24,25 effects to further +improve the efficiency, confirming again that Purcell enhance- +ment is beneficial in the SPS engineering. +However, this paradigm has been challenged by new broad- +band SPS geometries, such as the circular Bragg grating or "bulls- +eye" design16,27,28, for which high collection efficiency is ob- +tained in a wavelength range significantly broader27 (∼ 100 nm) +than the typical resonance linewidth (∼ 10 nm). Similar char- +acteristics were observed very recently for the nanowire optical +nanocavity or "nanopost" design29 shown in Fig. (1), for which a +significant Purcell factor Fp of 5.6 enabled by the ultrasmall mode +volume of the nanocavity was experimentally demonstrated. Ad- +ditionally, a surprisingly high collection efficiency of 0.35 was +measured29, which was attributed to a breakdown30 of the single +mode model εs = βγ for the efficiency. +In this work, we investigate this surprising breakdown by per- +forming a detailed quantitative analysis of the physics governing +1–18 | 1 +arXiv:2301.02556v1 [quant-ph] 5 Jan 2023 + +Gold mirror +SiO2 +Substrate +GaAs +ht +hb +tSiO2 +Fig. +1 Sketch of the "nanopost" nanowire optical nanocavity and the +geometrical parameters. The resulting in-plane electrical field profile of +a QD placed inside the nanopost is shown as an inset. The length of the +white scale bar in the inset is 100 nm. +the nanopost geometry. We show that the single mode model fails +to describe the physics of both the Purcell enhancement and the +collection efficiency due to a decoupling between the two: The +computed efficiency is significantly higher than the prediction of +the single-mode model thanks to additional transmission chan- +nels to the far-field, whose beneficial contributions are dominat- +ing over the resonant cavity effect. We show not only that max- +imum Purcell enhancement and maximum collection efficiency +are obtained for entirely different design parameters, but also +that maximum efficiency is obtained off-resonance. The analysis +is performed using a Fourier Modal Method31, allowing for di- +rect insight into the beneficial interplay beyond the single-mode +model with the continuum of radiation modes. +This article is organized as follows: In Section 2, we present +the nanopost and its performance in terms of Fp and ε, and we +demonstrate the breakdown of the single-mode model. In Sec- +tion 3, we present our theoretical framework based on the Fourier +Modal Method, which we subsequently use to analyze the com- +plex interplay with radiation mode channels in Section 4 and its +influence on the collection efficiency and the Purcell factor. In +Section 5 we put the nanopost physics into perspective and dis- +cuss its impact on SPS engineering, followed by our conclusion. +Additional simulation results are presented in the Supplementary +Information. +2 +The nanopost geometry and the break- +down +of +the +single-mode +Fabry-Pérot +model +The nanopost shown in Fig. (1) consists of a truncated GaAs +nanowire with diameter D on top of a SiO2-Au mirror. The top +of the nanowire is flat, and the surrounding medium is air. The +SiO2 layer, located between the nanowire and the gold, has a +thickness indicated by tSiO2. The QD, modelled as a dipole, is +placed on-axis inside the nanowire at a position hb from the bot- +tom interface and ht from the top interface. The refractive in- +dices of the materials are chosen as nGaAs = 3.46, nSiO2 = 1.5 and +nAu = 0.201+5.85i at λ = 930nm and assumed to be constant as a +function of wavelength. In the inset of Fig. (1), a dipole with an +emission wavelength of 930nm is placed inside the nanopost, and +the resulting in-plane electrical field, simulated using the FMM, +is shown. Three antinodes can be seen in the field profile, cor- +responding to the order 3 cavity mode, and they are enumerated +from the bottom mirror as the 1st, 2nd and 3rd antinode. The +field profile generated by the dipole is independent of the vertical +position of the dipole, only the intensity changes. The intensity is +not the same at the three antinodes due to the breakdown of the +SMM. +We now present the performance of the nanopost as a func- +tion of the diameter, D, and the silica layer thickness, tSiO2. We +have scanned the parameter ranges D = 196nm to D = 300nm and +tSiO2 = 0nm to tSiO2 = 25nm and chosen a design wavelength of +λd = 930nm. The height of the structure and the position of the +QD are dynamically changed to keep the order 3 cavity mode +resonance at λr = 930nm for the QD at the 2nd antinode. The +required procedure is presented in Supplementary 7.1. +3 +4 +5 +5 +5 +6 +6 +6 +7 +7 +7 +7.5 +7.5 +7.75 +2 +3 +3 +4 +4 +4 +5 +5 +5 +6 +6 +6 +6.5 +6.5 +2 +3 +4 +5 +6 +7 +2 +3 +3 +4 +5 +5 +5 +6 +6 +6 +7 +7 +7.5 +Fig. 2 Purcell factor, Fp, for a QD at the 2nd antinode (a), 1st antinode +(b) and any antinode using the SMM (c) as a function of the diameter, +D, and the silica layer thickness, tSiO2. +The first quantity of interest is the spontaneous emission rate +ΓT. For the high-β structures investigated here, the total nor- +malized rate ΓT/ΓBulk and the Purcell factor Fp are similar, and +2 | +1–18 + +100 nmwe will in the following refer to the normalized total rate as the +"Purcell factor Fp". In Fig. (2a,2b) the Purcell factor is shown as +a function of the diameter and the silica layer thickness for a QD +placed in the 2nd and 1st antinode. In the entire parameter space, +the Purcell factor is larger for the 2nd antinode and a maximum +value of Fp = 7.9 is reached at D = 250nm and tSiO2 = 13nm. This +discrepancy between the two antinodes also demonstrates the de- +viations of the SMM. Overall, the tendency of the Purcell factor +is similar at the 2 antinodes with one peak value. The minimum +is located in the corner of no silica and the smallest diameter. In +Fig. (2c) the Purcell factor is now shown using the SMM. Com- +paring the SMM to the full model for the two antinodes, there +are both positive and negative deviations across most of the pa- +rameter space. Compared to the 2nd antinode, the SMM also +predicts a slightly lower value for the maximum Purcell factor +of Fp = 7.5, but at a very different position of D = 242nm and +tSiO2 = 8nm. However, compared to the 1st antinode the SMM +predicts a larger value for the maxmimum Purcell factor. This is +an invitation to obtain a better description and understanding of +the physics responsible for the Purcell factor, which we will pro- +vide in this paper. +0.3 +0.3 +0.4 +0.4 +0.5 +0.5 +0.6 +0.6 +0.65 +0.65 +0.68 +0.2 +0.3 +0.3 +0.4 +0.4 +0.5 +0.5 +0.6 +0.6 +0.65 +0 +0.2 +0.4 +0.6 +0.8 +1 +0.25 +0.25 +0.3 +0.3 +0.3 +0.4 +Fig. 3 Collection efficiency, ε (NA = 0.75), for a QD at the 2nd antinode +(a), 1st antinode (b) and any antinode using the SMM (c) as a function +of the diameter, D, and the silica layer thickness, tSiO2. +The source collection efficiency, ε, for the 2nd antinode, the +1st antinode and the SMM are shown in Fig. (3) for a numerical +aperture of NA = 0.75. Here the overall performance for the 2nd +antinode is slightly better than for the 1st antinode, and a maxi- +mum value of ε = 0.69 is reached at D = 238nm and tSiO2 = 0nm. +Despite the very simple geometry of the nanopost, a surprisingly +high collection efficiency of ε = 0.69 is achievable. In general, it +can be ascertained that no silica layer is more beneficial for the +efficiency, which is surprising compared to the traditional pho- +tonic nanowire, where the silica layer enhances the modal re- +flectivity22 and thus the collection efficiency. +Now comparing +the efficiency of the SMM to the full model, there are substan- +tial differences. This is unlike the micropillar and the photonic +nanowire geometries for which the single-mode model εs = βγ +is an excellent12–14 approximation. For the nanopost, the SMM +efficiency is much smaller in the entire parameter space except +for the smallest diameters. This clearly shows that there are im- +portant physical mechanisms which are not accounted for in the +SMM. Furthermore, comparing the figures for the Purcell factors +and the efficiencies, there seems to be no apparent correlation +between the two. This is also a surprising result compared to tra- +ditional Fabry-Pérot cavities and indicates that there are different +physical mechanisms at play which govern the Purcell factor and +the efficiency. +In Supplementary 7.2, we vary the numerical aperture and +present its influence on the collection efficiency. In Supplemen- +tary 7.3, we present the collection efficiency taking into consider- +ation the overlap with a Gaussian profile. +3 +Theory +3.1 +Method +We use an eigenmode method combined with a standard scatter- +ing matrix formalism32 in the frequency domain. In this method, +the structure is divided into layers of uniform permittivity along +the propagation direction. +In each layer, the electrical field is +expanded on the eigenmodes, and the scattering matrices are +used to connect the eigenmodes at the interfaces between the +layers. +The eigenmodes are obtained using the Fourier modal +method (FMM) with open boundary conditions31, which provides +direct access to the modes needed to understand the physics. The +nanopost structure is split into four layers: the bottom gold sub- +strate, the silica layer, the nanowire and the top air region. Due to +the cylindrical symmetry of the nanowire, a cylindrical coordinate +system is used. The QD is modelled as a classical point dipole, and +we make use of the relationship Γ/ΓBulk = P/PBulk to calculate the +Purcell factor33. P is the emitted power of the dipole, and PBulk is +the power emitted in a bulk medium, and thus the Purcell factor +is defined as Fp = PT/PBulk. The second quantity of interest is the +collection efficiency defined as ε = Pcollected/PT, where Pcollected is +the power collected in the far-field of a lens with numerical aper- +ture NA. +3.1.1 +Eigenmodes +The electrical field for an eigenmode has the following expres- +sion: +Ej(r,φ,z) = ej(r,φ)exp(iβ jz), +(2) +where j refers to eigenmode index, ej(r,φ) is the mode profile for +the given eigenmode and βj is the propagation constant. For three +out of the four layers, the permittivity profile is constant over the +entire plane. In this case, the eigenmodes are simply cylindri- +cal plane waves. These eigenmodes exist as a continuum where +1–18 | 3 + +the propagation constant takes the value β = +� +(nlayerk0)2 −k2 +⊥, +where nlayer is the refractive index, k0 is the free-space wavenum- +ber and k⊥ is the in-plane k-value. +k⊥ can take any value +k⊥ ∈ [0,∞] and eigenmodes exist for all the values. For each β, +two orthogonal solutions exist, which can be separated into TE +modes (ez = 0) and TM modes (hz = 0). +For layers with real- +valued refractive index, this continuum can be separated into +radiation modes satisfying 0 < (β)2 ≤ (nlayerk0)2 and evanescent +modes (β)2 < 0 which decays exponentially and carry no power +according to the Poynting vector. Specifically for the air layer, +the propagation constant of the radiation modes can directly be +interpreted as the propagation angle with respect to the z-axis us- +ing the expression θ = arccos(β/k0). Expanding a forward prop- +agating electrical field on the eigenmodes will have the following +expression: +E(r,φ,z) = +2 +∑ +s=1 +� ∞ +0 as(k⊥)es(r,φ,k⊥)exp(iβ(k⊥)z)dk⊥, +(3) +where s refers to the two solutions and as(k⊥) is the expansion co- +efficient. Numerically, the continuum is discretized into N modes +and truncated with a cut-off value for k⊥ 31,34, which will lead to +the following expression: +E(r,φ,z) = +N +∑ +j=1 +a jej(r,φ)exp(iβ jz), +(4) +where s is absorbed into j, and the ∆k⊥ that would appear due +to the discretization is absorbed into definition of the eigenmode +profile. +The first class of eigenmodes for the nanowire are the guided +modes for which the propagation constants satisfy the condition +(nairk0)2 < (βj)2 < (nGaAsk0)2. The guided mode is confined to +the core of the nanowire, and outside the nanowire, the field +decays. +There is a finite number of guided modes, and there +will always be at least one guided mode, the fundamental HE11 +mode. The description of guided modes can be found in various +textbooks35. The second class of eigenmodes is the background +continuum, very similar to the continuum of the air layer. These +eigenmodes can be viewed as perturbed versions of the cylindri- +cal plane waves of the air layer and be separated into radiation +and evanescent modes the exact same way. There are also two +orthogonal solutions, but these can no longer be separated into +pure TE and TM modes. Studies of this class of eigenmodes are +plentiful in the literature36–45. +3.1.2 +Dipole emission in an infinite structure +The QD is modelled as a classical point dipole with in-plane ori- +entation and harmonic time dependence at the frequency ω. The +corresponding current density is J(r) = −iωpδ(r − rd), where rd +is the position of the QD and p is the dipole moment. The emitted +power of the dipole can be calculated as33: +P = −1 +2 +� +V Re[J∗(r)·E(r)]dV = ω +2 Im[p·E(rd)]. +(5) +The total emitted power, PT, can be calculated by evaluating the +total field, ET(rd), but also the power into individual modes, Pj, +by evaluating E j(rd). +Placing the dipole on-axis inside an in- +finitely long nanowire with position zJ will result in the following +electrical field: +ET(r) = ∑ +j +aJ +je+ +j (r⊥)exp(iβ j(z−zJ)) (z > zJ) +(6) +ET(r) = ∑ +j +bJ +je− +j (r⊥)exp(−iβj(z−zJ)) (z < zJ), +(7) +where the superscript + refers to forward propagating and − +refers to backward propagating. For the electrical fields, the for- +ward and backwards propagating fields are identical: ej = e+ +j = +e− +j . aJ +j and bJ +j are the field expansion coefficients and in an in- +finite structure, such as the nanowire, have the following simple +expressions32: +aJ +j = − +−iωp·e+ +j (rd) +2 +(8) +bJ +j = − +−iωp·e− +j (rd) +2 +. +(9) +The expansion coefficients can then be represented by vectors: +a∞NW = +� +aJ +1 +aJ +2 +··· +aJ +N +� +(10) +b∞NW = +� +bJ +1 +bJ +2 +··· +bJ +N +� +, +(11) +where subscript ∞NW refers to the infinite nanowire. +3.1.3 +Multilayered structures and scattering matrices +The reflection and transmission matrices are used to connect the +field at the interfaces between the layers. +These matrices are +derived from the boundary condition that the tangential compo- +nents of the electric and magnetic field are continuous across an +interface32. The coefficients in the matrices describe how a given +mode is transmitted or reflected into another mode. A reflection +matrix is shown in Eq. (12). +R = +� +����� +r11 +r12 +r13 +... +r1N +r21 +r22 +r23 +... +r2N +... +... +... +... +... +rN1 +rN2 +rN3 +... +rNN +� +����� +. +(12) +The r11 coefficient represents the reflection of the fundamental +mode back into itself, while the remaining part of the first column +represents the reflection of the fundamental mode into all other +modes and so forth. Propagation matrices are used to propagate +the field inside a layer and are defined in the following way: +P(z) = +� +����� +eiβ1z +0 +0 +... +0 +0 +eiβ2z +0 +... +0 +... +... +... +... +... +0 +0 +0 +... +eiβNz +� +����� +. +(13) +4 | +1–18 + +The total field inside the structure can then be calculated by tak- +ing into account the round-trips which the initially emitted light +takes inside the cavity. Above the emitter, the field takes the fol- +lowing expression32: +ET(r) = ∑ +j +aJ +tot, je+ +j (r⊥)exp(iβ j(z−zJ)) ++btot, je− +j (r⊥)exp(−iβj(z−zJ)) +(z > zJ), +(14) +where the new expansion coefficients are calculated using the fol- +lowing equation32: +aJ +tot = (I−P(hb)RbotP(hb)P(ht)RtopP(ht))−1 +(a∞NW +P(hb)RbotP(hb)b∞NW) +(15) +and +btot = P(ht)RtopP(ht)aJ +tot. +(16) +Now the Purcell factor can be calculated by evaluating the total +field at the dipole position using Eq. (5). +3.1.4 +Far-field and efficiency +To obtain the field in the air above the structure, we apply the +propagation and transmission matrix on the forward propagating +light in the cavity and thus obtain the expansion coefficients. +aair = TtopP(ht)aJ +tot. +(17) +To calculate the collected power in a lens with some numerical +aperture, a near- to far-field transformation is used46. The far- +fields EFF(R,θ,φ) and HFF(R,θ,φ) are calculated on the surface of +a sphere with radius R, and the radial component of the resulting +Poynting vector is: +SFF(R,θ,φ) = (E∗ +FF,θHFF,φ −E∗ +FF,φHFF,θ). +(18) +The collected power in the far-field is then: +PFF(NA) = 1 +2R2 +� 2π +0 +� θNA +0 +SFF(R,θ,φ)sin(θ)dθdφ += +� 2π +0 +� θNA +0 +pFF(θ,φ)sin(θ)dθdφ, +(19) +where pFF(θ,φ) is the power per unit solid angle in the far-field +and θNA is determined by the NA (NA = sin(θNA)). The R depen- +dence cancels out as the Poynting vector scales as 1/R2. +3.1.5 +Single-mode Fabry-Pérot model +When calculating the Purcell factor using the SMM, we only con- +sider the fundamental mode. The SMM equations equivalent to +Eq. (15-16) are +aJ +tot,SMM = aJ +1 +1+r11,botei2hbβ1 +1−r11,botr11,topei2htotalβ1 , +(20) +and +btot,SMM = aJ +tot,SMMr11,topei2htβ1. +(21) +The collected power in the far-field is then calculated by inserting +Eq. (20) into Eq. (17), and the SMM efficiency is then defined as +εs = Pcollected,SMM/PT, equivalent to the definition in the introduc- +tion. +3.2 +Mode-coupling and emission channels +Important coupling effects take place at the top and bottom in- +terfaces of the nanowire. At both interfaces, all the modes couple +to each other, i.e. all the elements in the reflection matrices are +non-zero; however, some modes and elements are more impor- +tant than others. In Fig. (4), different examples of mode coupling +are shown along with the emission channels that will contribute +to the far-field. The sketch is divided into two parts: the main +channels and the background channels. The main channels con- +sist of all the light that originated as the fundamental mode, αt +and αb, which is indicated by the red arrowheads. This is the +propagating mode that experiences sufficiently large reflections +at both interfaces such that it is Purcell enhanced. +c1 +Main channels +Background channels +c2 +c3 +t +b +t +b +t,r +b,r +t,� +b, +� +γt +γb +γt,α +γb,α +αt,γ +αb,γ +loss +γt,β +Fig. 4 Sketch of the nanopost and the different emission channels and +examples of mode-coupling. The emission channels are separated into the +main channels and the background channels. The red color corresponds +to the fundamental mode, the blue color corresponds to radiation modes +and the green color corresponds to evanescent modes. +Some arrows +have two colors, where the color of the arrowhead corresponds to the +original channel, but the 2nd color on the shaft signifies the current +mode classification. As an example consider c1 which is the transmission +of the fundamental mode into the air. It has a partly blue shaft as it is +now classified as radiation, but it originated as the fundamental mode +(red arrowhead). See main text for more information. +1–18 | 5 + +3.2.1 +Path of the fundamental mode +Let us now follow the path of the fundamental mode. The light +emitted into the fundamental mode will propagate upwards and +downwards indicated by αt and αb. At the top interface the fun- +damental mode is: +• Transmitted into the air indicated by c1. +• Reflected, indicated by αt,r (the channel responsible for Pur- +cell enhancement). +• Scattered into radiation that propagates downwards indi- +cated by the red/blue arrow pointing towards the bottom +mirror. This radiation will then be reflected by the bottom +mirror and then be transmitted into the air indicated by c2. +• Coupled to evanescent modes indicated by αt,γ. +At the bottom interface the fundamental mode is: +• Transmitted into the mirror and lost indicated by the +green/red arrow pointing downwards at the very bottom. +• Reflected, indicated by αb,r (the channel responsible for Pur- +cell enhancement). +• Scattered into radiation that propagates towards the air in- +dicated by c3. +• Coupled to evanescent modes indicated by αb,γ. +As we will demonstrate in the following, the three radiation +channels c1, c2 and c3 are the main channels that will contribute +to the far-field. +3.2.2 +Path of the background emission +Let us now consider the background emission channels. +First, +we have the light directly emitted into radiation, indicated by +the blue arrows of βt and βb. The radiation can both be emitted +upwards or downwards and then reflected by the mirror. At both +interfaces, a small part of the radiation modes can also couple to +the fundamental mode indicated by the blue/red arrows of βt,α +and βb,α. We also have light coupled to the evanescent modes, +which is indicated by the long green arrows pointing upwards +and downwards of γt and γb. These modes do not propagate in +a traditional sense, but at the interfaces, they can scatter into +the fundamental mode indicated by the green/red arrows of γt,α +and γb,α at the top and bottom interfaces. At the top interface, the +evanescent modes can also couple to radiation and be transmitted +indicated by the green/blue arrow pointing upwards, γt,β . +4 +Analysis of the Purcell factor and effi- +ciency +The starting point of the analysis is the emission rates in the in- +finitely long GaAs nanowire, which directly represent the initial +coefficients of Eq. (10-11) through Eq. (5-7). In Fig. (5), the +emission rates for the present guided modes, the radiation modes +and the total emission are shown as a function of the nanowire +diameter. The black dotted vertical lines represent the interval +which is used in the full simulations of the nanopost. +In this +interval, the infinite nanowire only contains one guided mode, +the fundamental HE11 mode, and most of the power is emitted +into this mode. The emission into radiation is suppressed in most +of the interval and only begins to increase when the diameter +reaches 300nm before the EH11 mode appears. The emission rates +thus show that the radiation background channel (the initial coef- +ficients for the radiation) presumably only has a minor influence +on the Purcell factor and efficiency as long as the cavity does not +suppress the fundamental mode. Fig. (5) cannot be used to quan- +tify the importance of the evanescent background channels, as +the field components of the evanescent eigenmodes only have a +real part. Later on, in the third subsection, models will be used +to quantify the effect of the evanescent background channels the +finite-length nanopost structure. +4.1 +Scattering of the fundamental mode at the interfaces +Here, we will study reflection and transmission at the top inter- +face between the GaAs nanowire and the air above. +The fun- +damental mode is launched towards the interface, and then the +reflection and transmission coefficients are calculated. The power +reflection coefficient for reflection of mode n into mode m is +Rm,n = |rm,n|2, and the power transmission coefficients are defined +similarly. The total power reflection of the fundamental into ra- +diation is then defined as Rrad,1 = ∑2 +s=1 +� k0 +0 |rs,1(k⊥)|2dk⊥, and in +the discretized regime Rrad,1 = ∑ +Nk0,wire+1 +n=2 +|rn,1|2, where the index +Nk0,wire corresponds to the total number of radiation modes in the +nanowire. +Thus the total power reflection of the fundamental +mode is Rtotal,1 = R1,1 +Rrad,1. The total power transmission of the +fundamental mode is then defined as Ttotal,1 = ∑ +Nk0,air +n=1 |tn,1|2 and +due to power conservation we have Rtotal,1 +Ttotal,1 = 1. +In Fig.(6), the power reflections, along with the power trans- +mission of the fundamental mode, are shown as a function of +the nanowire diameter D. By comparing the magnitudes of the +modal reflection (R1,1) and the reflection into radiation (Rrad,1), +it is clear that the reflection into radiation (c2 in Fig.(4)) is es- +sential for the far-field. This indicates why the SMM fails to de- +scribe the efficiency. This mechanism will have a much smaller +influence on the Purcell factor as the radiation modes only has +a small field amplitude at the center of the nanowire shown in +the previous section (Fig. (5)). However, the reflection matrix is +approximately symmetric, such that the first column and the first +row is identical, r1,n ≈ rn,1. This means that a small part of the +radiation will actually scatter back into the fundamental mode +at the top interface, and this will have an influence on the Pur- +cell factor, which we will show later. In general for Fig. (6), we +observe small modal reflections for small diameters and this will +lead to a limited cavity effect and thus a lower Purcell factor. +Prad +PHE11 +Fig. +5 Power emission in the infinite nanowire as a function of the +diameter, D. A sketch of the emission in an infinite nanowire with an +embedded QD is shown in the right part of the figure. +6 | +1–18 + +Fig. 6 Reflection and transmission of the fundamental mode as a function +of the diameter, D, at the top interface. A sketch of top interface and +the reflection and transmission of the fundamental mode is seen to the +right. +We now consider the reflections at the bottom interface be- +tween the GaAs nanowire and the silica-gold mirror. Compared +to the top interface, there is now an additional parameter, namely +the thickness of the silica layer, tSiO2. +The purpose of the sil- +ica layer is to increase the reflection of the fundamental mode +and avoid coupling to surface plasmons which would decrease +the reflection47. In Fig. (7), the bottom reflection coefficients are +shown as a function of the diameter and the silica layer thickness. +In the parameter ranges where the modal reflection is large ∼ 0.9 +(strong cavity effect), the reflection into radiation is small ∼ 0.01. +Here we do not expect a significant contribution of the scattering +into radiation, c3. However, as the modal reflection decreases, the +reflection into radiation increases to larger values ∼ 0.1, and here +c3 will contribute to the far-field. For small diameters and low +values of the silica layer thickness, the modal reflection is small +(weak cavity effect) and the scattering into radiation is very large +∼ 0.2. This behavior has been described in the literature47. +Fig. 7 (a) Modal reflection, R1,1 and (b) reflection into radiation of the +fundamental mode, Rrad,1, as a function of the diameter, D, and silica +layer thickness, tSiO2, at the bottom interface. A sketch of the bottom +interface is shown in the top part of the figure. +Due to the scale invariance of Maxwell’s equations, both the top +and bottom reflections are broadband, which gives the potential +for broadband Purcell enhancement. +4.2 +Enhanced efficiency +In the first part of this subsection, we present different methods +to model the efficiency to show how important the different emis- +sion channels are. Then we will apply the modelling methods on +the structure with the largest efficiency, namely D = 238nm and +tSiO2 = 0nm with a QD placed at the 2nd antinode. +4.2.1 +Efficiency contributions of the emission channels +We wish to separate and quantify the efficiency contributions of +the main channels and the background channels shown in Fig. +(4). We also wish to separate and quantify the direct emission +of c1 and the scattered channels of c2 and c3 shown in Fig. (4), +and therefore we need two different methods. Recall that the +efficiency is calculated as ε = Pcollected/PT. For both methods, PT is +calculated using the full model, but Pcollected is calculated such that +we can either 1. separate the main channels and the background +channels or 2. separate the direct emission of c1 and the scattered +channels of c2 and c3. +In the first method, the reflection and transmission matrices are +unchanged; however, a varying number of the initial coefficients, +a∞NW and b∞NW of Eq. (10-11), are included when aJ +tot (Eq. (15)) +and thus Pcollected is calculated. For instance, if we only include +the first element of the initial coefficients, aJ +1 and bJ +1, and put +the remaining elements to zero, then we only include the main +channels in Pcollected, which originated as the fundamental mode. +As we increase the number of initial coefficients included, the +background channels are added starting from the first radiation +mode until the last evanescent mode. ε can then be plotted as a +function of the included initial coefficients and if this curve is flat, +then the main channels dominate the efficiency. +In the second method, the reflection and transmission matrices +are also unchanged, all initial coefficients are included, but in- +stead, a varying number of the final coefficients, aJ +tot of Eq. (15), +are included when Pcollected is calculated. The first element of aJ +tot +represents all the light that ended up in the fundamental mode, +where the main contribution is from the fundamental mode it- +self, but also includes contributions of the background channels +which have scattered into the fundamental mode such as βt,α, +βb,α, γt,α and γb,α seen in Fig. (4). However, by using the first +method, we can quantify how strong these contributions are. If +these contributions are weak, then the main channels are dom- +inating. Thus if only the first element of the final coefficients is +included in Pcollected, then only the direct transmission of the fun- +damental mode is included in the far-field, namely c1 in Fig.(4). +Then as we increase the number of elements of the final coeffi- +cients, the contributions of c2 and c3 are included. In principle, +channels such as βt and βb are also included, but again we use +the previous method to quantify these. ε can then be plotted as +a function of the included final coefficients, and if this curve is +increasing, then the channels c2 and c3 are contributing to the +efficiency. +Finally, we wish to visualize the interference between the direct +1–18 | 7 + +transmission of the fundamental mode and the scattered channels +by calculating the transmission of the fundamental mode and the +entire background continuum separately: +aair,HE11 = TtopP(ht)aJ +tot,HE11 +(22) +and +aair,BG = TtopP(ht)aJ +tot,BG, +(23) +where +aJ +tot,HE11 = +� +atot,1 +0 +··· +0 +� +(24) +and +aJ +tot,BG = +� +0 +atot,2 +··· +atot,N +� +. +(25) +Then the phase difference between the two contributions can be +calculated: +∆φ = arg(aair,HE11)−arg(aair,BG). +(26) +The phase difference will be separated into TE and TM modes. +Along with the phase difference, the far-field plots of the di- +rect transmission of the fundamental mode (aair,HE11), the entire +background continuum (aair,BG) and the total field (aair) will be +shown. +4.2.2 +Influence of scattered radiation on collection effi- +ciency +Fig. 8 (a) Efficiency, ε (NA = 0.75), as a function of the initial coefficients +expressed with the propagation constant (β/k0)2. (b) Efficiency, ε (NA = +75), as a function of the final coefficients expressed with the propagation +constant (β/k0)2. +We will now apply the modelling methods for the structure with +the largest efficiency, D = 238nm and tSiO2 = 0nm. In Fig. (8) the +efficiency is shown as a function of the initial coefficients (Fig. +(8a)) and the final coefficients (Fig. (8b)), expressed with the +propagation constant (β/k0)2. This corresponds to using the two +different methods for calculating the efficiency presented in the +previous subsection. The very first red point in Fig. (8a) corre- +sponds to the fundamental mode (main channels) and includes +all scattering channels of the fundamental mode (c1, c2 and c3). +This is sufficient to describe most of the efficiency. Then the initial +background radiation modes (blue) are added one by one, start- +ing from larger values of (β/k0)2, i.e. from predominantly vertical +emission. This part of the curve is very flat, which means that the +initial background radiation modes (channels originating from βt +and βb) are not crucial for the efficiency, which was also indicated +by the low emission rates of the radiation modes in the infinite +nanowire. Finally, the evanescent background modes (green) are +included from small negative values of (β/k0)2, i.e. slowly de- +caying evanescent modes, to large negative values of (β/k0)2, i.e. +fast decaying evanescent modes. Here there is a small increase +due to the slowly decaying evanescent modes. Now in Fig. (8b), +the very first red point also corresponds to the fundamental mode +but only includes the direct transmission, c1. Here the efficiency +is only ∼ 0.2, much smaller than the total efficiency. Then the +scattered radiation modes (blue) corresponding to c2 and c3 are +added one by one, also starting from larger values of (β/k0)2, i.e. +from predominantly vertical emission. Here there is a massive +increase in the efficiency, which proves the importance of c2 and +c3 to the efficiency. At some point, there is a kink in the blue +part of the curve, which is due to the limited numerical aper- +ture, as an NA of 0.75 corresponds to (β/k0)2 = 0.36. However, +the remaining part of the curve is not completely flat, and this is +due to the non-perfect transmission of the radiation modes of the +nanowire to the radiation modes in the air. The radiation modes +of the nanowire mainly transmit into radiation modes with the +same value of β, but there is some scattering into the other radia- +tion modes. Finally, there is a tiny decrease due to the evanescent +modes, as very few of the evanescent modes transmit into radi- +ation at the top interface. By comparing the evanescent parts of +Fig. (8a) and Fig. (8b), we see that the effect of the evanescent +modes is mainly back scattering into other modes at the interfaces +rather than direct transmission scattering. To summarize, the key +point in the comparison between Fig. (8a) and Fig. (8b) is that +the scattering into radiation of the fundamental mode is crucial +for the efficiency. +Now we will visualize the interference between the direct trans- +mission of the fundamental mode and the scattered channels by +inspecting the phase changes and the far-fields. In Fig. (9a), the +phase difference in the air layer between the direct transmission +of the fundamental mode and the entire background is shown as +a function of the propagation constant for TE and TM modes. For +the light that propagates vertically, the phase difference is close +to zero, such that there is constructive interference between the +direct transmission and the background radiation. For the light +that propagates horizontally, the phase difference is closer to π, +and thus there is destructive interference. In Fig. (9b), Fig. (9c) +and Fig. (9d) the far-fields of the direct transmission of the fun- +damental mode, the background radiation and the total field are +shown. Here we can directly observe the effect caused by the +phase difference. In the center of the total far-field, the field is +enhanced due to the constructive interference, but for the light +that propagates horizontally there is destructive interference. As +such, the interference between the direct emission and the radia- +tion focuses the far-field. +8 | +1–18 + +Fig. 9 (a) The phase difference between the direct transmission of the +fundamental mode and the background continuum for TE and TM modes +as a function of the propagation constant, β/k0. (b) The far-field of the +fundamental mode. (c) The far-field of the background continuum. (d) +The total far-field. The white dotted line indicates NA = 0.75. Be aware +of the different color scales that have been used for the far-fields. +A similar analysis of the efficiency for the structure with the +largest Purcell factor is included in Supplementary 7.4. +4.3 +Enhanced Purcell factor +As shown in Fig. (2), there are deviations between the full model +and the SMM for the Purcell factor, and we wish to understand +where these deviations appear. +Therefore we will introduce a +model which can identify where these deviations appear and ap- +ply the model on the structure with the largest Purcell factor, +namely D = 250nm and tSiO2 = 13nm. +4.3.1 +Purcell factor contributions from the emission chan- +nels +To gain physical insight into the physics of the Purcell factor, we +will use a model which stepwise increases the complexity. At each +step the Purcell factor is calculated Fp = PT/PBulk along with the +power into the fundamental mode, PHE11/PBulk. The starting point +is the SMM, where only the fundamental mode is included; then, +in seven steps, the complexity increases until the full model is +reached. Specifically, the initial inputs (a∞NW and b∞NW) and the +top and bottom reflections (Rtop and Rbot) will be manipulated at +each step. Each step has a direct physical interpretation. We will +now list the 7 steps in the model and for each step, write up the +physical effect that is now included. This means that for each step +in the model, all previous effects are also included. +1. SMM +2. Scattering of the fundamental mode at the top interface +3. Scattering of the fundamental mode at the bottom interface +4. Back-scattering of the background continuum to the funda- +mental mode at the top interface +5. Back-scattering of the background continuum to the funda- +mental mode at the bottom interface +6. Scattering of the background continuum to itself at both in- +terfaces +7. Including initial background continuum +Steps number 4 and 5 correspond to the process HE11 → +radiation/evanescent → HE11 which is a recycling effect, and +we will show the importance of this process for the Pur- +cell +factor. +Step +number +6 +corresponds +to +the +process +radiation/evanescent → radiation/evanescent, which also opens +up for further scattering channels such as radiation/evanescent → +radiation/evanescent → HE11. +The 7 steps can then be translated to the initial inputs and the +top and bottom reflections in the following schematic way: +a(α) +∞NW = +� +1 +7 +··· +7 +� +(27) +b(α) +∞NW = +� +1 +7 +··· +7 +� +(28) +R(α) +top = +� +������� +1 +4 +4 +... +4 +2 +2 +6 +... +6 +2 +6 +2 +... +6 +... +... +... +... +... +2 +6 +6 +... +2 +� +������� +top +(29) +R(α) +bot = +� +������� +1 +5 +5 +... +5 +3 +2 +6 +... +6 +3 +6 +2 +... +6 +... +... +... +... +... +3 +6 +6 +... +2 +� +������� +bot +(30) +Here, the superscript α represents the seven complexity steps. +For a given step α, each matrix entry > α is set to zero, while +entries ≤ α keep their original value. We will use the step α = 4 +as an example: +a(4) +∞NW = +� +a1 +0 +··· +0 +� +(31) +b(4) +∞NW = +� +b1 +0 +··· +0 +� +(32) +R(4) +top = +� +������� +r11 +r12 +r13 +... +r1N +r21 +r22 +0 +... +0 +r31 +0 +r33 +... +0 +... +... +... +... +... +rN1 +0 +0 +... +rNN +� +������� +top +(33) +R(4) +bot = +� +������� +r11 +0 +0 +... +0 +r21 +r22 +0 +... +0 +r31 +0 +r33 +... +0 +... +... +... +... +... +rN1 +0 +0 +... +rNN +� +������� +bot +(34) +At each step in the previous model building, the entire back- +ground continuum was added, i.e. the entire column or row was +1–18 | 9 + +1 +(a) +/ +0.5 +-TE +0 +TM +5HEi1 FF +0.20 +0.5 +1 +β/ko +Scattered FF +Normalized units +0.2 +0Total FF +(p +0.5added at once. As such, it is difficult to quantify which part of +the background continuum that is important. Therefore we will +also present the models where the elements for the background +continuum are increased one at a time. +In this way it can be +quantified how the different parts of the background continuum +contribute to the Purcell factor. +4.3.2 +Influence of radiation modes on the Purcell factor +We will now apply the model for the Purcell factor for the struc- +ture with the largest Purcell factor, D = 250nm and tSiO2 = 13nm. +In Fig. (10), the Purcell factor is shown as a function of the dipole +position from the bottom, hb. The agreement between the SMM +and the full model for the 1st antinode is good, but there are devi- +ations for the 2nd and 3rd antinodes. Nevertheless, we will focus +on the analysis of the 1st and the 2nd antinode. +In Fig. (11a) and Fig. (11b), the Purcell factor and the power +enhancement of the fundamental mode are shown as a function +of the model complexity progression for the 1st and 2nd antinode +(model complexity number α will be shortened n. α). Evidently, +the analysis of the Purcell factor is complicated due to contri- +butions of the entire background continuum, multiple scattering +channels and feedback mechanism. Therefore there are changes +in the Purcell factor for all steps in the model complexity, which +makes it challenging to model the Purcell factor using only a few +modes. The continuum of radiation modes can be modelled using +leaky modes, which can enable the modelling using only a few +modes. This has been demonstrated in photonic crystal micro- +cavities where strong feedback mechanisms also were present48. +However, by using the presented model, we will obtain an in- +depth physical insight. +The Purcell factor starts at the same value with the SMM for +both antinodes. By including scattering of the fundamental mode +at the top interface (n. 2) there is a significant increase for the +2nd antinode but a very small decrease for the 1st antinode. Al- +ready now, the deviations compared to the SMM have started to +appear. By including the scattering at the bottom interface (n. 3) +the 2nd antinode is almost unaffected, but a small decrease ap- +pears for the 1st antinode. Now including the back-scattering at +the top interface (n. 4), there is a large decrease in the Purcell +factor at both antinodes. This decrease is directly represented +in PHE11/PBulk. Interestingly, when including the back-scattering +at the bottom interface (n. 5), there is now a large increase in +the Purcell factor at both antinodes, which is also represented +Fig. 10 Purcell factor, Fp, computed using the full model and SMM as a +function of the dipole position from the bottom interface, hb. D = 250nm +and tSiO2 = 13nm. +Fig. +11 Purcell factor (PT/PBulk) and fundamental mode enhancement +(PHE11/PBulk) for the 1st (a) and 2nd (b) antinode as a function of the +model complexity progression. In (c) and (d) the background continuum +is continuously included between each model complexity. +in PHE11/PBulk. This also shows that the recycling effect HE11 → +radiation/evanescent → HE11 can provide both negative and pos- +itive contributions. When the background is allowed to scatter +to itself, i.e. radiation/evanescent → radiation/evanescent (n. 6), +there is an increase for both antinodes. This increase is directly +represented in PHE11/PBulk, which in fact means that the process +radiation/evanescent → radiation/evanescent → HE11 is dominat- +ing compared to radiation/evanescent → radiation/evanescent it- +self. Finally, by including the initial background (n. 7), there is +an increase for the 2nd antinode but a decrease for the 1st antin- +ode. These changes also correspond to the change in PHE11/PBulk, +which means it is the process of radiation/evanescent → HE11 that +is important for the initial background. +The main differences between the two antinodes appear, when +the fundamental mode scatters at the top interface (n. 2) and +when the initial background (n. 7) is included. To better under- +stand which part of the background continuum is important, we +will also consider the continuous steps of the model complexity. +This is now shown in Fig. (11c) and Fig. (11d). Here we observe +that the main positive contributions at the 2nd antinode are due +to the slowly decaying evanescent modes. This is the process of +HE11 → evanescent at the top interface and the process of the ini- +tial background evanescent → HE11. For the back-scattering at +the interfaces (n. 4 and n. 5), we observe that the propagating +radiation modes can also significantly affect the Purcell factor. +An additional important observation is that PHE11/PBulk exceeds +PT/PBulk for the first antinode due to the negative contributions. +This would in fact result in β factors above 1 using the definition +βHE11 = PHE11/PT. This indicates that the β factor might not be +a suitable figure of merit for structures where the SMM breaks +down, or at least one should be very careful in the definition of the +β factor. Alternatively, the typical interpretation of the β factor as +10 | +1–18 + +Fig. +12 (a) and (b) Purcell factor, Fp, of the two antinodes for the +two structures as a function of wavelength, λ. (c) and (d) efficiency, ε +(NA = 0.75), of the two antinodes for the two structures as a function of +wavelength, λ. The numerical uncertainty is represented by the thickness +of the curves. +a standard power fraction should be reconsidered in the regime +of the SMM breakdown. +The analysis of the Purcell factor for the structure with the +largest efficiency, D = 238nm and tSiO2 = 0nm, is included in Sup- +plementary 7.5. +4.4 +Wavelength dependence +In this section, we will present the broadband performance of the +nanopost. The focus will be on the designs with the largest Purcell +factor and efficiency, respectively. +In Fig. (12a), the Purcell factor is shown as a function of the +wavelength for the 1st and 2nd antinode of the structure with +the largest Purcell factor. The Purcell factor of the 2nd antin- +ode performs better than the 1st antinode close to the resonance +wavelength, and the spectral width at FWHM (full width half +maximum) of the antinodes are approximately ∆λ2nd = 27nm and +∆λ1st = 28nm showcasing the broadband performance. The spec- +trum for the two antinodes is not completely symmetric, and the +two curves for the antinodes also cross further away from the res- +onance. In Fig. (12b), the Purcell factor is shown as a function of +the wavelength for the 1st and 2nd antinode of the structure with +the largest efficiency (at resonance). Here the Purcell factor is +much lower and the spectral width much broader at ∆λ2nd = 52nm +and ∆λ1st = 56nm. Furthermore, the resonance wavelength for +the 1st antinode is slightly shifted to λr = 931.5nm. This can be +explained by the low Q factor and the neighbouring low Q factor +cavity modes, i.e. the cavity modes with 2 and 4 antinodes. Due +to the low Q factor, there is a small spectral overlap causing the +slight shift. In the Supplementary 7.6, we present a nanopost de- +sign where the resonance wavelength shift is more pronounced +between the two antinodes. +In Fig. (12c) and Fig. (12d) the efficiencies (NA = 0.75) are +shown as a function of the wavelength for the 1st and 2nd antin- +Fig. 13 βHE11 = PHE11/PT as a function of the wavelength, λ, for the 2nd +and 1st antinode and the infinite nanowire. The numerical uncertainty +is represented by the thickness of the curves. +Fig. 14 Efficiency, ε (NA = 0.75), as a function of the wavelength, λ, for +the 2nd antinode (a) and 1st antinode (b). The full model is compared +to only using c1, c2 and c3. D = 238nm and tSiO2 = 0nm. The width of +the curves represents the numerical uncertainty. +ode for the two structures. The main characteristic of the effi- +ciency is that it does not follow the Purcell factor, which is sur- +prising compared to traditional Fabry-Pérot cavities. Instead, the +efficiency changes roughly linearly across the resonance, and in +general, the slope for the 2nd antinode is negative and positive +for the 1st antinode. This means that the maximum efficiency +is in fact achieved off-resonance, ε2nd,NA=0.75(λ = 880nm) = 0.71, +but at a much smaller Purcell factor. +4.4.1 +Analysis of the broadband collection efficiency +In the previous work on the nanopost29, the broadband efficiency +was attributed to the broadband β factor, which again was at- +tributed to the dielectric screening effect. This can also be seen in +Fig. (5), where the emission into radiation modes is suppressed +in most of the interval. Here we will define the β factor for the +fundamental mode as βHE11 = PHE11/PT, even though it exceeds 1 +as we have already shown. The focus is on the structure with the +largest efficiency at resonance, D = 238nm and tSiO2 = 0nm. +In Fig. (13), βHE11 is shown as a function of the wavelength +for the 2nd and 1st antinode and the infinite nanowire. βHE11 is +close to 1 (both above and below) in the entire interval, which +indicates that the fundamental mode is still the dominating con- +tribution. Due to the broadband βHE11 of the infinite nanowire, +it is not required to be on resonance to obtain large values of +βHE11, and the QD would need to be very close to a node before +βHE11 would decrease. Regardless, the fundamental mode scatters +into radiation which affects PT such that βHE11 does not follow a +Lorentzian curve like the Purcell factor. +To quantify how dominating the fundamental mode is for the +1–18 | 11 + +Fig. 15 Efficiency, ε (NA = 0.75), as a function of the initial coefficients +expressed with the propagation constant (β/k0)2 for λ = 890nm (a) and +λ = 970nm (b) at the 2nd antinode. +efficiency, we will use the first method presented in Section 4.2.1 +to separate the main channels (c1, c2 and c3) from the background +channels. In Fig. (14), we compare the efficiency of the full model +to only including c1, c2 and c3 for both antinodes. By comparing +to Fig. (13) we observe that the curves for the main channels +directly follow the trend of βHE11. However, there is a discrep- +ancy between the efficiency of the full model and only using the +main channels of c1, c2 and c3. This discrepancy is not at a min- +imum on resonance (λ = 930nm), but almost at the minimum +when βHE11 = 1 for both antinodes. We observe that for βHE11 < 1, +the background channels provide a positive contribution to the +efficiency, while for βHE11 > 1 the background channels provide a +negative contribution. This shows that the β factor is still useful +in the analysis but not necessarily a figure of merit for SPSs in the +breakdown of the SMM. Furthermore, Fig. (14) shows that the +background channels still interfere with the main channels caus- +ing this discrepancy and even changing the slope of the curves for +the efficiency. As an example we will consider two wavelengths +for the 2nd antinode and use the first method presented in Sec- +tion 4.2.1: +In Fig. (15) the efficiency is shown as a function of the ini- +tial coefficients expressed with the propagation constant (β/k0)2 +for the two wavelengths λ = 890nm and λ = 970nm at the 2nd +antinode. Again the first red point corresponds to including all +the main channels, and then background channels are added. +For λ = 890nm, the initial propagating background radiation pro- +vides a significant increase to the efficiency, while the evanescent +modes provide no change. On the other hand, for λ = 970nm, the +initial propagating background radiation provides a significant +decrease to the efficiency, while the evanescent modes provide +a positive increase to the efficiency. This also showcases the com- +plex interplay between the main channels and the background +channels. +5 +Perspective +We have shown that contributions from multiple scattering chan- +nels influence both the Purcell factor and especially the collection +efficiency. Unlike traditional Fabry-Pérot cavities, where scatter- +ing of the light is viewed simply as a loss mechanism, this scat- +tering is in fact beneficial for the performance of the nanopost +SPS. Importantly, this scattering mechanism decouples the effi- +ciency from the Purcell factor directly challenging a well-known +design paradigm that maximum collection efficiency is obtained +on resonance. The identification of this mechanism opens a door +to unconventional SPS design approaches, especially in the non- +resonant regime where the scattering coefficients are no longer +analysed and optimized with respect to the fundamental HE11 +mode alone, and where definitions of fundamental performance +parameters such as the spontaneous emission β factor need to be +revisited. +For the nanopost itself, this invites to a new optimization of +the collection efficiency with respect to all geometrical param- +eters. The maximum collection efficiency will be obtained for a +reduced Purcell factor due to the new trade-off between efficiency +and Purcell enhancement. Potential future work on the nanopost +design could also be to explore the properties of cavity modes +with different orders than 3. Additionally, structuring the bottom +mirror could also lead to increased performance. Adding rings +around the nanopost could positively alter the scattering mecha- +nism while also bridging the gap to the closely related bullseye de- +sign16,27,28, which also features broadband collection efficiency +independently of the Purcell factor. Despite flourishing literature +on the bullseye design, the physical mechanisms underlying the +performance is still unclear. The analysis of the bullseye will be +more challenging as the inner mesa/nanowire features larger di- +ameters resulting in additional guided modes. The rings around +the inner mesa will also heavily influence the radiation modes +and their mode profiles. This will, in turn, lead to changes in +the emission rates and the reflection matrices and, thus, the scat- +tering channels. In typical bullseye structures16,27,28, which are +numerically optimized, the silica layer is hundred of nanometers +thick. Here we anticipate that such large thicknesses will also +result in increased mode coupling as light diverges when propa- +gating in the silica. +6 +Conclusions +We have shown that the traditional Fabry-Pérot single-mode +model, which typically provides an excellent description12–14 of +the physics for cavity-based single-photon sources, significantly +underestimates the achievable performance of the nanopost struc- +ture. Using a modal expansion method, we have performed a +detailed analysis of the emission channels. We have shown that +in particular the collection efficiency benefits significantly from a +contribution from light scattered to radiation modes, which often +is simply considered a loss mechanism. This scattering into ra- +diation modes not only allows for improved collection efficiency +but also decouples the collection efficiency from the Purcell fac- +tor, such that optimum performance is obtained off-resonance. +Our parameter scan of the nanopost structure reveals an achiev- +able Purcell factor Fp of 7.9 or a collection efficiency ε of 0.69 +obtained for two very different parameter sets. Our work invites +further exploration of unconventional SPS design mechanisms, +especially in the non-resonant regime. +Conflicts of interest +There are no conflicts to declare. +12 | +1–18 + +Acknowledgements +We thank Battulga Munkhbat for his assistance in creating the +sketch of the nanopost. This work is funded by the European +Research Council (ERC-CoG “UNITY,” Grant No. 865230), the +French National Research Agency (Grant No. +ANR-19-CE47- +0009-02), the European Union’s Horizon 2020 Research and In- +novation Programme under the Marie Skłodowska-Curie Grant +(Agreement No. 861097), and by the Independent Research Fund +Denmark (Grant No. DFF-9041-00046B). +References +1 J.-W. Pan, Z.-B. Chen, C.-Y. Lu, H. Weinfurter, A. Zeilinger and +M. Zukowski, Rev. Mod. Phys., 2012, 84, 777–838. +2 J. L. O’Brien, A. Furusawa and J. Vuˇckovi´c, Nat. Photonics, +2009, 3, 687–695. +3 Y. Arakawa and M. J. Holmes, Appl. Phys. Rev., 2020, 7, +021309. +4 I. Aharonovich, D. Englund and M. Toth, Nat. Photonics, 2016, +10, 631–641. +5 N. Gregersen, P. Kaer and J. Mørk, IEEE J. Sel. Top. Quantum +Electron., 2013, 19, 9000516. +6 N. Gregersen, D. P. S. McCutcheon and J. 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Thus ht and hb will be fixed +to ensure a cavity with 3 antinodes and a resonance wavelength +of λr = 930nm at the 2nd antinode. A procedure is required such +that ht and hb satisfies these conditions. The first step is to use the +SMM (phase conditions of the fundamental mode) to determine +the initial total height h: +h = (2π −arg(rbot,11))/(2β1)+(2π −arg(rtop,11))/(2β1). +(35) +This is under the conditions arg(rbot,11) < 0 and arg(rtop,11) > 0. +Then we place the dipole in the second antinode from the bottom: +hb = (2π −arg(rbot,11))/(2β1) +(36) +and +ht = (2π −arg(rtop,11))/(2β1). +(37) +However, due to the background continuum and mode-coupling, +this method does not ensure that the dipole is placed exactly at +an antinode nor that the resonant wavelength of the cavity corre- +sponds exactly to the design wavelength, ( dFp +dλ )λ=λd = 0. To solve +this problem, the dipole is first adjusted to the exact position of +the antinode by plotting |Er(z)|2 and locating the peak. Then the +height is slightly adjusted h = h ± δh, while the dipole is moved +to the exact position of the antinode for each adjustment until +( dFp +dλ )λ=λd = 0 is satisfied. Finally, the position of the 1st antinode +(from the bottom) can also be identified by plotting |Er(z)|2. +Finally, we will provide the total height, htotal of the structure +along with the height deviation between the initial height ob- +tained from the SMM and the final height, hdiff = htotal −hinitial. +450 +500 +550 +600 +650 +700 +400 +400 +550 +700 +-4 +-4 +-2 +-2 +0 +0 +2 +2 +5 +5 +10 +0 +20 +40 +Fig. +16 (a) Total height of the structure, htotal, as a function of the +diameter, D, and the silica layer thickness, tSiO2. (b) The height difference +between the initial height and the final height, hdiff, as a function of the +diameter, D, and the silica layer thickness, tSiO2. +In Fig. (16a), the total height of the structure is shown as a +function of the diameter and the silica layer thickness. The to- +tal height mostly depends on the diameter as the diameter deter- +mines the propagation constant β1 and thus the primary influence +of the phase. In Fig. (16b), the height difference of the final total +height compared to the SMM is shown. For most of the parame- +ters, the difference is small in the range −5nm to 5nm. However, +for the small diameters and thin silica layer thicknesses, there is a +very large difference up to 50nm. This is the same parameter re- +gion where the modal reflection at the bottom interface is small. +Thus the phase of the fundamental mode is less dominating com- +pared to the contributions of the radiation and evanescent modes. +Oscillations can also be observed in the height difference due to +numerical noise. The exact resonance wavelength is sensitive to +the height of the structure, but these oscillations are on the scale +of less than 1nm, and the uncertainty in the resonance wavelength +will be on a similar scale. +7.2 +Influence of the numerical aperture +In all the simulations of the efficiency a numerical aperture of +NA = 0.75 has been used. Here we will investigate the influence +of varying the numerical aperture. +0.05 +0.1 +0.1 +0.15 +0.2 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +0 +0.5 +1 +Fig. +17 Efficiency, ε, at the 2nd antinode for a numerical aperture +of NA = 0.4 (a) and NA = 1.00 (b). (c) Efficiency, ε, as a function of +the numerical aperture, NA, for three different parameters at the 2nd +antinode. +(d) Efficiency, ε, as a function of wavelength, λ, for four +different values of the NA. +In Fig.(17a) and Fig.(17b) the efficiency is shown as a function +of the diameter and the silica layer thickness similar to Fig. (3a), +but for a numerical aperture NA = 0.4 and NA = 1. Lowering the +numerical aperture to NA = 0.4 drastically reduces the efficiency +and the maximum is barely above ε = 0.2. This shows that a large +numerical aperture is crucial for the good performance of the +nanopost. By increasing the numerical aperture from NA = 0.75 +1–18 | 15 + +0.2 +0.2 +0.3 +0.3 +0.4 +0.4 +0.5 +0.5 +0.57 +0 +0.5 +1 +Fig. 18 Gaussian collection efficiency, εg, as a function of the diameter, +D, and the silica layer thickness, tSiO2, for the 2nd antinode. +(θ ≈ 49◦) to NA = 1.00 (θ = 90◦) there is roughly a 20% increase +in the efficiency, so there is still some light lost at angles above +θ ≈ 49◦. Furthermore, for a numerical aperture of NA = 1, the +efficiency directly represents the losses to the bottom mirror. For +diameters above D = 210nm, an increased silica layer thickness +increases the losses to the bottom mirror even though the Pur- +cell factor increases. In Fig. (17c) the efficiency for the structures +with the largest efficiency and Purcell factor, at the 2nd antinode, +are shown as a function of the numerical aperture. The steepest +part of the curves is roughly in the interval NA = 0.4 to NA = 0.75, +which is the reason for the huge difference in efficiency between +NA = 0.4 and NA = 0.75. The curves also start to flatten out as the +NA reaches 1. In Fig. (17d) the efficiency is plotted for four dif- +ferent values of the numerical aperture as a function of the wave- +length. The numerical aperture does not influence the curvature +of the efficiency as a function of the wavelength. This means that +being on resonance does not focus the far-field compared to being +off resonance. +7.3 +Gaussian collection efficiency +So far the efficiency has been evaluated by calculating the to- +tal power collected in the lens with some numerical aperture. +However, in many applications the light will couple to a fiber +afterwards. Therefore we have also calculated the power over- +lap between the emitted far-field and the far-field of a Gaussian +representative for the fundamental mode in many single-mode +fibers49. The applied method is identical to the one presented in +the appendix of14, and the Gaussian collection efficiency is de- +fined as εg = Pcollected,Gaussian/PT, where Pcollected,Gaussian is defined +as the overlap with a Gaussian profile. In Fig. (18) the Gaussian +efficiency is shown for the 2nd antinode. Compared to the stan- +dard efficiency in Fig. (3a), the difference is approximately 0.1 +over the entire parameter space, showcasing the Gaussian shaped +profile of the far-field. +7.4 +Efficiency analysis for the structure with maximum Pur- +cell factor +We will now apply the efficiency analysis for the structure with the +largest Purcell factor with the parameters D = 250nm and tSiO2 = +13nm and an efficiency of ε = 0.41. +In Fig. (19) the efficiency is shown as a function of the initial +coefficients (Fig. (19a)) and the final coefficients (Fig. (19b)), +expressed with the propagation constant (β/k0)2, just as for the +structure with maximum efficiency. Again the curve in Fig. (19a) +is flat and the channels of the fundamental mode dominates the +efficiency. The efficiency increase by adding the final coefficients, +i.e. c2 and c3, seen in Fig. (19b), is still significant, but much +smaller compared to the structure with maximum efficiency. Here +the increase is from approximately ε = 0.3 to ε = 0.41. The curve +in Fig. (19b) also flattens out completely due to the numerical +aperture. +Fig. +19 (a) Efficiency, ε (NA = 0.75), as a function of the initial coef- +ficients expressed with the propagation constant (β/k0)2. (b) Efficiency, +ε (NA = 75), as a function of the final coefficients expressed with the +propagation constant (β/k0)2. +Fig. +20 (a) The phase difference between the direct transmission of +the fundamental mode and the background continuum for TE and TM +modes as a function of the propagation constant. (b) The far-field of the +fundamental mode. (c) The far-field of the background continuum. (d) +The total far-field. The white dotted line indicates NA = 0.75. Be aware +of the different color scales that have been used for the far-fields. +In Fig. (20a) the phase difference in the air layer between the +direct transmission of the fundamental mode and the entire back- +ground is shown as a function of the propagation constant for +TE and TM modes. We observe similar features as before, i.e. +constructive (destructive) interference for light propagating ver- +tically (horizontally). Though at β/k0 = 1, the phase difference +is larger compared to the previous structure. In Fig. (20b), Fig. +(20c) and Fig. (20d) the far-fields of the direction transmission +16 | +1–18 + +1 +(a) +/ +0.5 +0 +TE +TM +5HEi1 FF +0.50 +0.5 +1 +β/ ko +Scattered FF +Normalized units +c +0.5 +0Total FF +(p +0.5 +0of the fundamental mode, the background radiation and the total +field is shown. Compared to the structure with maximum effi- +ciency, the far-field of the background radiation is significantly +different. Here, the far-field is mainly focused towards horizontal +angles and the intensity is much smaller compared to the far-field +of HE11. As such the constructive contribution at smaller angles +is not as significant and less of the radiation will be captured by +the lens, due to the numerical aperture. This explains why the +efficiency increase in Fig. (19b) is much smaller compared the +structure with the maximum efficiency. However, there is still de- +structive interference for the light that propagates horizontally. +As such the interference between the direction emission and the +radiation focuses the far-field, but not to the same degree as for +the structure with the maximum efficiency. +7.5 +Purcell factor analysis for the structure with maximum +collection efficiency +We will now apply the model for the Purcell factor for the struc- +ture with the largest efficiency, D = 238nm and tSiO2 = 0nm and +Fp = 4.8. +Fig. 21 Purcell factor, Fp, computed using the full model and the SMM +as a function of the dipole position from the bottom interface, hb. D = +238nm and tSiO2 = 0nm. +In Fig. (21) the Purcell factor is shown as a function of the +dipole position throughout the cavity, both the full model (n. 7) +and the SMM (n. 1) are used. Here the 3 antinodes can be ob- +served and the SMM predicts a larger Purcell factor for all 3 antin- +odes compared to the full model. The positions of the antinodes +are almost identical between the full model and the SMM. The +Purcell factor increases drastically when the dipole is placed close +to the metal mirror due to non-radiative decay processes33. +In Fig. (22a) and Fig. (22b) the Purcell factor and the power en- +hancement of the fundamental mode is shown as a function of the +model complexityfor the 1st and 2nd antinode. Compared to the +structure with D = 250nm and tSiO2 = 13nm, there are a few differ- +ences. The SMM predicts a smaller Purcell factor, which is simply +caused by the lower modal reflection at the bottom. There is a +large negative contribution when including the back-scattering at +the bottom interface (n 5.) at both antinodes. This is caused by +the change of the silica layer thickness and as seen in Fig. (22c) +and Fig. (22d). The propagating radiation modes are responsible +for this decrease. Furthermore, by including the scattering of the +background to itself (n. 6), there is now a small decrease for both +antinodes. These are the differences between the two structures. +The differences between the 1st and 2nd antinode are exactly the +same for the two structures, where the scattering into evanescent +modes at the top interface (n. 2) and the initial evanescent modes +provide a positive contribution at the 2nd antinode. +Fig. +22 Purcell factor (PT/PBulk) and fundamental mode enhancement +(PHE11/PBulk) for the 1st (a) and 2nd (b) antinode as a function of the +model number. In (c) and (d) the background continuum is continuously +included between each model number. +7.6 +Asymmetric wavelength dependence for the two antin- +odes +To further study the resonance shift between the 1st and 2nd +antinodes, we choose a nanopost design of D = 202nm and tSiO2 = +5nm, where the shift is more pronounced. +Fig. +23 Purcell factor, Fp, as a function of wavelength, λ, for the two +antinodes. The parameters are D = 202nm and tSiO2 = 5nm. +In Fig. (23) the Purcell factor is shown for the two antin- +odes as a function of the wavelength. The peak positions of the +Purcell factors (resonance wavelength) are λ2nd,r = 930nm and +λ1st,r = 933nm. By observing the curve for the 1st antinode, this +shift is caused by another broad resonance at approximately λ = +1050nm. To gain further insight into the resonances of the struc- +ture and verify our results, we have performed a quasi-normal +mode (QNM) simulation50 of the nanopost. In this simulation +15 QNMs are found and the complex eigenfrequencies, +˜ωµ = +ωµ − iγµ, of the 3 important QNMs are ˜ωQNM1 = 2.0237×1015 − +i4.4904×1013Hz, +˜ωQNM2 = 1.7595×1015 − i1.4654×1014Hz and +1–18 | 17 + +˜ωQNM3 = 2.2809×1015−i2.0703×1013Hz. The corresponding real +parts of the complex wavelength are λQNM1 = 930.3nm, λQNM2 = +1063.2nm and λQNM3 = 825.8nm. +The Q factors of the QNMs +can also be calculated using Qµ = ωµ/(2γµ)50, and we obtain +QQNM1 = 22.5, QQNM2 = 6.0 and QQNM3 = 55.1. +Fig. +24 Comparison of the Purcell factor between the FMM and the +QNM simulation for the 1st antinode (a) and the 2nd antinode (b). +In-plane electrical field profiles of the 3 QNMs at their resonance wave- +lengths are shown in (c), (d) and (e). The green star corresponds to +the position of the 1st antinode, and the red star corresponds to the +position of the 2nd antinode. The white scale bar in (a) corresponds to +100 nm. The intensity is scaled in each field plot and should not be used +for comparison. +In Fig. (24a,24b), the comparison of the Purcell factor between +the FMM and the QNM simulation is shown for the two antinodes. +Overall, the quantitative agreement between the FMM and QNM +simulations is good with some small deviations. In Fig. (24a), +the individual contributions of 3 QNMs are plotted along with +their sum and the result of the FMM for the 1st antinode. These +3 QNMs provide a good description of the overall Purcell factor +and they directly correspond to the peaks in the spectrum. QNM1 +and QNM2 also overlap in the spectrum due to the low Q factor +of QNM2, which slightly shifts the peak position of the total Pur- +cell factor. In Fig. (24b), the individual contributions of 2 QNMs +are plotted along with their sum and the result of the FMM for +the 2nd antinode. Here QNM1 is almost sufficient to describe the +entire spectrum, and we do not observe any other peaks than the +one at λ = 930nm, besides a small bump at longer wavelengths. +Now, consider the in-plane electrical field profiles of the 3 QNMs +shown in Fig. (24c,24d,24e). QNM1 has 3 antinodes, QNM2 has 2 +antinodes and QNM3 has 4 antinodes. The green star corresponds +to the position of the 1st antinode, where the QD is placed, and +this position is very close to an antinode for QNM2 and QNM3. +Therefore the contributions of these QNMs appear in the spec- +trum. However, the position of the 2nd antinode (red star) is +much closer to a node for QNM2 and QNM3, and therefore they +do not influence the spectrum. +18 | +1–18 + +lst antinode +-FMM +(b) +3 +QNM - 3 modes +QNM1 +QNM2 +2 +QNM3 +12nd antinode +-FMM +-QNM - 2 modes +-QNM1 +QNM20 +900 +1000 +1100 +90 +入(nm) +QNM1 +QNM2 +米00 +1000 +1100 +入(nm) +QNM3 \ No newline at end of file diff --git a/ptE0T4oBgHgl3EQfqwEz/content/tmp_files/load_file.txt b/ptE0T4oBgHgl3EQfqwEz/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ed1d2353eb0772184ab1adba3dd141263315095d --- /dev/null +++ b/ptE0T4oBgHgl3EQfqwEz/content/tmp_files/load_file.txt @@ -0,0 +1,1438 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf,len=1437 +page_content='Performance of the nanopost single-photon source: beyond the single-mode model Martin Arentoft Jacobsen,∗a Yujing Wang,a Luca Vannucci,a Julien Claudonb, Jean-Michel Gérard,b and Niels Gregersena We present a detailed analysis of the physics governing the collection efficiency and the Purcell enhancement of the nanopost single- photon source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' We show that a standard single-mode Fabry-Pérot model is insufficient to describe the device performance, which benefits significantly from scattering from the fundamental mode to radiation modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' We show how the scattering mechanism decouples the collection efficiency from the Purcell enhancement, such that maximum collection efficiency is obtained off-resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Finally, we discuss how this scattering mechanism can be beneficial for future single-photon source designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 1 Introduction The construction of scalable optical quantum technologies1,2 re- lies on the development of sources of single indistinguishable photons3–6 and of entangled photon pairs7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' The ideal single- photon source (SPS) should be deterministic and feature pure emission of single photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' The main figure of merit6 is the col- lection efficiency ε defined as the number of photons detected in the out-coupling channel per trigger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' In a multi-photon interfer- ence experiment8 with N photons, the success probability P scales as P = εN, and increasing ε towards 1 is thus critical to achieve scalable optical quantum information processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' The sponta- neous parametric down conversion process9 is a straight-forward technique widely used within the quantum optics community for production of highly indistinguishable photons, however its prob- abilistic nature limits the efficiency of pure photon emission to a few percent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' For this reason, the community has turned its attention to- wards two level systems, in particular the semiconductor quan- tum dot3,4,10 (QD), capable of deterministic emission of single photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' For a QD in a bulk material, ε is limited to a few percent this time due to the large index contrast at the semiconductor-air interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' It is thus necessary to place the QD inside a photonic nanostructure5,6 directing the light towards the collection optics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' A main strategy for controlling the light emission is to place the QD inside a micro cavity and exploit cavity quantum electrody- namics (CQED) in the weak coupling regime to selectively en- hance the light emission into the optical mode of the microcavity using the Purcell effect11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Detailed understanding of the CQED physics governing the collection efficiency can be obtained using a single-mode Fabry-Pérot description12–14 of the light emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Here, the spontaneous emission β factor describes the emission rate ΓC of the QD into a fundamental HE11 cavity mode divided by the total emission rate ΓT = ΓC + ΓB including a contribution ΓB to background radiation modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' The rate ΓC into the cavity mode normalized to the rate ΓBulk in a bulk medium is quantified by the Purcell15 factor Fp = ΓC/ΓBulk = 3 4π2 Q Vn at resonance, where Q is the cavity quality factor and Vn is the mode volume in units of material cubic wavelengths (λ/n)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' The spontaneous emission a DTU Electro, Department of Electrical and Photonics Engineering, Technical University of Denmark, DK-2800 Kongens Lyngby, Denmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' E-mail: maaja@dtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='dk b Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Grenoble Alpes, CEA, Grenoble INP, IRIG, PHELIQS, “Nanophysique et Semicon- ducteurs” Group, F-38000 Grenoble, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' β factor can then be written in terms of the Purcell factor as β = ΓC ΓC +ΓB = Fp Fp +ΓB/ΓBulk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (1) Furthermore, we define the transmission γ as the fraction of power in the cavity mode detected by the collection optics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Fi- nally, we can then define a single-mode Fabry-Pérot model (SMM) εs for the efficiency as εs = βγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (1), we observe that increasing the Purcell factor Fp will improve the collection effi- ciency, and maximum efficiency is thus expected for a QD on res- onance with the cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Indeed, this design paradigm that Purcell enhancement is ben- eficial for achieving high collection efficiency is well-established within SPS engineering: The most succesful SPS design strategies today include the microcavity pillar14,16–18 and the open cavity approach19 demonstrating up to ε ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='6 into a first lens16 and into a fiber19, respectively, combined with highly indistinguish- able photon emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' These narrowband approaches, for which the single mode model εs = βγ is an excellent approximation14, rely critically on resonant Purcell enhancement and thus on con- trol of the spectral alignment17 to achieve high efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' On the other hand, broadband approaches including the photonic nanowire13,20–23 and the photonic crystal waveguide24–26 de- signs exploit suppression of the background emission rate using the dielectric screening effect20,24,25 to non-resonantly maximize the β factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Even so, these broadband approaches also benefit from resonant cavity13,21 and slow-light24,25 effects to further improve the efficiency, confirming again that Purcell enhance- ment is beneficial in the SPS engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' However, this paradigm has been challenged by new broad- band SPS geometries, such as the circular Bragg grating or "bulls- eye" design16,27,28, for which high collection efficiency is ob- tained in a wavelength range significantly broader27 (∼ 100 nm) than the typical resonance linewidth (∼ 10 nm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Similar char- acteristics were observed very recently for the nanowire optical nanocavity or "nanopost" design29 shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (1), for which a significant Purcell factor Fp of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='6 enabled by the ultrasmall mode volume of the nanocavity was experimentally demonstrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Ad- ditionally, a surprisingly high collection efficiency of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='35 was measured29, which was attributed to a breakdown30 of the single mode model εs = βγ for the efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' In this work, we investigate this surprising breakdown by per- forming a detailed quantitative analysis of the physics governing 1–18 | 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='02556v1 [quant-ph] 5 Jan 2023 Gold mirror SiO2 Substrate GaAs ht hb tSiO2 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 1 Sketch of the "nanopost" nanowire optical nanocavity and the geometrical parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' The resulting in-plane electrical field profile of a QD placed inside the nanopost is shown as an inset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' The length of the white scale bar in the inset is 100 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' the nanopost geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' We show that the single mode model fails to describe the physics of both the Purcell enhancement and the collection efficiency due to a decoupling between the two: The computed efficiency is significantly higher than the prediction of the single-mode model thanks to additional transmission chan- nels to the far-field, whose beneficial contributions are dominat- ing over the resonant cavity effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' We show not only that max- imum Purcell enhancement and maximum collection efficiency are obtained for entirely different design parameters, but also that maximum efficiency is obtained off-resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' The analysis is performed using a Fourier Modal Method31, allowing for di- rect insight into the beneficial interplay beyond the single-mode model with the continuum of radiation modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' This article is organized as follows: In Section 2, we present the nanopost and its performance in terms of Fp and ε, and we demonstrate the breakdown of the single-mode model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' In Sec- tion 3, we present our theoretical framework based on the Fourier Modal Method, which we subsequently use to analyze the com- plex interplay with radiation mode channels in Section 4 and its influence on the collection efficiency and the Purcell factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' In Section 5 we put the nanopost physics into perspective and dis- cuss its impact on SPS engineering, followed by our conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Additional simulation results are presented in the Supplementary Information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 2 The nanopost geometry and the break- down of the single-mode Fabry-Pérot model The nanopost shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (1) consists of a truncated GaAs nanowire with diameter D on top of a SiO2-Au mirror.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' The top of the nanowire is flat, and the surrounding medium is air.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' The SiO2 layer, located between the nanowire and the gold, has a thickness indicated by tSiO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' The QD, modelled as a dipole, is placed on-axis inside the nanowire at a position hb from the bot- tom interface and ht from the top interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' The refractive in- dices of the materials are chosen as nGaAs = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='46, nSiO2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='5 and nAu = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='201+5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='85i at λ = 930nm and assumed to be constant as a function of wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' In the inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (1), a dipole with an emission wavelength of 930nm is placed inside the nanopost, and the resulting in-plane electrical field, simulated using the FMM, is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Three antinodes can be seen in the field profile, cor- responding to the order 3 cavity mode, and they are enumerated from the bottom mirror as the 1st, 2nd and 3rd antinode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' The field profile generated by the dipole is independent of the vertical position of the dipole, only the intensity changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' The intensity is not the same at the three antinodes due to the breakdown of the SMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' We now present the performance of the nanopost as a func- tion of the diameter, D, and the silica layer thickness, tSiO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' We have scanned the parameter ranges D = 196nm to D = 300nm and tSiO2 = 0nm to tSiO2 = 25nm and chosen a design wavelength of λd = 930nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' The height of the structure and the position of the QD are dynamically changed to keep the order 3 cavity mode resonance at λr = 930nm for the QD at the 2nd antinode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' The required procedure is presented in Supplementary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 3 4 5 5 5 6 6 6 7 7 7 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='75 2 3 3 4 4 4 5 5 5 6 6 6 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='5 2 3 4 5 6 7 2 3 3 4 5 5 5 6 6 6 7 7 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 2 Purcell factor, Fp, for a QD at the 2nd antinode (a), 1st antinode (b) and any antinode using the SMM (c) as a function of the diameter, D, and the silica layer thickness, tSiO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' The first quantity of interest is the spontaneous emission rate ΓT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' For the high-β structures investigated here, the total nor- malized rate ΓT/ΓBulk and the Purcell factor Fp are similar, and 2 | 1–18 100 nmwe will in the following refer to the normalized total rate as the "Purcell factor Fp".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (2a,2b) the Purcell factor is shown as a function of the diameter and the silica layer thickness for a QD placed in the 2nd and 1st antinode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' In the entire parameter space, the Purcell factor is larger for the 2nd antinode and a maximum value of Fp = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='9 is reached at D = 250nm and tSiO2 = 13nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' This discrepancy between the two antinodes also demonstrates the de- viations of the SMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Overall, the tendency of the Purcell factor is similar at the 2 antinodes with one peak value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' The minimum is located in the corner of no silica and the smallest diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (2c) the Purcell factor is now shown using the SMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Com- paring the SMM to the full model for the two antinodes, there are both positive and negative deviations across most of the pa- rameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Compared to the 2nd antinode, the SMM also predicts a slightly lower value for the maximum Purcell factor of Fp = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='5, but at a very different position of D = 242nm and tSiO2 = 8nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' However, compared to the 1st antinode the SMM predicts a larger value for the maxmimum Purcell factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' This is an invitation to obtain a better description and understanding of the physics responsible for the Purcell factor, which we will pro- vide in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='68 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='65 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='8 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='4 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 3 Collection efficiency, ε (NA = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='75), for a QD at the 2nd antinode (a), 1st antinode (b) and any antinode using the SMM (c) as a function of the diameter, D, and the silica layer thickness, tSiO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' The source collection efficiency, ε, for the 2nd antinode, the 1st antinode and the SMM are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (3) for a numerical aperture of NA = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Here the overall performance for the 2nd antinode is slightly better than for the 1st antinode, and a maxi- mum value of ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='69 is reached at D = 238nm and tSiO2 = 0nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Despite the very simple geometry of the nanopost, a surprisingly high collection efficiency of ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='69 is achievable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' In general, it can be ascertained that no silica layer is more beneficial for the efficiency, which is surprising compared to the traditional pho- tonic nanowire, where the silica layer enhances the modal re- flectivity22 and thus the collection efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Now comparing the efficiency of the SMM to the full model, there are substan- tial differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' This is unlike the micropillar and the photonic nanowire geometries for which the single-mode model εs = βγ is an excellent12–14 approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' For the nanopost, the SMM efficiency is much smaller in the entire parameter space except for the smallest diameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' This clearly shows that there are im- portant physical mechanisms which are not accounted for in the SMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Furthermore, comparing the figures for the Purcell factors and the efficiencies, there seems to be no apparent correlation between the two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' This is also a surprising result compared to tra- ditional Fabry-Pérot cavities and indicates that there are different physical mechanisms at play which govern the Purcell factor and the efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' In Supplementary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='2, we vary the numerical aperture and present its influence on the collection efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' In Supplemen- tary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='3, we present the collection efficiency taking into consider- ation the overlap with a Gaussian profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 3 Theory 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='1 Method We use an eigenmode method combined with a standard scatter- ing matrix formalism32 in the frequency domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' In this method, the structure is divided into layers of uniform permittivity along the propagation direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' In each layer, the electrical field is expanded on the eigenmodes, and the scattering matrices are used to connect the eigenmodes at the interfaces between the layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' The eigenmodes are obtained using the Fourier modal method (FMM) with open boundary conditions31, which provides direct access to the modes needed to understand the physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' The nanopost structure is split into four layers: the bottom gold sub- strate, the silica layer, the nanowire and the top air region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Due to the cylindrical symmetry of the nanowire, a cylindrical coordinate system is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' The QD is modelled as a classical point dipole, and we make use of the relationship Γ/ΓBulk = P/PBulk to calculate the Purcell factor33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' P is the emitted power of the dipole, and PBulk is the power emitted in a bulk medium, and thus the Purcell factor is defined as Fp = PT/PBulk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' The second quantity of interest is the collection efficiency defined as ε = Pcollected/PT, where Pcollected is the power collected in the far-field of a lens with numerical aper- ture NA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='1 Eigenmodes The electrical field for an eigenmode has the following expres- sion: Ej(r,φ,z) = ej(r,φ)exp(iβ jz), (2) where j refers to eigenmode index, ej(r,φ) is the mode profile for the given eigenmode and βj is the propagation constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' For three out of the four layers, the permittivity profile is constant over the entire plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' In this case, the eigenmodes are simply cylindri- cal plane waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' These eigenmodes exist as a continuum where 1–18 | 3 the propagation constant takes the value β = � (nlayerk0)2 −k2 ⊥, where nlayer is the refractive index, k0 is the free-space wavenum- ber and k⊥ is the in-plane k-value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' k⊥ can take any value k⊥ ∈ [0,∞] and eigenmodes exist for all the values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' For each β, two orthogonal solutions exist, which can be separated into TE modes (ez = 0) and TM modes (hz = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' For layers with real- valued refractive index, this continuum can be separated into radiation modes satisfying 0 < (β)2 ≤ (nlayerk0)2 and evanescent modes (β)2 < 0 which decays exponentially and carry no power according to the Poynting vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Specifically for the air layer, the propagation constant of the radiation modes can directly be interpreted as the propagation angle with respect to the z-axis us- ing the expression θ = arccos(β/k0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Expanding a forward prop- agating electrical field on the eigenmodes will have the following expression: E(r,φ,z) = 2 ∑ s=1 � ∞ 0 as(k⊥)es(r,φ,k⊥)exp(iβ(k⊥)z)dk⊥, (3) where s refers to the two solutions and as(k⊥) is the expansion co- efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Numerically, the continuum is discretized into N modes and truncated with a cut-off value for k⊥ 31,34, which will lead to the following expression: E(r,φ,z) = N ∑ j=1 a jej(r,φ)exp(iβ jz), (4) where s is absorbed into j, and the ∆k⊥ that would appear due to the discretization is absorbed into definition of the eigenmode profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' The first class of eigenmodes for the nanowire are the guided modes for which the propagation constants satisfy the condition (nairk0)2 < (βj)2 < (nGaAsk0)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' The guided mode is confined to the core of the nanowire, and outside the nanowire, the field decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' There is a finite number of guided modes, and there will always be at least one guided mode, the fundamental HE11 mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' The description of guided modes can be found in various textbooks35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' The second class of eigenmodes is the background continuum, very similar to the continuum of the air layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' These eigenmodes can be viewed as perturbed versions of the cylindri- cal plane waves of the air layer and be separated into radiation and evanescent modes the exact same way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' There are also two orthogonal solutions, but these can no longer be separated into pure TE and TM modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Studies of this class of eigenmodes are plentiful in the literature36–45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='2 Dipole emission in an infinite structure The QD is modelled as a classical point dipole with in-plane ori- entation and harmonic time dependence at the frequency ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' The corresponding current density is J(r) = −iωpδ(r − rd), where rd is the position of the QD and p is the dipole moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' The emitted power of the dipole can be calculated as33: P = −1 2 � V Re[J∗(r)·E(r)]dV = ω 2 Im[p·E(rd)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (5) The total emitted power, PT, can be calculated by evaluating the total field, ET(rd), but also the power into individual modes, Pj, by evaluating E j(rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Placing the dipole on-axis inside an in- finitely long nanowire with position zJ will result in the following electrical field: ET(r) = ∑ j aJ je+ j (r⊥)exp(iβ j(z−zJ)) (z > zJ) (6) ET(r) = ∑ j bJ je− j (r⊥)exp(−iβj(z−zJ)) (z < zJ), (7) where the superscript + refers to forward propagating and − refers to backward propagating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' For the electrical fields, the for- ward and backwards propagating fields are identical: ej = e+ j = e− j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' aJ j and bJ j are the field expansion coefficients and in an in- finite structure, such as the nanowire, have the following simple expressions32: aJ j = − −iωp·e+ j (rd) 2 (8) bJ j = − −iωp·e− j (rd) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (9) The expansion coefficients can then be represented by vectors: a∞NW = � aJ 1 aJ 2 ··· aJ N � (10) b∞NW = � bJ 1 bJ 2 ··· bJ N � , (11) where subscript ∞NW refers to the infinite nanowire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='3 Multilayered structures and scattering matrices The reflection and transmission matrices are used to connect the field at the interfaces between the layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' These matrices are derived from the boundary condition that the tangential compo- nents of the electric and magnetic field are continuous across an interface32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' The coefficients in the matrices describe how a given mode is transmitted or reflected into another mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' A reflection matrix is shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' R = � ����� r11 r12 r13 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' r1N r21 r22 r23 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' r2N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' rN1 rN2 rN3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' rNN � ����� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (12) The r11 coefficient represents the reflection of the fundamental mode back into itself, while the remaining part of the first column represents the reflection of the fundamental mode into all other modes and so forth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Propagation matrices are used to propagate the field inside a layer and are defined in the following way: P(z) = � ����� eiβ1z 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 0 0 eiβ2z 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 0 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' eiβNz � ����� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (13) 4 | 1–18 The total field inside the structure can then be calculated by tak- ing into account the round-trips which the initially emitted light takes inside the cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Above the emitter, the field takes the fol- lowing expression32: ET(r) = ∑ j aJ tot, je+ j (r⊥)exp(iβ j(z−zJ)) +btot, je− j (r⊥)exp(−iβj(z−zJ)) (z > zJ), (14) where the new expansion coefficients are calculated using the fol- lowing equation32: aJ tot = (I−P(hb)RbotP(hb)P(ht)RtopP(ht))−1 (a∞NW +P(hb)RbotP(hb)b∞NW) (15) and btot = P(ht)RtopP(ht)aJ tot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (16) Now the Purcell factor can be calculated by evaluating the total field at the dipole position using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='4 Far-field and efficiency To obtain the field in the air above the structure, we apply the propagation and transmission matrix on the forward propagating light in the cavity and thus obtain the expansion coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' aair = TtopP(ht)aJ tot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (17) To calculate the collected power in a lens with some numerical aperture, a near- to far-field transformation is used46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' The far- fields EFF(R,θ,φ) and HFF(R,θ,φ) are calculated on the surface of a sphere with radius R, and the radial component of the resulting Poynting vector is: SFF(R,θ,φ) = (E∗ FF,θHFF,φ −E∗ FF,φHFF,θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (18) The collected power in the far-field is then: PFF(NA) = 1 2R2 � 2π 0 � θNA 0 SFF(R,θ,φ)sin(θ)dθdφ = � 2π 0 � θNA 0 pFF(θ,φ)sin(θ)dθdφ, (19) where pFF(θ,φ) is the power per unit solid angle in the far-field and θNA is determined by the NA (NA = sin(θNA)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' The R depen- dence cancels out as the Poynting vector scales as 1/R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='5 Single-mode Fabry-Pérot model When calculating the Purcell factor using the SMM, we only con- sider the fundamental mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' The SMM equations equivalent to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (15-16) are aJ tot,SMM = aJ 1 1+r11,botei2hbβ1 1−r11,botr11,topei2htotalβ1 , (20) and btot,SMM = aJ tot,SMMr11,topei2htβ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (21) The collected power in the far-field is then calculated by inserting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (20) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (17), and the SMM efficiency is then defined as εs = Pcollected,SMM/PT, equivalent to the definition in the introduc- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='2 Mode-coupling and emission channels Important coupling effects take place at the top and bottom in- terfaces of the nanowire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' At both interfaces, all the modes couple to each other, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' all the elements in the reflection matrices are non-zero;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' however, some modes and elements are more impor- tant than others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (4), different examples of mode coupling are shown along with the emission channels that will contribute to the far-field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' The sketch is divided into two parts: the main channels and the background channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' The main channels con- sist of all the light that originated as the fundamental mode, αt and αb, which is indicated by the red arrowheads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' This is the propagating mode that experiences sufficiently large reflections at both interfaces such that it is Purcell enhanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' c1 Main channels Background channels c2 c3 t b t b t,r b,r t,� b, � γt γb γt,α γb,α αt,γ αb,γ loss γt,β Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 4 Sketch of the nanopost and the different emission channels and examples of mode-coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' The emission channels are separated into the main channels and the background channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' The red color corresponds to the fundamental mode, the blue color corresponds to radiation modes and the green color corresponds to evanescent modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Some arrows have two colors, where the color of the arrowhead corresponds to the original channel, but the 2nd color on the shaft signifies the current mode classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' As an example consider c1 which is the transmission of the fundamental mode into the air.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' It has a partly blue shaft as it is now classified as radiation, but it originated as the fundamental mode (red arrowhead).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' See main text for more information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 1–18 | 5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='1 Path of the fundamental mode Let us now follow the path of the fundamental mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' The light emitted into the fundamental mode will propagate upwards and downwards indicated by αt and αb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' At the top interface the fun- damental mode is: Transmitted into the air indicated by c1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Reflected, indicated by αt,r (the channel responsible for Pur- cell enhancement).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Scattered into radiation that propagates downwards indi- cated by the red/blue arrow pointing towards the bottom mirror.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' This radiation will then be reflected by the bottom mirror and then be transmitted into the air indicated by c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Coupled to evanescent modes indicated by αt,γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' At the bottom interface the fundamental mode is: Transmitted into the mirror and lost indicated by the green/red arrow pointing downwards at the very bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Reflected, indicated by αb,r (the channel responsible for Pur- cell enhancement).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Scattered into radiation that propagates towards the air in- dicated by c3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Coupled to evanescent modes indicated by αb,γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' As we will demonstrate in the following, the three radiation channels c1, c2 and c3 are the main channels that will contribute to the far-field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='2 Path of the background emission Let us now consider the background emission channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' First, we have the light directly emitted into radiation, indicated by the blue arrows of βt and βb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' The radiation can both be emitted upwards or downwards and then reflected by the mirror.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' At both interfaces, a small part of the radiation modes can also couple to the fundamental mode indicated by the blue/red arrows of βt,α and βb,α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' We also have light coupled to the evanescent modes, which is indicated by the long green arrows pointing upwards and downwards of γt and γb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' These modes do not propagate in a traditional sense, but at the interfaces, they can scatter into the fundamental mode indicated by the green/red arrows of γt,α and γb,α at the top and bottom interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' At the top interface, the evanescent modes can also couple to radiation and be transmitted indicated by the green/blue arrow pointing upwards, γt,β .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 4 Analysis of the Purcell factor and effi- ciency The starting point of the analysis is the emission rates in the in- finitely long GaAs nanowire, which directly represent the initial coefficients of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (10-11) through Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (5-7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (5), the emission rates for the present guided modes, the radiation modes and the total emission are shown as a function of the nanowire diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' The black dotted vertical lines represent the interval which is used in the full simulations of the nanopost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' In this interval, the infinite nanowire only contains one guided mode, the fundamental HE11 mode, and most of the power is emitted into this mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' The emission into radiation is suppressed in most of the interval and only begins to increase when the diameter reaches 300nm before the EH11 mode appears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' The emission rates thus show that the radiation background channel (the initial coef- ficients for the radiation) presumably only has a minor influence on the Purcell factor and efficiency as long as the cavity does not suppress the fundamental mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (5) cannot be used to quan- tify the importance of the evanescent background channels, as the field components of the evanescent eigenmodes only have a real part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Later on, in the third subsection, models will be used to quantify the effect of the evanescent background channels the finite-length nanopost structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='1 Scattering of the fundamental mode at the interfaces Here, we will study reflection and transmission at the top inter- face between the GaAs nanowire and the air above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' The fun- damental mode is launched towards the interface, and then the reflection and transmission coefficients are calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' The power reflection coefficient for reflection of mode n into mode m is Rm,n = |rm,n|2, and the power transmission coefficients are defined similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' The total power reflection of the fundamental into ra- diation is then defined as Rrad,1 = ∑2 s=1 � k0 0 |rs,1(k⊥)|2dk⊥, and in the discretized regime Rrad,1 = ∑ Nk0,wire+1 n=2 |rn,1|2, where the index Nk0,wire corresponds to the total number of radiation modes in the nanowire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Thus the total power reflection of the fundamental mode is Rtotal,1 = R1,1 +Rrad,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' The total power transmission of the fundamental mode is then defined as Ttotal,1 = ∑ Nk0,air n=1 |tn,1|2 and due to power conservation we have Rtotal,1 +Ttotal,1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (6), the power reflections, along with the power trans- mission of the fundamental mode, are shown as a function of the nanowire diameter D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' By comparing the magnitudes of the modal reflection (R1,1) and the reflection into radiation (Rrad,1), it is clear that the reflection into radiation (c2 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (4)) is es- sential for the far-field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' This indicates why the SMM fails to de- scribe the efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' This mechanism will have a much smaller influence on the Purcell factor as the radiation modes only has a small field amplitude at the center of the nanowire shown in the previous section (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (5)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' However, the reflection matrix is approximately symmetric, such that the first column and the first row is identical, r1,n ≈ rn,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' This means that a small part of the radiation will actually scatter back into the fundamental mode at the top interface, and this will have an influence on the Pur- cell factor, which we will show later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' In general for Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (6), we observe small modal reflections for small diameters and this will lead to a limited cavity effect and thus a lower Purcell factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Prad PHE11 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 5 Power emission in the infinite nanowire as a function of the diameter, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' A sketch of the emission in an infinite nanowire with an embedded QD is shown in the right part of the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 6 | 1–18 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 6 Reflection and transmission of the fundamental mode as a function of the diameter, D, at the top interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' A sketch of top interface and the reflection and transmission of the fundamental mode is seen to the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' We now consider the reflections at the bottom interface be- tween the GaAs nanowire and the silica-gold mirror.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Compared to the top interface, there is now an additional parameter, namely the thickness of the silica layer, tSiO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' The purpose of the sil- ica layer is to increase the reflection of the fundamental mode and avoid coupling to surface plasmons which would decrease the reflection47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (7), the bottom reflection coefficients are shown as a function of the diameter and the silica layer thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' In the parameter ranges where the modal reflection is large ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='9 (strong cavity effect), the reflection into radiation is small ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Here we do not expect a significant contribution of the scattering into radiation, c3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' However, as the modal reflection decreases, the reflection into radiation increases to larger values ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='1, and here c3 will contribute to the far-field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' For small diameters and low values of the silica layer thickness, the modal reflection is small (weak cavity effect) and the scattering into radiation is very large ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' This behavior has been described in the literature47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 7 (a) Modal reflection, R1,1 and (b) reflection into radiation of the fundamental mode, Rrad,1, as a function of the diameter, D, and silica layer thickness, tSiO2, at the bottom interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' A sketch of the bottom interface is shown in the top part of the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Due to the scale invariance of Maxwell’s equations, both the top and bottom reflections are broadband, which gives the potential for broadband Purcell enhancement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='2 Enhanced efficiency In the first part of this subsection, we present different methods to model the efficiency to show how important the different emis- sion channels are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Then we will apply the modelling methods on the structure with the largest efficiency, namely D = 238nm and tSiO2 = 0nm with a QD placed at the 2nd antinode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='1 Efficiency contributions of the emission channels We wish to separate and quantify the efficiency contributions of the main channels and the background channels shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' We also wish to separate and quantify the direct emission of c1 and the scattered channels of c2 and c3 shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (4), and therefore we need two different methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Recall that the efficiency is calculated as ε = Pcollected/PT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' For both methods, PT is calculated using the full model, but Pcollected is calculated such that we can either 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' separate the main channels and the background channels or 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' separate the direct emission of c1 and the scattered channels of c2 and c3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' In the first method, the reflection and transmission matrices are unchanged;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' however, a varying number of the initial coefficients, a∞NW and b∞NW of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (10-11), are included when aJ tot (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (15)) and thus Pcollected is calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' For instance, if we only include the first element of the initial coefficients, aJ 1 and bJ 1, and put the remaining elements to zero, then we only include the main channels in Pcollected, which originated as the fundamental mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' As we increase the number of initial coefficients included, the background channels are added starting from the first radiation mode until the last evanescent mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' ε can then be plotted as a function of the included initial coefficients and if this curve is flat, then the main channels dominate the efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' In the second method, the reflection and transmission matrices are also unchanged, all initial coefficients are included, but in- stead, a varying number of the final coefficients, aJ tot of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (15), are included when Pcollected is calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' The first element of aJ tot represents all the light that ended up in the fundamental mode, where the main contribution is from the fundamental mode it- self, but also includes contributions of the background channels which have scattered into the fundamental mode such as βt,α, βb,α, γt,α and γb,α seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' However, by using the first method, we can quantify how strong these contributions are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' If these contributions are weak, then the main channels are dom- inating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Thus if only the first element of the final coefficients is included in Pcollected, then only the direct transmission of the fun- damental mode is included in the far-field, namely c1 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Then as we increase the number of elements of the final coeffi- cients, the contributions of c2 and c3 are included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' In principle, channels such as βt and βb are also included, but again we use the previous method to quantify these.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' ε can then be plotted as a function of the included final coefficients, and if this curve is increasing, then the channels c2 and c3 are contributing to the efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Finally, we wish to visualize the interference between the direct 1–18 | 7 transmission of the fundamental mode and the scattered channels by calculating the transmission of the fundamental mode and the entire background continuum separately: aair,HE11 = TtopP(ht)aJ tot,HE11 (22) and aair,BG = TtopP(ht)aJ tot,BG, (23) where aJ tot,HE11 = � atot,1 0 ··· 0 � (24) and aJ tot,BG = � 0 atot,2 ··· atot,N � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (25) Then the phase difference between the two contributions can be calculated: ∆φ = arg(aair,HE11)−arg(aair,BG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (26) The phase difference will be separated into TE and TM modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Along with the phase difference, the far-field plots of the di- rect transmission of the fundamental mode (aair,HE11), the entire background continuum (aair,BG) and the total field (aair) will be shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='2 Influence of scattered radiation on collection effi- ciency Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 8 (a) Efficiency, ε (NA = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='75), as a function of the initial coefficients expressed with the propagation constant (β/k0)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (b) Efficiency, ε (NA = 75), as a function of the final coefficients expressed with the propagation constant (β/k0)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' We will now apply the modelling methods for the structure with the largest efficiency, D = 238nm and tSiO2 = 0nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (8) the efficiency is shown as a function of the initial coefficients (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (8a)) and the final coefficients (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (8b)), expressed with the propagation constant (β/k0)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' This corresponds to using the two different methods for calculating the efficiency presented in the previous subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' The very first red point in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (8a) corre- sponds to the fundamental mode (main channels) and includes all scattering channels of the fundamental mode (c1, c2 and c3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' This is sufficient to describe most of the efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Then the initial background radiation modes (blue) are added one by one, start- ing from larger values of (β/k0)2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' from predominantly vertical emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' This part of the curve is very flat, which means that the initial background radiation modes (channels originating from βt and βb) are not crucial for the efficiency, which was also indicated by the low emission rates of the radiation modes in the infinite nanowire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Finally, the evanescent background modes (green) are included from small negative values of (β/k0)2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' slowly de- caying evanescent modes, to large negative values of (β/k0)2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' fast decaying evanescent modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Here there is a small increase due to the slowly decaying evanescent modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Now in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (8b), the very first red point also corresponds to the fundamental mode but only includes the direct transmission, c1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Here the efficiency is only ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='2, much smaller than the total efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Then the scattered radiation modes (blue) corresponding to c2 and c3 are added one by one, also starting from larger values of (β/k0)2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' from predominantly vertical emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Here there is a massive increase in the efficiency, which proves the importance of c2 and c3 to the efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' At some point, there is a kink in the blue part of the curve, which is due to the limited numerical aper- ture, as an NA of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='75 corresponds to (β/k0)2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' However, the remaining part of the curve is not completely flat, and this is due to the non-perfect transmission of the radiation modes of the nanowire to the radiation modes in the air.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' The radiation modes of the nanowire mainly transmit into radiation modes with the same value of β, but there is some scattering into the other radia- tion modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Finally, there is a tiny decrease due to the evanescent modes, as very few of the evanescent modes transmit into radi- ation at the top interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' By comparing the evanescent parts of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (8a) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (8b), we see that the effect of the evanescent modes is mainly back scattering into other modes at the interfaces rather than direct transmission scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' To summarize, the key point in the comparison between Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (8a) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (8b) is that the scattering into radiation of the fundamental mode is crucial for the efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Now we will visualize the interference between the direct trans- mission of the fundamental mode and the scattered channels by inspecting the phase changes and the far-fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (9a), the phase difference in the air layer between the direct transmission of the fundamental mode and the entire background is shown as a function of the propagation constant for TE and TM modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' For the light that propagates vertically, the phase difference is close to zero, such that there is constructive interference between the direct transmission and the background radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' For the light that propagates horizontally, the phase difference is closer to π, and thus there is destructive interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (9b), Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (9c) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (9d) the far-fields of the direct transmission of the fun- damental mode, the background radiation and the total field are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Here we can directly observe the effect caused by the phase difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' In the center of the total far-field, the field is enhanced due to the constructive interference, but for the light that propagates horizontally there is destructive interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' As such, the interference between the direct emission and the radia- tion focuses the far-field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 8 | 1–18 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 9 (a) The phase difference between the direct transmission of the fundamental mode and the background continuum for TE and TM modes as a function of the propagation constant, β/k0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (b) The far-field of the fundamental mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (c) The far-field of the background continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (d) The total far-field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' The white dotted line indicates NA = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Be aware of the different color scales that have been used for the far-fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' A similar analysis of the efficiency for the structure with the largest Purcell factor is included in Supplementary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='3 Enhanced Purcell factor As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (2), there are deviations between the full model and the SMM for the Purcell factor, and we wish to understand where these deviations appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Therefore we will introduce a model which can identify where these deviations appear and ap- ply the model on the structure with the largest Purcell factor, namely D = 250nm and tSiO2 = 13nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='1 Purcell factor contributions from the emission chan- nels To gain physical insight into the physics of the Purcell factor, we will use a model which stepwise increases the complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' At each step the Purcell factor is calculated Fp = PT/PBulk along with the power into the fundamental mode, PHE11/PBulk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' The starting point is the SMM, where only the fundamental mode is included;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' then, in seven steps, the complexity increases until the full model is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Specifically, the initial inputs (a∞NW and b∞NW) and the top and bottom reflections (Rtop and Rbot) will be manipulated at each step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Each step has a direct physical interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' We will now list the 7 steps in the model and for each step, write up the physical effect that is now included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' This means that for each step in the model, all previous effects are also included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' SMM 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Scattering of the fundamental mode at the top interface 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Scattering of the fundamental mode at the bottom interface 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Back-scattering of the background continuum to the funda- mental mode at the top interface 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Back-scattering of the background continuum to the funda- mental mode at the bottom interface 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Scattering of the background continuum to itself at both in- terfaces 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Including initial background continuum Steps number 4 and 5 correspond to the process HE11 → radiation/evanescent → HE11 which is a recycling effect, and we will show the importance of this process for the Pur- cell factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Step number 6 corresponds to the process radiation/evanescent → radiation/evanescent, which also opens up for further scattering channels such as radiation/evanescent → radiation/evanescent → HE11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' The 7 steps can then be translated to the initial inputs and the top and bottom reflections in the following schematic way: a(α) ∞NW = � 1 7 ··· 7 � (27) b(α) ∞NW = � 1 7 ··· 7 � (28) R(α) top = � ������� 1 4 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 4 2 2 6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 6 2 6 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 2 6 6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 2 � ������� top (29) R(α) bot = � ������� 1 5 5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 5 3 2 6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 6 3 6 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 3 6 6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 2 � ������� bot (30) Here, the superscript α represents the seven complexity steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' For a given step α, each matrix entry > α is set to zero, while entries ≤ α keep their original value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' We will use the step α = 4 as an example: a(4) ∞NW = � a1 0 ··· 0 � (31) b(4) ∞NW = � b1 0 ··· 0 � (32) R(4) top = � ������� r11 r12 r13 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' r1N r21 r22 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 0 r31 0 r33 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' rN1 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' rNN � ������� top (33) R(4) bot = � ������� r11 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 0 r21 r22 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 0 r31 0 r33 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' rN1 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' rNN � ������� bot (34) At each step in the previous model building, the entire back- ground continuum was added, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' the entire column or row was 1–18 | 9 1 (a) / 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='5 TE 0 TM 5HEi1 FF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='5 1 β/ko Scattered FF Normalized units 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='2 0Total FF (p 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='5added at once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' As such, it is difficult to quantify which part of the background continuum that is important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Therefore we will also present the models where the elements for the background continuum are increased one at a time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' In this way it can be quantified how the different parts of the background continuum contribute to the Purcell factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='2 Influence of radiation modes on the Purcell factor We will now apply the model for the Purcell factor for the struc- ture with the largest Purcell factor, D = 250nm and tSiO2 = 13nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (10), the Purcell factor is shown as a function of the dipole position from the bottom, hb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' The agreement between the SMM and the full model for the 1st antinode is good, but there are devi- ations for the 2nd and 3rd antinodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Nevertheless, we will focus on the analysis of the 1st and the 2nd antinode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (11a) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (11b), the Purcell factor and the power enhancement of the fundamental mode are shown as a function of the model complexity progression for the 1st and 2nd antinode (model complexity number α will be shortened n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Evidently, the analysis of the Purcell factor is complicated due to contri- butions of the entire background continuum, multiple scattering channels and feedback mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Therefore there are changes in the Purcell factor for all steps in the model complexity, which makes it challenging to model the Purcell factor using only a few modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' The continuum of radiation modes can be modelled using leaky modes, which can enable the modelling using only a few modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' This has been demonstrated in photonic crystal micro- cavities where strong feedback mechanisms also were present48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' However, by using the presented model, we will obtain an in- depth physical insight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' The Purcell factor starts at the same value with the SMM for both antinodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' By including scattering of the fundamental mode at the top interface (n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 2) there is a significant increase for the 2nd antinode but a very small decrease for the 1st antinode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Al- ready now, the deviations compared to the SMM have started to appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' By including the scattering at the bottom interface (n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 3) the 2nd antinode is almost unaffected, but a small decrease ap- pears for the 1st antinode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Now including the back-scattering at the top interface (n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 4), there is a large decrease in the Purcell factor at both antinodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' This decrease is directly represented in PHE11/PBulk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Interestingly, when including the back-scattering at the bottom interface (n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 5), there is now a large increase in the Purcell factor at both antinodes, which is also represented Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 10 Purcell factor, Fp, computed using the full model and SMM as a function of the dipole position from the bottom interface, hb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' D = 250nm and tSiO2 = 13nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 11 Purcell factor (PT/PBulk) and fundamental mode enhancement (PHE11/PBulk) for the 1st (a) and 2nd (b) antinode as a function of the model complexity progression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' In (c) and (d) the background continuum is continuously included between each model complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' in PHE11/PBulk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' This also shows that the recycling effect HE11 → radiation/evanescent → HE11 can provide both negative and pos- itive contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' When the background is allowed to scatter to itself, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' radiation/evanescent → radiation/evanescent (n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 6), there is an increase for both antinodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' This increase is directly represented in PHE11/PBulk, which in fact means that the process radiation/evanescent → radiation/evanescent → HE11 is dominat- ing compared to radiation/evanescent → radiation/evanescent it- self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Finally, by including the initial background (n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 7), there is an increase for the 2nd antinode but a decrease for the 1st antin- ode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' These changes also correspond to the change in PHE11/PBulk, which means it is the process of radiation/evanescent → HE11 that is important for the initial background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' The main differences between the two antinodes appear, when the fundamental mode scatters at the top interface (n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 2) and when the initial background (n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 7) is included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' To better under- stand which part of the background continuum is important, we will also consider the continuous steps of the model complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' This is now shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (11c) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (11d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Here we observe that the main positive contributions at the 2nd antinode are due to the slowly decaying evanescent modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' This is the process of HE11 → evanescent at the top interface and the process of the ini- tial background evanescent → HE11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' For the back-scattering at the interfaces (n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 4 and n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 5), we observe that the propagating radiation modes can also significantly affect the Purcell factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' An additional important observation is that PHE11/PBulk exceeds PT/PBulk for the first antinode due to the negative contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' This would in fact result in β factors above 1 using the definition βHE11 = PHE11/PT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' This indicates that the β factor might not be a suitable figure of merit for structures where the SMM breaks down, or at least one should be very careful in the definition of the β factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Alternatively, the typical interpretation of the β factor as 10 | 1–18 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 12 (a) and (b) Purcell factor, Fp, of the two antinodes for the two structures as a function of wavelength, λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (c) and (d) efficiency, ε (NA = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='75), of the two antinodes for the two structures as a function of wavelength, λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' The numerical uncertainty is represented by the thickness of the curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' a standard power fraction should be reconsidered in the regime of the SMM breakdown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' The analysis of the Purcell factor for the structure with the largest efficiency, D = 238nm and tSiO2 = 0nm, is included in Sup- plementary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='4 Wavelength dependence In this section, we will present the broadband performance of the nanopost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' The focus will be on the designs with the largest Purcell factor and efficiency, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (12a), the Purcell factor is shown as a function of the wavelength for the 1st and 2nd antinode of the structure with the largest Purcell factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' The Purcell factor of the 2nd antin- ode performs better than the 1st antinode close to the resonance wavelength, and the spectral width at FWHM (full width half maximum) of the antinodes are approximately ∆λ2nd = 27nm and ∆λ1st = 28nm showcasing the broadband performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' The spec- trum for the two antinodes is not completely symmetric, and the two curves for the antinodes also cross further away from the res- onance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (12b), the Purcell factor is shown as a function of the wavelength for the 1st and 2nd antinode of the structure with the largest efficiency (at resonance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Here the Purcell factor is much lower and the spectral width much broader at ∆λ2nd = 52nm and ∆λ1st = 56nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Furthermore, the resonance wavelength for the 1st antinode is slightly shifted to λr = 931.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='5nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' This can be explained by the low Q factor and the neighbouring low Q factor cavity modes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' the cavity modes with 2 and 4 antinodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Due to the low Q factor, there is a small spectral overlap causing the slight shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' In the Supplementary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='6, we present a nanopost de- sign where the resonance wavelength shift is more pronounced between the two antinodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (12c) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (12d) the efficiencies (NA = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='75) are shown as a function of the wavelength for the 1st and 2nd antin- Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 13 βHE11 = PHE11/PT as a function of the wavelength, λ, for the 2nd and 1st antinode and the infinite nanowire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' The numerical uncertainty is represented by the thickness of the curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 14 Efficiency, ε (NA = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='75), as a function of the wavelength, λ, for the 2nd antinode (a) and 1st antinode (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' The full model is compared to only using c1, c2 and c3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' D = 238nm and tSiO2 = 0nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' The width of the curves represents the numerical uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' ode for the two structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' The main characteristic of the effi- ciency is that it does not follow the Purcell factor, which is sur- prising compared to traditional Fabry-Pérot cavities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Instead, the efficiency changes roughly linearly across the resonance, and in general, the slope for the 2nd antinode is negative and positive for the 1st antinode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' This means that the maximum efficiency is in fact achieved off-resonance, ε2nd,NA=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='75(λ = 880nm) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='71, but at a much smaller Purcell factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='1 Analysis of the broadband collection efficiency In the previous work on the nanopost29, the broadband efficiency was attributed to the broadband β factor, which again was at- tributed to the dielectric screening effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' This can also be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (5), where the emission into radiation modes is suppressed in most of the interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Here we will define the β factor for the fundamental mode as βHE11 = PHE11/PT, even though it exceeds 1 as we have already shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' The focus is on the structure with the largest efficiency at resonance, D = 238nm and tSiO2 = 0nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (13), βHE11 is shown as a function of the wavelength for the 2nd and 1st antinode and the infinite nanowire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' βHE11 is close to 1 (both above and below) in the entire interval, which indicates that the fundamental mode is still the dominating con- tribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Due to the broadband βHE11 of the infinite nanowire, it is not required to be on resonance to obtain large values of βHE11, and the QD would need to be very close to a node before βHE11 would decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Regardless, the fundamental mode scatters into radiation which affects PT such that βHE11 does not follow a Lorentzian curve like the Purcell factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' To quantify how dominating the fundamental mode is for the 1–18 | 11 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 15 Efficiency, ε (NA = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='75), as a function of the initial coefficients expressed with the propagation constant (β/k0)2 for λ = 890nm (a) and λ = 970nm (b) at the 2nd antinode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' efficiency, we will use the first method presented in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='1 to separate the main channels (c1, c2 and c3) from the background channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (14), we compare the efficiency of the full model to only including c1, c2 and c3 for both antinodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' By comparing to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (13) we observe that the curves for the main channels directly follow the trend of βHE11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' However, there is a discrep- ancy between the efficiency of the full model and only using the main channels of c1, c2 and c3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' This discrepancy is not at a min- imum on resonance (λ = 930nm), but almost at the minimum when βHE11 = 1 for both antinodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' We observe that for βHE11 < 1, the background channels provide a positive contribution to the efficiency, while for βHE11 > 1 the background channels provide a negative contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' This shows that the β factor is still useful in the analysis but not necessarily a figure of merit for SPSs in the breakdown of the SMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Furthermore, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (14) shows that the background channels still interfere with the main channels caus- ing this discrepancy and even changing the slope of the curves for the efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' As an example we will consider two wavelengths for the 2nd antinode and use the first method presented in Sec- tion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='1: In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (15) the efficiency is shown as a function of the ini- tial coefficients expressed with the propagation constant (β/k0)2 for the two wavelengths λ = 890nm and λ = 970nm at the 2nd antinode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Again the first red point corresponds to including all the main channels, and then background channels are added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' For λ = 890nm, the initial propagating background radiation pro- vides a significant increase to the efficiency, while the evanescent modes provide no change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' On the other hand, for λ = 970nm, the initial propagating background radiation provides a significant decrease to the efficiency, while the evanescent modes provide a positive increase to the efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' This also showcases the com- plex interplay between the main channels and the background channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 5 Perspective We have shown that contributions from multiple scattering chan- nels influence both the Purcell factor and especially the collection efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Unlike traditional Fabry-Pérot cavities, where scatter- ing of the light is viewed simply as a loss mechanism, this scat- tering is in fact beneficial for the performance of the nanopost SPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Importantly, this scattering mechanism decouples the effi- ciency from the Purcell factor directly challenging a well-known design paradigm that maximum collection efficiency is obtained on resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' The identification of this mechanism opens a door to unconventional SPS design approaches, especially in the non- resonant regime where the scattering coefficients are no longer analysed and optimized with respect to the fundamental HE11 mode alone, and where definitions of fundamental performance parameters such as the spontaneous emission β factor need to be revisited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' For the nanopost itself, this invites to a new optimization of the collection efficiency with respect to all geometrical param- eters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' The maximum collection efficiency will be obtained for a reduced Purcell factor due to the new trade-off between efficiency and Purcell enhancement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Potential future work on the nanopost design could also be to explore the properties of cavity modes with different orders than 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Additionally, structuring the bottom mirror could also lead to increased performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Adding rings around the nanopost could positively alter the scattering mecha- nism while also bridging the gap to the closely related bullseye de- sign16,27,28, which also features broadband collection efficiency independently of the Purcell factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Despite flourishing literature on the bullseye design, the physical mechanisms underlying the performance is still unclear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' The analysis of the bullseye will be more challenging as the inner mesa/nanowire features larger di- ameters resulting in additional guided modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' The rings around the inner mesa will also heavily influence the radiation modes and their mode profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' This will, in turn, lead to changes in the emission rates and the reflection matrices and, thus, the scat- tering channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' In typical bullseye structures16,27,28, which are numerically optimized, the silica layer is hundred of nanometers thick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Here we anticipate that such large thicknesses will also result in increased mode coupling as light diverges when propa- gating in the silica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 6 Conclusions We have shown that the traditional Fabry-Pérot single-mode model, which typically provides an excellent description12–14 of the physics for cavity-based single-photon sources, significantly underestimates the achievable performance of the nanopost struc- ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Using a modal expansion method, we have performed a detailed analysis of the emission channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' We have shown that in particular the collection efficiency benefits significantly from a contribution from light scattered to radiation modes, which often is simply considered a loss mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' This scattering into ra- diation modes not only allows for improved collection efficiency but also decouples the collection efficiency from the Purcell fac- tor, such that optimum performance is obtained off-resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Our parameter scan of the nanopost structure reveals an achiev- able Purcell factor Fp of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='9 or a collection efficiency ε of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='69 obtained for two very different parameter sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Our work invites further exploration of unconventional SPS design mechanisms, especially in the non-resonant regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Conflicts of interest There are no conflicts to declare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 12 | 1–18 Acknowledgements We thank Battulga Munkhbat for his assistance in creating the sketch of the nanopost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' This work is funded by the European Research Council (ERC-CoG “UNITY,” Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 865230), the French National Research Agency (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' ANR-19-CE47- 0009-02), the European Union’s Horizon 2020 Research and In- novation Programme under the Marie Skłodowska-Curie Grant (Agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 861097), and by the Independent Research Fund Denmark (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' DFF-9041-00046B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' References 1 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Pan, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 8 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Qin, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Ding, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Chen, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Chen, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' You, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' He, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Jiang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' You, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Wang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Schneider, J.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Denning, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Gür, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Lu and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Gregersen, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' B, 2020, 102, 125301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 15 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Purcell, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=', 1946, 69, 681.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 16 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' He, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Chung, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Hu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Yu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Chen, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Ding, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Chen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Qin, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Yang, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='-Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Liu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Duan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Li, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Gerhardt, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Winkler, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Jurkat, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Wang, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Gregersen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Huo, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Dai, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Yu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Höfling, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Lu and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Pan, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Photonics, 2019, 13, 770–775.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 17 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Somaschi, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Giesz, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' De Santis, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Loredo, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Almeida, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Hornecker, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Portalupi, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Grange, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' An- tón, J.' metadata={'source': 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A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' White, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Lanco and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Senel- lart, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Photonics, 2016, 10, 340–345.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 18 X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Ding, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' He, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Duan, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Gregersen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Chen, S.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Pan, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=', 2016, 116, 020401.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 19 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Tomm, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Javadi, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Antoniadis, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Najer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Löbl, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Korsch, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Schott, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Valentin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Wieck, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Ludwig and 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Munsch, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Malik, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Dupuy, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Delga, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Bleuse, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Gérard, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Claudon, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Gregersen and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Mørk, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=', 2013, 110, 177402.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 50 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' de Lasson, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Kristensen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Mørk and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Gregersen, Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=', 2015, 40, 5790–5793.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 14 | 1–18 7 Supplementary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='1 Height adjustment procedure In principle, the dipole position ht and hb would also be free pa- rameters if we wanted to perform a complete optimization of the nanocavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' However, this would be a very demanding task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' In- stead, we have chosen to study cavities with 3 antinodes as our test simulations have shown that the 2nd antinode in a 3 antinode cavity provides a good performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Thus ht and hb will be fixed to ensure a cavity with 3 antinodes and a resonance wavelength of λr = 930nm at the 2nd antinode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' A procedure is required such that ht and hb satisfies these conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' The first step is to use the SMM (phase conditions of the fundamental mode) to determine the initial total height h: h = (2π −arg(rbot,11))/(2β1)+(2π −arg(rtop,11))/(2β1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (35) This is under the conditions arg(rbot,11) < 0 and arg(rtop,11) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Then we place the dipole in the second antinode from the bottom: hb = (2π −arg(rbot,11))/(2β1) (36) and ht = (2π −arg(rtop,11))/(2β1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (37) However, due to the background continuum and mode-coupling, this method does not ensure that the dipole is placed exactly at an antinode nor that the resonant wavelength of the cavity corre- sponds exactly to the design wavelength, ( dFp dλ )λ=λd = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' To solve this problem, the dipole is first adjusted to the exact position of the antinode by plotting |Er(z)|2 and locating the peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Then the height is slightly adjusted h = h ± δh, while the dipole is moved to the exact position of the antinode for each adjustment until ( dFp dλ )λ=λd = 0 is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Finally, the position of the 1st antinode (from the bottom) can also be identified by plotting |Er(z)|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Finally, we will provide the total height, htotal of the structure along with the height deviation between the initial height ob- tained from the SMM and the final height, hdiff = htotal −hinitial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 450 500 550 600 650 700 400 400 550 700 4 4 2 2 0 0 2 2 5 5 10 0 20 40 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 16 (a) Total height of the structure, htotal, as a function of the diameter, D, and the silica layer thickness, tSiO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (b) The height difference between the initial height and the final height, hdiff, as a function of the diameter, D, and the silica layer thickness, tSiO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (16a), the total height of the structure is shown as a function of the diameter and the silica layer thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' The to- tal height mostly depends on the diameter as the diameter deter- mines the propagation constant β1 and thus the primary influence of the phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (16b), the height difference of the final total height compared to the SMM is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' For most of the parame- ters, the difference is small in the range −5nm to 5nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' However, for the small diameters and thin silica layer thicknesses, there is a very large difference up to 50nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' This is the same parameter re- gion where the modal reflection at the bottom interface is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Thus the phase of the fundamental mode is less dominating com- pared to the contributions of the radiation and evanescent modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Oscillations can also be observed in the height difference due to numerical noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' The exact resonance wavelength is sensitive to the height of the structure, but these oscillations are on the scale of less than 1nm, and the uncertainty in the resonance wavelength will be on a similar scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='2 Influence of the numerical aperture In all the simulations of the efficiency a numerical aperture of NA = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='75 has been used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Here we will investigate the influence of varying the numerical aperture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='9 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='5 1 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 17 Efficiency, ε, at the 2nd antinode for a numerical aperture of NA = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='4 (a) and NA = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='00 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (c) Efficiency, ε, as a function of the numerical aperture, NA, for three different parameters at the 2nd antinode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (d) Efficiency, ε, as a function of wavelength, λ, for four different values of the NA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (17a) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (17b) the efficiency is shown as a function of the diameter and the silica layer thickness similar to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (3a), but for a numerical aperture NA = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='4 and NA = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Lowering the numerical aperture to NA = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='4 drastically reduces the efficiency and the maximum is barely above ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' This shows that a large numerical aperture is crucial for the good performance of the nanopost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' By increasing the numerical aperture from NA = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='75 1–18 | 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='57 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='5 1 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 18 Gaussian collection efficiency, εg, as a function of the diameter, D, and the silica layer thickness, tSiO2, for the 2nd antinode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (θ ≈ 49◦) to NA = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='00 (θ = 90◦) there is roughly a 20% increase in the efficiency, so there is still some light lost at angles above θ ≈ 49◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Furthermore, for a numerical aperture of NA = 1, the efficiency directly represents the losses to the bottom mirror.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' For diameters above D = 210nm, an increased silica layer thickness increases the losses to the bottom mirror even though the Pur- cell factor increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (17c) the efficiency for the structures with the largest efficiency and Purcell factor, at the 2nd antinode, are shown as a function of the numerical aperture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' The steepest part of the curves is roughly in the interval NA = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='4 to NA = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='75, which is the reason for the huge difference in efficiency between NA = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='4 and NA = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' The curves also start to flatten out as the NA reaches 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (17d) the efficiency is plotted for four dif- ferent values of the numerical aperture as a function of the wave- length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' The numerical aperture does not influence the curvature of the efficiency as a function of the wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' This means that being on resonance does not focus the far-field compared to being off resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='3 Gaussian collection efficiency So far the efficiency has been evaluated by calculating the to- tal power collected in the lens with some numerical aperture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' However, in many applications the light will couple to a fiber afterwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Therefore we have also calculated the power over- lap between the emitted far-field and the far-field of a Gaussian representative for the fundamental mode in many single-mode fibers49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' The applied method is identical to the one presented in the appendix of14, and the Gaussian collection efficiency is de- fined as εg = Pcollected,Gaussian/PT, where Pcollected,Gaussian is defined as the overlap with a Gaussian profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (18) the Gaussian efficiency is shown for the 2nd antinode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Compared to the stan- dard efficiency in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (3a), the difference is approximately 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='1 over the entire parameter space, showcasing the Gaussian shaped profile of the far-field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='4 Efficiency analysis for the structure with maximum Pur- cell factor We will now apply the efficiency analysis for the structure with the largest Purcell factor with the parameters D = 250nm and tSiO2 = 13nm and an efficiency of ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (19) the efficiency is shown as a function of the initial coefficients (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (19a)) and the final coefficients (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (19b)), expressed with the propagation constant (β/k0)2, just as for the structure with maximum efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Again the curve in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (19a) is flat and the channels of the fundamental mode dominates the efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' The efficiency increase by adding the final coefficients, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' c2 and c3, seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (19b), is still significant, but much smaller compared to the structure with maximum efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Here the increase is from approximately ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='3 to ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' The curve in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (19b) also flattens out completely due to the numerical aperture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 19 (a) Efficiency, ε (NA = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='75), as a function of the initial coef- ficients expressed with the propagation constant (β/k0)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (b) Efficiency, ε (NA = 75), as a function of the final coefficients expressed with the propagation constant (β/k0)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 20 (a) The phase difference between the direct transmission of the fundamental mode and the background continuum for TE and TM modes as a function of the propagation constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (b) The far-field of the fundamental mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (c) The far-field of the background continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (d) The total far-field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' The white dotted line indicates NA = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Be aware of the different color scales that have been used for the far-fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (20a) the phase difference in the air layer between the direct transmission of the fundamental mode and the entire back- ground is shown as a function of the propagation constant for TE and TM modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' We observe similar features as before, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' constructive (destructive) interference for light propagating ver- tically (horizontally).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Though at β/k0 = 1, the phase difference is larger compared to the previous structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (20b), Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (20c) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (20d) the far-fields of the direction transmission 16 | 1–18 1 (a) / 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='5 0 TE TM 5HEi1 FF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='5 1 β/ ko Scattered FF Normalized units c 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='5 0Total FF (p 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='5 0of the fundamental mode, the background radiation and the total field is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Compared to the structure with maximum effi- ciency, the far-field of the background radiation is significantly different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Here, the far-field is mainly focused towards horizontal angles and the intensity is much smaller compared to the far-field of HE11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' As such the constructive contribution at smaller angles is not as significant and less of the radiation will be captured by the lens, due to the numerical aperture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' This explains why the efficiency increase in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (19b) is much smaller compared the structure with the maximum efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' However, there is still de- structive interference for the light that propagates horizontally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' As such the interference between the direction emission and the radiation focuses the far-field, but not to the same degree as for the structure with the maximum efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='5 Purcell factor analysis for the structure with maximum collection efficiency We will now apply the model for the Purcell factor for the struc- ture with the largest efficiency, D = 238nm and tSiO2 = 0nm and Fp = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 21 Purcell factor, Fp, computed using the full model and the SMM as a function of the dipole position from the bottom interface, hb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' D = 238nm and tSiO2 = 0nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (21) the Purcell factor is shown as a function of the dipole position throughout the cavity, both the full model (n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 7) and the SMM (n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 1) are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Here the 3 antinodes can be ob- served and the SMM predicts a larger Purcell factor for all 3 antin- odes compared to the full model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' The positions of the antinodes are almost identical between the full model and the SMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' The Purcell factor increases drastically when the dipole is placed close to the metal mirror due to non-radiative decay processes33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (22a) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (22b) the Purcell factor and the power en- hancement of the fundamental mode is shown as a function of the model complexityfor the 1st and 2nd antinode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Compared to the structure with D = 250nm and tSiO2 = 13nm, there are a few differ- ences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' The SMM predicts a smaller Purcell factor, which is simply caused by the lower modal reflection at the bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' There is a large negative contribution when including the back-scattering at the bottom interface (n 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=') at both antinodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' This is caused by the change of the silica layer thickness and as seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (22c) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (22d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' The propagating radiation modes are responsible for this decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Furthermore, by including the scattering of the background to itself (n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 6), there is now a small decrease for both antinodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' These are the differences between the two structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' The differences between the 1st and 2nd antinode are exactly the same for the two structures, where the scattering into evanescent modes at the top interface (n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 2) and the initial evanescent modes provide a positive contribution at the 2nd antinode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 22 Purcell factor (PT/PBulk) and fundamental mode enhancement (PHE11/PBulk) for the 1st (a) and 2nd (b) antinode as a function of the model number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' In (c) and (d) the background continuum is continuously included between each model number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='6 Asymmetric wavelength dependence for the two antin- odes To further study the resonance shift between the 1st and 2nd antinodes, we choose a nanopost design of D = 202nm and tSiO2 = 5nm, where the shift is more pronounced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 23 Purcell factor, Fp, as a function of wavelength, λ, for the two antinodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' The parameters are D = 202nm and tSiO2 = 5nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (23) the Purcell factor is shown for the two antin- odes as a function of the wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' The peak positions of the Purcell factors (resonance wavelength) are λ2nd,r = 930nm and λ1st,r = 933nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' By observing the curve for the 1st antinode, this shift is caused by another broad resonance at approximately λ = 1050nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' To gain further insight into the resonances of the struc- ture and verify our results, we have performed a quasi-normal mode (QNM) simulation50 of the nanopost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' In this simulation 15 QNMs are found and the complex eigenfrequencies, ˜ωµ = ωµ − iγµ, of the 3 important QNMs are ˜ωQNM1 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='0237×1015 − i4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='4904×1013Hz, ˜ωQNM2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='7595×1015 − i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='4654×1014Hz and 1–18 | 17 ˜ωQNM3 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='2809×1015−i2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='0703×1013Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' The corresponding real parts of the complex wavelength are λQNM1 = 930.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='3nm, λQNM2 = 1063.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='2nm and λQNM3 = 825.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='8nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' The Q factors of the QNMs can also be calculated using Qµ = ωµ/(2γµ)50, and we obtain QQNM1 = 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='5, QQNM2 = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='0 and QQNM3 = 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 24 Comparison of the Purcell factor between the FMM and the QNM simulation for the 1st antinode (a) and the 2nd antinode (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' In-plane electrical field profiles of the 3 QNMs at their resonance wave- lengths are shown in (c), (d) and (e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' The green star corresponds to the position of the 1st antinode, and the red star corresponds to the position of the 2nd antinode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' The white scale bar in (a) corresponds to 100 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' The intensity is scaled in each field plot and should not be used for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (24a,24b), the comparison of the Purcell factor between the FMM and the QNM simulation is shown for the two antinodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Overall, the quantitative agreement between the FMM and QNM simulations is good with some small deviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (24a), the individual contributions of 3 QNMs are plotted along with their sum and the result of the FMM for the 1st antinode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' These 3 QNMs provide a good description of the overall Purcell factor and they directly correspond to the peaks in the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' QNM1 and QNM2 also overlap in the spectrum due to the low Q factor of QNM2, which slightly shifts the peak position of the total Pur- cell factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (24b), the individual contributions of 2 QNMs are plotted along with their sum and the result of the FMM for the 2nd antinode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Here QNM1 is almost sufficient to describe the entire spectrum, and we do not observe any other peaks than the one at λ = 930nm, besides a small bump at longer wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Now, consider the in-plane electrical field profiles of the 3 QNMs shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' (24c,24d,24e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' QNM1 has 3 antinodes, QNM2 has 2 antinodes and QNM3 has 4 antinodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' The green star corresponds to the position of the 1st antinode, where the QD is placed, and this position is very close to an antinode for QNM2 and QNM3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' Therefore the contributions of these QNMs appear in the spec- trum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' However, the position of the 2nd antinode (red star) is much closer to a node for QNM2 and QNM3, and therefore they do not influence the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} +page_content=' 18 | 1–18 lst antinode FMM (b) 3 QNM - 3 modes QNM1 QNM2 2 QNM3 12nd antinode FMM QNM - 2 modes QNM1 QNM20 900 1000 1100 90 入(nm) QNM1 QNM2 米00 1000 1100 入(nm) QNM3' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfqwEz/content/2301.02556v1.pdf'} diff --git a/q9E5T4oBgHgl3EQflQ94/vector_store/index.faiss b/q9E5T4oBgHgl3EQflQ94/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..d109c260b8ea3a7970f9b292d9af2a3826fa49ab --- /dev/null +++ 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NV Sensors in Diamond-based High-pressure Devices +Kin On Ho,1, ∗ Man Yin Leung,2, ∗ Wenyan Wang,1, ∗ Jianyu Xie,1 King Yau Yip,1 +Jiahao Wu,2 Swee K. Goh,1, 3 Andrej Denisenko,4 J¨org Wrachtrup,4 and Sen Yang1, 2, † +1Department of Physics, The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong, China +2Department of Physics and the IAS Centre for Quantum Technologies, +The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China +3Shenzhen Research Institute, The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong, China +43rd Institute of Physics, University of Stuttgart and Institute for Quantum Science and Technology (IQST), +Pfaffenwaldring 57, D-70569, Stuttgart, Germany +(Dated: January 16, 2023) +High-pressure experiments are crucial in modern interdisciplinary research fields such as engineer- +ing quantum materials, yet local probing techniques remain restricted due to the tight confinement of +the pressure chamber in certain pressure devices. Recently, the negatively charged nitrogen-vacancy +(NV) center has emerged as a robust and versatile quantum sensor in pressurized environments. +There are two popular ways to implement NV sensing in a diamond anvil cell (DAC), which is +a conventional workhorse in the high-pressure community: create implanted NV centers (INVs) +at the diamond anvil tip or immerse NV-enriched nano-diamonds (NDs) in the pressure medium. +Nonetheless, there are limited studies on comparing the local stress environments experienced by +these sensor types as well as their performances as pressure gauges. In this work, by probing the NV +energy levels with the optically detected magnetic resonance (ODMR) method, we experimentally +reveal a dramatic difference in the partially reconstructed stress tensors of INVs and NDs incorpo- +rated in the same DAC. Our measurement results agree with computational simulations, concluding +that INVs perceive a more non-hydrostatic environment dominated by a uniaxial stress along the +DAC axis. This provides insights on the suitable choice of NV sensors for specific purposes and the +stress distribution in a DAC. We further propose some possible methods, such as using NDs and +diamond nanopillars, to extend the maximum working pressure of quantum sensing based on ODMR +spectroscopy, since the maximum working pressure could be restricted by non-hydrostaticity of the +pressure environment. Moreover, we explore more sensing applications of the NV center by study- +ing how pressure modifies different aspects of the NV system. We perform a photoluminescence +(PL) study using both INVs and NDs to determine the pressure dependence of the zero-phonon +line (ZPL), which helps developing an all-optical pressure sensing protocol with the NV center. We +also characterize the spin-lattice relaxation (T1) time of INVs under pressure to lay a foundation +for robust pulsed measurements with NV centers in pressurized environments. +I. +INTRODUCTION +Pressure is an important thermodynamic parameter +for engineering quantum materials because it allows one +to tune material properties without altering the chemical +composition, and some long-searched-for quantum phases +are expected to emerge under ultra-high pressure, such as +room-temperature superconductivity [1–4] and metallic +hydrogen [5, 6]. High-pressure experiments are, however, +non-trivial to perform since one needs robust pressure +devices and compatible measurement techniques. +One reliable pressure device is the diamond anvil cell +(DAC), which has been widely used in the high-pressure +community. +The pressure is achieved by mechanically +pressing two opposing diamond anvils towards a tightly +confined pressure chamber in the middle. The pressure +medium filling up the chamber remains hydrostatic be- +low its critical pressure Pc, and it undergoes solidification +or glass transition at Pc where pressure inhomogeneity +∗ These authors contributed equally to this work. +† phsyang@ust.hk +starts building up. It is crucial to understand whether +the medium is hydrostatic during the experiment, since +the subsequent data processing and interpretation may +be inappropriate if the artifacts from pressure inhomo- +geneity are not taken into account. +As to the suitable measurement techniques in high- +pressure experiments, quantum sensing with negatively +charged nitrogen-vacancy (NV−) centers has emerged as +a strong candidate. We always denote NV− as NV in +the rest of this paper. The NV center is a color defect +in diamond which consists of a substitutional nitrogen +atom, an adjacent atomic vacancy, and an extra electron. +Its ground state is an electron spin S = 1 system and +the spin sublevels are responsive to temperature, stress +field, magnetic field, electric field, and the surrounding +spin bath, making the NV center a versatile sensor for +these physical quantities [7–14]. +In practice, we mea- +sure the electron spin resonance (ESR) spectrum of the +NV center using the optically detected magnetic reso- +nance (ODMR) method, which relies on the spin-state- +dependent fluorescence rate of the NV center caused by +the spin-state-dependent decay route back to the ground +state (Fig. 1(a)). In ODMR spectroscopy, a green laser +is used for initialization and read-out of the NV spin +arXiv:2301.05462v1 [cond-mat.mes-hall] 13 Jan 2023 + +2 +state while a microwave (MW) is used for spin-state ma- +nipulation. The ODMR spectrum encodes information +about the NV energy structure and hence the environ- +ment around the NV center. Due to the superior reso- +lution and sensitivity, quantum sensing with NV centers +has become a promising experimental technique. +It has been demonstrated that NV sensing is highly +compatible with DACs, and NV centers have outstand- +ing sensing performance even under the demanding con- +ditions inside DACs [15–23]. There are mainly two ways +to incorporate NV sensors in DACs: (1) create a layer of +implanted NV centers (INVs) at a suitable depth inside +the diamond anvil tip [18, 19, 22], (2) drop-cast some NV- +enriched nano-diamonds (NDs) at the pressure medium +interface inside the pressure chamber [16, 17, 20]. +In +general, they are employed to study different kinds of +materials under pressure. INVs are commonly used to +probe 2D or 3D materials with flat surfaces, while NDs +are often applied to examine materials with irregular sur- +faces. Moreover, INVs and NDs have their own advan- +tages and drawbacks in sensing applications. INVs pro- +vide an easy way to detect vector fields because of the +known orientation of the bulk diamond crystal in the lab- +oratory frame, yet, the spatial resolution is restricted by +the optical diffraction limit and the spatial uniformity of +INVs is constrained by imperfections in the implantation +procedures; on the other hand, NDs present high spatial +resolution controlled by the ND size given the NDs are +sparsely distributed and NV centers in the NDs are in +close proximity to the sample, yet, the crystal orienta- +tions of NDs are random and require individual calibra- +tion in the laboratory frame and spin decoherence times +of NDs are generally shorter than INVs. Some obvious +pros and cons of INVs and NDs are long known, but to +the best of our knowledge, no studies have directly com- +pared the pressurized environments perceived by these +two types of NV sensors in a single DAC. This incom- +plete understanding of the pressurized environments at +different locations in a DAC may hinder the accurate +choice of NV sensors for different experimental purposes. +Another prevailing question from the NV community +is the maximum pressure that NV centers can work with +as quantum sensors, especially as magnetic field sensors +since the probing of local magnetic fields with high spa- +tial resolution is crucial for material research and phase +transition studies [17–19]. Ultra-high pressure can bring +detrimental effects on quantum sensing with NV centers, +including the quenching of ODMR contrasts due to the +spin-sublevel mixing in a non-hydrostatic environment. +To realize magnetic field sensing in pressurized systems, +some previous studies have demonstrated the use of a +bias magnetic field to overcome the effects of uniaxial +stresses [17–19]. Nonetheless, a strong bias field is re- +quired for large uniaxial stresses. This may impose tech- +nical difficulties on the experimental setup, and a strong +bias field may undesirably change the properties of the +material under investigation. Thus, it is of interest to +explore other complementary solutions for extending the +working pressure of NV sensing. +In this work, we first incorporate both INVs and NDs +in the same DAC and analyze the difference in effec- +tive pressure transmissions from the hydrostatic pressure +medium to these two types of sensors. We partially re- +construct the local stress tensors perceived by INVs and +NDs using information from the respective ODMR spec- +tra, and we also perform finite-element simulations to +cross-check our experimental findings. +These analyses +serve to calibrate the local pressurized environments of +the two sensor types, to compare their performances as +hydrostatic pressure gauges, and to determine their op- +timal working ranges. By comparing the pressure condi- +tions of the two sensor types, we demonstrate how non- +hydrostaticity restricts the maximum working pressure +of NV sensing, and we further propose some possible so- +lutions to conquer the non-hydrostaticity. Besides, thor- +oughly characterizing the stress responses of NV sensors +may pave the way for simultaneous detection of multiple +physical parameters via ODMR spectroscopy, like pres- +sure and temperature or pressure and magnetic field. +Next, we employ our ODMR-calibrated NV sensors to +investigate from different perspectives the pressure-tuned +energy structure of the NV center. We measure the pho- +toluminescence (PL) spectra of both INVs and NDs to +study the pressure dependence of the zero-phonon line +(ZPL), which represents the energy spacing between elec- +tronic orbitals of the NV center. We also measure the +spin-lattice relaxation (T1) time of INVs against pres- +sure to probe the electron-phonon coupling in the solid- +state defect system. +Combining various spectroscopic +techniques ranging from continuous-wave (cw) to pulsed +measurements and from ESR to PL measurements, we +hereby provide a multi-dimensional understanding of the +NV quantum system under high pressure, which helps +fostering more accurate and distinct applications of NV +sensing in extreme conditions. Such applications include +an all-optical pressure sensing protocol based on PL spec- +troscopy and robust implementation of pulse sequences +at high pressure. +II. +THEORETICAL BACKGROUND +In a single crystalline diamond with an ensemble of +NV centers, there are four possible spatial orientations +for the NV centers. We thus have five relevant reference +frames: the crystal frame (X, Y, Z) and the principal axis +frames for the four NV orientations (x, y, z)k, k ∈ {NV1, +NV2, NV3, NV4}. The four NV frames can be related +by simple rotation transformations due to the symmetry +of the diamond crystal. In this work, we follow Barfuss +et al.’s conventions of the five frames and the coordinate +transformations between them [24], and we always take +compressive stresses to be positive. +The NV center is a robust stress sensor due to the +spin-stress coupling effect [11, 14, 19–21, 24–37]. Under a +stress field affecting the spin-spin interaction, the ground- + +3 +state Hamiltonian for each NV orientation in its principal +axis frame can be written as [24–26] +Hk = (D0 + M k +z )S2 +z + M k +x(S2 +y − S2 +x) ++ M k +y {Sx, Sy} + N k +x{Sx, Sz} + N k +y {Sy, Sz}, (1) +where S is the spin-1 operator, D0 = 2870 MHz in am- +bient conditions, and in the hybrid representation, the +NV-frame quantities M k +x,y,z and N k +x,y can be expressed in +terms of the components σIJ of the crystal-frame stress +tensor σ. For NV1 along [111] crystal direction, +M NV1 +z += a1(σXX + σY Y + σZZ) ++ 2a2(σY Z + σXZ + σXY ), +(2) +M NV1 +x += b(2σZZ − σXX − σY Y ) ++ c(2σXY − σY Z − σXZ), +(3) +M NV1 +y += +√ +3b(σXX − σY Y ) + +√ +3c(σY Z − σXZ), +(4) +N NV1 +x += d(2σZZ − σXX − σY Y ) ++ e(2σXY − σY Z − σXZ), +(5) +N NV1 +y += +√ +3d(σXX − σY Y ) + +√ +3e(σY Z − σXZ), +(6) +where a1, a2, b, c, d, and e are the spin-stress coupling +constants in the hybrid representation. +To obtain the +above expressions for the other three NV orientations, we +need to replace σIJ by (Kl·σ·(Kl)T)IJ in Eqs. (2) to (6), +where Kl are the coordinate transformations from NV1 +to l ∈ {NV2, NV3, NV4} as defined in Ref. [24]. The re- +sulting expressions for NV2-4 are different from Eqs. (2) +to (6) only by sign flips in some of the off-diagonal tensor +components. See Supplementary Materials for details. +Experiments have found that a1 = 0.486±0.0002, a2 = +−0.37 ± 0.002, b = −0.147 ± 0.0002, c = 0.342 ± 0.0007 +MHz/kbar [19, 25], agreeing well with the theoretical +values from a density functional theory (DFT) study +[26]. +This DFT study also reports d = 0.012(1) and +e = −0.066(1) MHz/kbar. Since d and e are an order +of magnitude smaller than the rest of the coupling con- +stants, we can neglect the N k +x and N k +y terms in Eq. (1) for +our first-order discussion here, and the three eigenvalues +of the Hamiltonian Hk can thus be analytically solved as +follows, +f k +0 = 0, f k +± = D0 + M k +z ± +� +(M kx)2 + (M ky )2. +(7) +Hence, f k +± are the two resonance frequencies detectable +by ODMR spectroscopy, with their center and splitting +being Dk and 2Ek respectively (Fig. 1(a)). +In the regime of small shear stresses, the four NV ori- +entations have close f+’s and close f−’s, leading to two +overall resonances in the ODMR spectrum of the whole +NV ensemble. We further assume equal population for +the four NV orientations in the diamond crystal, such +that the two overall ODMR resonances should be aver- +ages of f k ++ and f k +− over k ∈ {NV1, NV2, NV3, NV4}, +with their center D and splitting 2E written respectively +k +k +FIG. 1. (a) A simplified energy level diagram of the NV cen- +ter showing the spin-state-dependent transitions. (b) An il- +lustration of the DAC configuration in our experiment. The +MW antenna is fabricated on the implanted diamond anvil +culet, while some 140-nm NDs are drop-casted on the other +un-implanted anvil culet. (c, d) ODMR responses of a 140-nm +ND (labeled as ND1) and a patch of INVs (labeled as INV1) +to the change in pressure of DAC1, respectively. The decrease +in their ODMR contrasts is due to the stress-induced mixing +of NV spin states and the degradation of the MW structure, +where the latter factor takes a heavier toll on ND1. +as +D = D0 + 1 +4 +� +k +M k +z , +(8) +E = 1 +4 +� +k +� +(M kx)2 + (M ky )2. +(9) +These expressions reveal that D scales with pressure, +while E results from the imbalance between uniaxial +stresses along the three orthogonal directions and the +presence of shear stresses, or in other words E is an indi- +cator of hydrostaticity. When we compress the diamond +crystal, both D and E will increase in general, i.e. the +ODMR resonances will shift to the right and split further +apart. +With Eqs. (8) and (9) in hand, we can employ ODMR +spectroscopy to partially reconstruct the crystal-frame +stress tensor σ perceived by the NV ensemble. This the- +ory section is applicable for both INVs and NDs, and +to have meaningful interpretations of the reconstructed +crystal-frame stress tensors, we must also understand +how the INV and ND crystal frames are related to the +laboratory frame, which we will discuss in Section III. + +(d)(arb. units) +1.0 +DAC1 ND1 +0.9 +Fluorescence +d +0.8 +2850 +2925 +3000 +Frequency (MHz)DAC1INV1 +(arb. +0.9 +Fluorescence +0.8 +2850 +2925 +3000 +Frequency (MHz)(a) +(b) +Excited +states +0 +Metastable +states +D +Ground +(c) +states +(d)(c)4 +III. +EXPERIMENTAL SETUP +Fig. 1(b) illustrates our customized DAC design where +both INVs and NDs are incorporated. We utilize (100)- +oriented diamond anvils, and the layer of INVs is located +at the culet of one of the anvils. This implanted anvil +is prepared by 9.8 keV 15N ion implantation at a dose +of ∼1012 N/cm2 and subsequent annealing at 950oC in +a high vacuum (P < 10−6 mbar) for 2 hours. The re- +sulting implantation area has a diameter of 200 µm and +is at a depth of ∼10 nm below the culet surface that +has a surface roughness of ∼1.5 nm. On the other hand, +some 140-nm NDs with nitrogen concentration of 3 ppm +are sparsely drop-casted on the culet surface of the other +un-implanted diamond anvil. To perform ODMR spec- +troscopy with these two types of NV sensors, a 150- +µm-radius Omega-shaped gold microstructure is fabri- +cated on the implanted anvil for MW transmission [38]. +As to the pressure chamber in our design, a 300-µm- +diameter hole is drilled in the middle of a beryllium- +copper gasket and the hole is completely filled with a 4:1 +methanol:ethanol mixture as the pressure medium. At +room temperature, this particular medium remains hy- +drostatic up to ∼100 kbar [39–43] which fully covers our +experimental pressure range, enabling us to compare the +local pressurized environments of INVs and NDs given +the medium is in an excellent hydrostatic condition. An- +other reason for choosing this medium is that most of +the common phase transitions tuned by pressure in con- +densed matter physics lie within 100 kbar. Therefore, it +is of technical significance to study the stress distribution +in a DAC, which is a popular pressure device in material +research, using a medium with the hydrostatic limit up +to 100 kbar. +We have prepared two DACs based on the above- +described design, where all the cell configurations are the +same except for the thickness of the pre-indented gasket +(150 µm in one DAC and 200 µm in the other). We will +denote these two DACs as DAC1 and DAC2 respectively +hereafter. In our experiments with the DACs, a home- +built confocal microscope with a 520-nm laser diode and a +long-working-distance objective (50X Mitutoyo Plan Apo +SL) is used to optically address the NV sensors, and the +local pressure is calibrated by ∂D/∂P = 1.49 MHz/kbar +[20] and the D value at ambient pressure measured by +the corresponding NV sensor (the ambient D values have +only tiny deviations from D0 = 2870 MHz). +Since the implanted anvil is (100)-oriented, it is nat- +ural to define the INV crystal frame (X, Y, Z) with the +X axis along the DAC axis. On the other hand, it is +not that trivial to determine how the crystal frames of +individual NDs are oriented with respect to the labora- +tory frame. We need to first apply a static magnetic field +along the DAC axis and measure the ODMR spectrum of +the target ND. Then by studying the Zeeman splittings +in the spectrum, we can obtain the projection angles of +the DAC axis onto the four NV orientations [44]. The +unit direction of the DAC axis in the ND crystal frame +can thus be computed by solving an effective problem of +the intersection of three cones (see Supplementary Mate- +rials for details). The subsequent stress analysis should +not depend on exactly how we assign the four angles to +the four NV orientations (NV1-4) under our assumption +of the equal population for the four orientations, and we +will explicitly check that this is the case in Section V. +For PL measurements, we use the 520-nm laser diode +to excite NV electrons from the electronic ground state +to the phonon band above the electronic excited state. +The NV electrons would decay to the zero-phonon mode +of the excited state via emitting infrared (IR) radiation, +then to the phonon band of the ground state via emit- +ting red PL, and finally to the zero-phonon mode of the +ground state via emitting IR radiation [45]. The ZPL +in the resulting PL spectrum is produced by NV elec- +trons that decay from the zero-phonon mode of the ex- +cited state directly back to the zero-phonon mode of the +ground state. The PL spectra of INVs and NDs are col- +lected using a commercial spectrometer (Princeton In- +strument IsoPlane SCT-320) with a 550-nm long-pass fil- +ter in front. To obtain a PL spectrum solely originating +from the NV centers in a targeted sensor, we subtract +the PL spectrum measured under an applied MW field +at one of the ODMR resonance frequencies from the spec- +trum without any exerted MW. This method makes use +of the spin-state dependence of the NV fluorescence. To +enhance the data quality, we choose to drive whichever +one of the two ODMR resonances with higher contrast. +IV. +COMPARISONS OF LOCAL +PRESSURIZED ENVIRONMENTS +DAC1 (DAC2) is pressurized in an ascending pressure +sequence from the ambient pressure p0 up to p6 (p5), ex- +cept that p4 (p5) is a reduced pressure point. Through- +out the experiment with DAC1 (DAC2), we have tracked +three (five) 140-nm NDs and six (six) 500-nm patches of +INVs. Note that our confocal microscope has a lateral +resolution of ∼500 nm, and we will number the tracked +sensors in DAC1 and DAC2 with Arabic numerals and +in alphabetical order respectively, e.g. ND1, INV1, NDa, +INVa. In general, the difference between the local pres- +surized environments of NDs and INVs becomes more +significant as we increase the DAC pressure. +Using data from DAC1 as examples, we present in +Fig. 1(c) and (d) how the raw ODMR spectra of ND1 +and INV1 change with the DAC1 pressure respectively. +Their spectral changes can be compared in terms of the +center D and splitting 2E of the ODMR resonances. At +p0, ND1 and INV1 agree well on D but ND1 has a larger +E than INV1, indicating a larger intrinsic lattice distor- +tion in ND1. +When DAC1 is pressurized to p2, ND1 +shows a greater rightward shift in D while INV1 exhibits +a more noticeable increase in E, and such differences in +their spectral responses become more pronounced at p6. +These reveal that when we press the diamond anvils to- + +5 +(a) +(b) +(c) +(d) +FIG. 2. (a, b) Plots of SD of pressure and average Enet against +average pressure for three (five) NDs and six (six) patches of +INVs in DAC1 (DAC2), where Enet is the net change in E +with respect to the ambient value. The two DACs reveal very +similar data trends. The NDs only show a tiny increase in SD +while the INVs have no observable SD at all, signifying the +pressure homogeneity at the pressure medium interface and +∼10 nm deep in the culet. Moreover, the INVs reveal a much +greater increase in average Enet, implying a more hydrostatic +environment around the NDs and a more anisotropic environ- +ment around the INVs. Here, the markers are joined in a way +to indicate the pressure sequences in the experiments, while +the purple arrows drawn are to emphasize the significantly +different behaviors of NDs and INVs at the highest pressure +point achieved. +(c, d) The differences in average D, aver- +age P, and average Enet between the three NDs and the six +patches of INVs in DAC1 along the pressure sequence. In gen- +eral, the NDs experience much stronger hydrostatic pressure +compared with the INVs. For all subfigures, one of the three +NDs in DAC1 is replaced by another ND for the statistics at +p4 and p6, due to the occasionally weakened fluorescence of +those NDs. +wards each other, ND1 experiences stronger local pres- +sure from a more hydrostatic environment at the pressure +medium interface, while INV1 is subjected to weaker lo- +cal pressure from a more directional stress environment +inside the anvil culet. The stress anisotropy around INV1 +may have produced a symmetry breaking between the +two ground-state sublevel transitions, as seen from the +increasingly unequal contrasts of the two ODMR reso- +nances at p2 and p6 in Fig. 1(d). On the other hand, +both ND1 and INV1 show decreases in D and E at the +reduced pressure point p4, reflecting the expected stress +relaxation when we loosen the diamond anvils. Note that +the decline in ODMR contrasts for ND1 and INV1 is +due to the stress-induced mixing of NV spin states and +the degradation of MW structure, where the latter factor +takes a heavier toll on ND1. Apart from the artifact of +the deteriorated MW structure, all the mentioned main +features in the ODMR responses of the two NV sensor +types can be reproduced in the independent experiment +with DAC2 (see Supplementary Materials). +To go beyond describing the raw spectra, we perform +statistical comparisons of the local environments per- +ceived by NDs and INVs. In Fig. 2(a) and (b), we plot +the standard deviation (SD) of pressure and average Enet +against average pressure for the tracked NDs and the +tracked patches of INVs in DAC1 and DAC2, where Enet +is the measured E offset by the ambient value. It is evi- +dent that the two DACs give rise to very similar results. +First, the NDs only show a tiny increase in the SD of pres- +sure while the patches of INVs have no observable SD at +all. This suggests we have highly homogeneous pressure +at both the medium interface and ∼10 nm deep in the +culet, and the small SD from the NDs also hints at an ex- +cellent hydrostatic condition of the pressure medium (4:1 +methanol:ethanol mixture) within the pressure range un- +der investigation [20, 43]. Second, the average pressure +detected by the NDs becomes increasingly greater than +that detected by the patches of INVs, and the patches of +INVs have a much more remarkable increase in the aver- +age Enet compared with the NDs. These statistics verify +our previous inference that a more hydrostatic environ- +ment exists at the medium interface to produce stronger +local pressure, while a more anisotropic environment ex- +ists inside the anvil culet to give weaker local pressure. +Third, at the reduced pressure points, the NDs and the +patches of INVs show much smaller differences in the av- +erage pressure, SD of pressure, and average Enet. This +implies relaxation of the DAC may tend to “unify” the +pressurized environments at the medium interface and +inside the culet. Note that for DAC2, the data of NDs +at p1 may be affected by the instability of the pressure +medium due to insufficient buffer time between the pres- +surization of the DAC and measurements. +As a more +direct comparison to supplement the above discussions, +we plot in Fig. 2(c) and (d) the quantitative differences +in average D, average pressure P, and average Enet be- +tween the NDs and the patches of INVs in DAC1 along +the pressure sequence. +V. +QUANTITATIVE STRESS TENSOR +ANALYSIS +In this section, we will consider net effects of the DAC +pressure on the stress tensors of the two NV sensor types. +First, we assume the stress tensors induced by the DAC +pressure to be quasi-hydrostatic, i.e. a hydrostatic pres- +sure P plus a first-order correction from a uniaxial stress +of magnitude λP along the DAC axis. This assumption +is intuitive since any non-hydrostaticity in the DAC is +likely to arise from the symmetry breaking due to the +external force applied along the DAC axis. Under this +assumption, the crystal-frame stress tensor of an ND can +be written as +σND = +� +� +P +0 +0 +0 P +0 +0 +0 P +� +� + U T +n +� +� +λP 0 0 +0 +0 0 +0 +0 0 +� +� Un, +(10) + + DAC1NDs +4 +DAC1NVs +(kbar) +- DAC2 NDs +- DAC2 NVs +2 +D +S +0 +12 +(ZHW) +8 +4 +V +0 +20 +40 +60 +80 +0 +Average pressure +(kbar)30 +20_ +一DAC1 +台 +M +20 +V +10. +P +10 +D +VV +>i +V +(kbar) +ND +0 + +(kbar)16 +DAC1 ND1 +( = 0.034 ± 0.005) +DAC2 NDI +(MHz) +12 +(入 = 0.007 ± 0.009) +8 +4 +0 +25 +75 +0 +50 +P (kbar)40 +(kbar) +30 +30 +(kbar) +, expt +0 +20 +0 +15 +30 +

(kbar) +V +6 +10 +0 +15 +25 +0 +10 +15 +20 +30 +

+(kbar)80 +(kbar) +50 +60 +^25 +(kbar) +expt +6 +40 +25 +50 +0 +

+(kbar) +V +6 +20 +0 +10 +20 +30 +40 +0 +50 +60 +

( +(kbar) +Oxx, expt +Oxx, sim +OzZ, sim +OYYIZZ, expt +OYY, sim +OxY, sim7 +substituting Eq. (12) into Eqs. (8) and (9), we can use the +average D and average Enet measured by the six patches +of INVs in each DAC to determine the σXX and σY Y/ZZ +perceived by INVs at each pressure point. The experi- +mentally derived results are presented using markers in +Fig. 3(c) and (d) for DAC1 and DAC2 respectively, where +the loading stress σXX,expt gradually becomes greater +than the lateral stress σY Y/ZZ,expt in both DACs. Quan- +titatively, the ratio of σXX,expt to σY Y/ZZ,expt increases +from 1 at p0 to 1.62 (1.38) at the highest pressure point +for DAC1 (DAC2). This demonstrates the accumulation +of non-hydrostaticity in the diamond anvil culet due to +the gradual dominance of the stress component along the +DAC axis, in accord with our previous claims. To show- +case the validity of our results, we further illustrate in +the insets of Fig. 3(c) and (d) that the average of our +derived diagonal stress components, < σII,expt >, indeed +gives the average pressure measured by the six patches +of INVs. +To cross-check the above INV crystal-frame stress ten- +sors derived from our experimental data, we perform +simulations using a finite-element analysis software. We +employ the solid mechanics module in the software to +study the steady-state problem at each pressure point for +our DACs. Fig. 4(a) shows the 2D axisymmetric model +used in our simulations, which consists of the two (100)- +oriented diamond anvils and the beryllium-copper gasket +(the bottom anvil is the implanted one). The anvil ge- +ometry follows the standard design of a Type IIas dia- +mond anvil from the manufacturer Megabar-Tech, with +the culet diameter being 800 µm. The X-axis of the INV +crystal frame is defined along the DAC axis as usual, +and we impose the following boundary conditions in the +simulation at each pressure point for our DACs (refer to +Fig. 4(a) for the naming of boundaries): +(1) Boundary loads: the base of the un-implanted anvil +(boundary BC) is loaded with our externally applied +force, while the pressure medium interfaces (boundaries +AU, UP, and PO) are loaded with the average pressure +measured by our NDs. +(2) Displacement constraints: +the base of the im- +planted anvil (boundary MN) is fixed, while the bound- +aries GH, HI, and IJ are prescribed to have no radial +displacements with respect to the DAC axis, such that +the symmetry axis in our model will not be shifted. +(3) Contact surfaces: +the static Coulomb friction +model is applied to simulate the contact between the +anvils and the gasket, where boundaries UT, TS, PQ, +and QR have a friction constant of 0.02 while boundaries +SF and RJ have a friction constant of 0.2 (constants taken +from Ref. [19]). +To clearly portray the stress features in our DACs, +the simulation results at the highest pressure point p6 of +DAC1 are summarized in Fig. 4(b) as examples, where +the color maps visualize the spatial distributions of the +loading stress σXX, the lateral stresses σY Y , σZZ, and +one of the shear stresses σXY . Note that the distributions +of σXX, σY Y , and σZZ are symmetric in the two anvils +FIG. 4. (a) The 2D axisymmetric model used in our compu- +tational simulations, with the blue regions representing the +(100)-oriented diamond anvils (the bottom one is the im- +planted anvil) and the orange-brown region representing the +beryllium-copper gasket. A zoom-in on the beveled diamond +culets is provided, and the orientation of INV crystal frame +with respect to the DAC model is specified. All the bound- +aries are properly named for discussing the boundary con- +ditions in our simulations. (b) Colour maps visualizing the +spatial distributions of the loading stress σXX, the lateral +stresses σY Y , σZZ, and the shear stress σXY from the simula- +tion at the highest pressure point p6 of DAC1. Compressive +stresses are taken to be positive. The distributions of σXX, +σY Y , and σZZ are symmetric in the two anvils but it is not +the case for σXY . Furthermore, σXY is considerably smaller +than the diagonal stress components. +but it is not the case for σXY . +This may be due to +the asymmetry of the gasket’s pre-indentation and the +fact that we compress the DAC from above. The shear +stress σXY is notably smaller than the diagonal stress +components, in agreement with the simulation in Ref. +[19]. +We notice, however, that in Ref. +[19], they can +reconstruct finite shear stresses from experimental data. +For each simulation, we average the simulated stress +tensor components over the spatial region where the INVs +are supposed to be in the DAC, i.e. 10 to 100 µm from the +DAC axis and 10 to 15 nm below the culet surface of the +implanted anvil. These simulated results of INV crystal- +frame stress tensor components are presented using lines +in Fig. 3(c) and (d) for DAC1 and DAC2 respectively, + +(a) E +P +C +D +E +G +H +A +S +A +UP +G +Q +R +0 +K +X +L +N +M +(b) +xx +QYY +80 +60 +60 +40 +NDS +(kbar) +20 +20 +INVs +0 +0 +60 +20 +40 +20 +NDS +(kbar) +0 +20 +INVs +10 +40 +-20 +60 +100 μm +zz +OxY8 +which reveal the same gradual dominance of the loading +stress over the lateral stresses as in our previous results +derived from experimental data. +Moreover, the simu- +lation results substantiate our assumptions in Eq. (12): +First, one of the shear stresses σXY,sim is negligible at all +the pressure points under investigation; second, although +the lateral stresses σY Y,sim and σZZ,sim exhibit different +spatial dependencies (Fig. 4(b) as an example), they have +very close averages over the INV region as shown by the +overlapping dashed and dotted lines in Fig. 3(c) and (d). +To conclude, the computational simulations uphold our +speculation that the non-hydrostaticity of the stress en- +vironment inside the anvil culet mainly emanates from +the dominant uniaxial stress along the DAC axis. +Results from Sections IV and V elucidate that NDs +perform better than INVs as hydrostatic pressure gauges. +Given a hydrostatic pressure medium in a DAC, NDs at +the medium interface efficiently receive the hydrostatic +pressure, while INVs inside the anvil culet are heavily +affected by the breaking of spatial symmetry due to the +externally applied force. Moreover, our results substan- +tiate that NDs have a longer working range for pres- +sure detection compared with INVs. +Throughout the +hydrostatic pressure range of the pressure medium, NDs +present tiny changes in Enet and contrast ratio of ODMR +resonances. However, as the DAC pressure is increased, +INVs show gradual suppression of one of the ODMR res- +onances due to non-hydrostaticity in the local stress en- +vironment, which hinders accurate pressure determina- +tion from the center of resonances. One way-out to ex- +tend the working range of INVs is to apply a magnetic +field B of at least 50 Gauss along [100] of the diamond +anvil, such that in the NV ground-state Hamiltonian, the +magnetic field term γ∥B∥ ≥ O(102) MHz is significantly +greater than the M k +x and M k +y terms which are related to +E ≤ O(101) MHz of INVs, where γ = 2.8 MHz/Gauss +is the gyromagnetic ratio for electrons. +Then, the lo- +cal non-hydrostaticity would bring negligible effects to +the ODMR spectrum of INVs and we would have two +ODMR resonances of similar contrasts. We can solve D +and hence P from this spectrum using equations from +the section of three-cone method in Supplementary Ma- +terials, with the known magnetic field projections on the +NV orientations. Thus, a well-controlled magnetic field is +necessary for INVs to work fine in the entire hydrostatic +pressure range of the medium, while NDs do not require +extra apparatus for robust pressure sensing. +VI. +ZPL AS AN ALTERNATIVE PRESSURE +GAUGE +Researchers have extensively studied the responses of +NV ground-state spin sublevels to external perturbations, +and developed the well-known ODMR spectroscopy for +quantum information technologies. In fact, not only spin +sublevels but also electronic orbitals of the NV center +would be adjusted by perturbations. +Here, we aim at +b +(a) +(b) +(c) +FIG. 5. (a) A sample NV PL spectrum obtained by the mea- +surement method stated in Section III. In general, there is a +narrow ZPL followed by broad phonon sidebands. (b) The +ZPL fitting procedures in our data analysis, which concen- +trates on the portion of the PL spectrum from 615 to 658 +nm. First, this piece of data is linearly interpolated and fitted +with a quadratic polynomial (Upper panel). Then, the fitted +polynomial is subtracted from the interpolated data and a +Lorentzian peak fitting is performed (Lower panel). For both +(a) and (b), the raw data from INV1 in DAC1 at p2 is used as +an example. (c) Linear fittings of ZPL energy versus ODMR- +calibrated local pressure for the INVs and NDs in DAC1 and +DAC2. The data points for INVs are averages from the six +patches in the corresponding DAC. The five fittings reveal +similar pressure dependencies of the ZPL, from 0.56 to 0.59 +meV/kbar with errors on the order of 0.01 meV/kbar. Our +experimental results agree well with the literature. +quantifying the pressure-induced change in the energy +spacing between electronic ground and excited states of +the NV center, via measuring the pressure dependence of +ZPL in the PL spectra of INVs and NDs. We expect the +two types of NV sensors would concur on the dependence +as long as their respective local pressures are calibrated +by ODMR spectroscopy. +By manipulating the NV spin state as described in +Section III, we measure the NV PL spectra from the six +(six) patches of INVs and two (one) of the NDs in DAC1 +(DAC2) along the pressure sequence. A sample NV PL +spectrum is shown in Fig. 5(a). In general, the PL spec- +trum of an NV ensemble consists of broad phonon side- +bands trailing behind a narrow ZPL which undergoes a +weak blue shift under pressure. Without delving into the +complicated fitting of the entire PL spectrum, we con- +sistently extract the ZPL position from each measured + +629 N +1970 +PL +(meV) +1960 +DAC1 INVs +ler +DAC2 INVs +1950 +DAC1 ND2 +ZPL +637 +DAC1 ND3 +DAC2 ND +639 +1940 +0 +10 +30 +40 +20 +50 +ODMR pressure +(kbar)0.8 +Interpolated data +its) +Polynomial fitting +0.0 +Intensity +Subtracted data +Lorentzian fitting +0.2 +0.0 +620 +640 +660 +Wavelength (nm)units) +DAC1 +INV1 (p2) +0.8 +qu +e +Intensity +0.4 +0.0 +600 +800 +1000 +Wavelength (nm)9 +spectrum in the following steps: (i) focus on the data +between 615 and 658 nm which fully captures the ZPL +evolution in our experimental pressure range, (ii) linearly +interpolate this portion of data and perform a quadratic +polynomial fitting (see the upper panel in Fig. 5(b)), +(iii) subtract the fitted polynomial from the interpolated +data and perform a Lorentzian peak fitting (see the lower +panel in Fig. 5(b)). Note that Fig. 5(b) depicts the ZPL +fitting procedures using the raw data in Fig. 5(a) as an +example. +With the ZPL positions distilled out, we linearly fit +the curves of ZPL energy versus local pressure for our +targeted sensors in DAC1 and DAC2 (see Fig. 5(c)), +where the ZPL energy is converted from the extracted +ZPL wavelength and the local pressure is determined by +ODMR method. The data points for INVs are averages +from the six patches in the corresponding DAC. The five +fittings in Fig. 5(c) reveal similar slopes ranging from +0.56 to 0.59 meV/kbar, with errors on the order of 0.01 +meV/kbar. This fulfills our earlier expectation that well- +calibrated NV sensors should agree on the pressure de- +pendence of the ZPL, and our results can match with the +previously reported values [15, 33, 34]. Moreover, it can +be seen that our measured ZPL wavelengths are around +638 nm under ambient conditions, slightly off from the +literature value of 637 nm. +This may be due to some +finite intrinsic stresses in the implanted diamond anvil. +Note that each fitting in Fig. 5(c) only considers the pres- +sure points where the corresponding sensor has sufficient +ODMR contrasts to yield satisfactory NV PL spectra. +Besides, the reduced pressure points are purposely ex- +cluded from the fittings in Fig. 5(c) for the same reason +as before that relaxation of a DAC is not simply the re- +verse process of pressurization. +Our experimentally determined blue shift of the ZPL +indicates a repulsion between NV electronic ground and +excited states caused by pressure. In fact, it is of practical +importance to confirm the slope of ZPL energy against +pressure. +By doing so, PL spectroscopy can be devel- +oped into an alternative to the ODMR method for pres- +sure sensing with the NV center, and we can choose to +utilize the spin or orbital degree of freedom in the NV en- +ergy structure for different experimental situations. PL +spectroscopy is particularly useful if one does not want +to introduce electrical feedthroughs into the DAC. An +all-optical pressure sensing protocol is possible with PL +spectroscopy, where one can obtain the NV PL spectrum +by subtracting the spectrum measured near the NV sen- +sor from the spectrum measured precisely at the location +of the NV sensor, under an assumption of spatially uni- +form background PL signals. This assumption is valid if +no components in the DAC other than NV sensors would +emit red fluorescence under green laser excitation. With +ODMR and PL spectroscopies, the NV center can be a +resilient pressure sensor that caters to different experi- +mental conditions. +π +I +𝜏 +𝜏 +𝜏 +-𝜏/ +Before +After +(a) +(b) +(c) +(d) +(e) +FIG. 6. (a, b) Pulse sequences for measuring the relaxation +curves of the bright |0⟩ state and the dark |+⟩ state respec- +tively. The 520-nm laser is for initialization to the |0⟩ state +and readout of the final state, while the MW π pulse is to +flip |0⟩ to |+⟩. Furthermore, τ is the free precession time to +be varied, and the π pulse is half the Rabi period of driving +the right-hand ODMR resonance f+. (c) Exponential decay +fitting of the net relaxation curve obtained by subtracting the +|+⟩ curve from the |0⟩ curve. The fitted decay time is taken +as the experimental spin-lattice relaxation time T1. The data +from INVI in DAC3 at p1 is used as an example here. (d) T1 +time versus ODMR-calibrated local pressure for three patches +of INVs in DAC3, with the ambient-condition data from a +patch of INVs close to the six tracked patches as a reference. +No significant changes in T1 time are observed. (e) The Rabi +oscillations at different pressure points before and after tuning +the MW power fed into the DAC. (Upper panel) If the input +MW power is kept the same in the experiment, the Rabi pe- +riod changes with the DAC pressure, which is expected due to +the varying MW transmission efficiency through the Omega- +shaped antenna when the DAC is pressurized. (Lower panel) +After appropriately tuning the input MW power, the Rabi +period can be fixed at 582 ns such that the INVs receive the +same MW power at all pressure points. Here, the data from +INVI in DAC3 is used as an example, and the Rabi oscillation +amplitudes are normalized for easier comparison of periods. +VII. +PULSED MEASUREMENTS WITH A +HYDROSTATIC PRESSURE MEDIUM +Pulsed measurements are key to enhancing the sensi- +tivity and realizing complex sensing schemes [8]. In real +life, spin decoherence creates difficulties in implementing +pulse sequences. The decoherence occurs via two chan- +nels: (i) the relaxation in z-direction of the Bloch sphere +due to electron-phonon coupling between NV centers and +the lattice; (ii) the dephasing in x-y plane of the Bloch + +Ambient +12 +DAC3 INVI +DAC3NVI +DAC3 INV +8 +S +T +4 +T +0 +0 +20 +40 +60 +80 +Pressure +(kbar)Fluorescence (arb. units) +0 +1 +pl +-p2 +p3 +p4 +0 +500 +1000 +Time (ns)<+I +520 nm +MW10> +520 nm +MWDAC3 INVa (p,) +units) +Aexp(-t / T,) fitting +2000 +(arb. +Fluorescence +1000 +0 +0 +2 +4 +6 +8 +10 +(sw) 210 +sphere due to spin-spin interactions. These two decoher- +ence channels are characterized by the T1 and T2 times, +respectively. In recent years, some promising pulsed sens- +ing protocols have been demonstrated in either ambient +or pressurized conditions [19, 23, 46–49]. +Nonetheless, +little attention has been paid to the hydrostaticity of the +pressure medium and the characterization of NV decoher- +ence times using a hydrostatic medium. These concerns +are important for high-fidelity NV sensing and NV-based +quantum computing to be robustly performed in extreme +conditions. To address these concerns, we examine the +INV T1 time over the course of pressurizing our DACs, +where we check the hydrostaticity of the pressure medium +with great caution. +Knowing the results in Sections IV and V, one may +wonder if the local stress anisotropy in the diamond +anvil culet would induce peculiar crystal deformations +and modify the system’s electron-phonon coupling which +in turn affects the T1 time of the INVs. Thus, we would +focus on the INVs rather than the NDs in this subsec- +tion. Refer to Section II, if the shear stresses are negli- +gible (suggested by the simulations in Section V), all the +four NV orientations would have the same eigenfrequen- +cies, f0 = 0 and f±, and the same eigenstates, |ms = 0⟩ +and |±⟩, where |±⟩ are superpositions of |ms = +1⟩ and +|ms = −1⟩. In this case, we would like to study the T1 +time of the two-level system spanned by |0⟩ and |+⟩ with +transition frequency f+, which can be obtained by fitting +the right-hand ODMR peak at zero magnetic field. With +the pulse sequences depicted in Fig. 6(a) and (b), we can +measure the relaxation curves of the bright |0⟩ state and +the dark |+⟩ state respectively. Note that the π pulse in +Fig. 6(b) is half the Rabi period of driving the resonance +f+. To extract the T1 time, we subtract the relaxation +curve of |+⟩ from that of |0⟩, and fit the resulting curve +with an exponential decay A exp(−τ/T1) (see Fig. 6(c) +as an example), where the amplitude A is limited by +the Rabi contrast. +Here, we focus on the right-hand +ODMR resonance because the left-hand ODMR contrast +of INVs is suppressed when the DAC pressure is increased +as shown in Fig. 1(d). +We perform the T1 measurement using a new DAC +(named as DAC3) which has the same cell configura- +tions as DAC2 but with 99.5% glycerol as the pressure +medium. Compared with 4:1 methanol:ethanol mixture, +glycerol is a more common medium since it is not a strong +solvent, and it is chemically inert. The ascending pres- +sure sequence for DAC3 is from the ambient pressure p0 +to p4 without any reduced pressure points. Through in- +specting the SD of pressure among NDs, we find that our +prepared glycerol in DAC3 has a critical pressure Pc at +around 80 kbar (see Supplementary Materials), which is +in good agreement with [17]. Our medium is therefore +perfectly hydrostatic before p4 and quasi-hydrostatic at +p4. +By tracking two NDs and six patches of INVs in +DAC3, we can reproduce the results in Figs. 1 and 2 +(see Supplementary Materials), proving that our previous +claims are independent of the choice of pressure medium +as long as it is in the hydrostatic regime. For DAC3, we +will number the tracked sensors using Roman numerals, +e.g. NDI, INVI. +We have monitored the T1 time for three patches of +INVs in DAC3 from p1 to p4, with the ambient-condition +data from a patch of INVs close to the six tracked patches +as a reference (INVs in the implanted region exhibit con- +sistent ODMR properties from previous experience with +DAC1 and DAC2). The measurement results are shown +in Fig. 6(d), where no significant changes in the T1 times +are observed. +Note that the efficiency of MW trans- +mission through the Omega-shaped antenna is inevitably +changed when we increase the DAC pressure. To ensure +a constant MW power received by the INVs at differ- +ent pressure points for fair comparison of the T1 times +in Fig. 6(d), we have fixed the Rabi period to be 582 ns +for all patches of INVs at all pressure points by tuning +the MW power fed into the DAC. We assume negligi- +ble detuning in the Rabi oscillations here. As examples, +Fig. 6(e) shows the Rabi oscillations of a patch of INVs in +DAC3 at different pressure points before and after tuning +the input MW power. +Our measurement results indicate that possible mod- +ifications to the electron-phonon coupling by the local +stress anisotropy in the anvil culet are tiny and within +our experimental errors, given the medium is in a good +hydrostatic condition. This demonstrates the stability of +NV properties under extreme conditions and once again +proves the robustness of NV sensing. A natural exten- +sion of our work is to monitor the T2 time of INVs under +pressure by implementing the Hahn-echo pulse sequence, +but to do so, a well-controlled magnetic field is a req- +uisite for aligning a magnetic field along one of the NV +orientations. Our preliminary results (data not shown) +under zero magnetic field in DAC3 seem to indicate no +observable changes in the T2 time up to 80 kbar. +VIII. +PROPOSAL OF DIAMOND +NANOPILLARS IN A DAC +In addition to using NDs and applying a bias mag- +netic field, we propose here the third method to miti- +gate the adverse effects of the non-hydrostaticity in the +pressurized environment and to extend the working pres- +sure of NV centers as quantum sensors. +The fabrica- +tion and characterization of diamond nanopillars have +been discussed in the literature [50–53], and NV sensing +using nanopillars has been demonstrated under ambient +conditions [50]. We therefore suggest the integration of +nanopillars into a DAC so that the INVs embedded in the +nanopillars can perform reliable quantum sensing in a hy- +drostatic stress environment. We conduct finite-element +simulations to support our proposal, where we modify +the 2D axisymmetric model in Fig. 4(a) by adding one +nanopillar to the center of the implanted anvil culet (the +bottom anvil in Fig. 4(a)). For the 2D model, we use +the dimensions of DAC1 and try two different nanopillar + +11 +r +h +r = 70 nm +h = 150 nm +r +h +0 +12 +10 +8 +6 +4 +2 +ENV1 (MHz) +r = 100 nm +h = 1500 nm +2r +h +(a) +(b) +(c) +FIG. 7. (a) The modified 2D axisymmetric DAC model with a +diamond nanopillar at the center of the implanted anvil culet. +The zoom-in is a schematic diagram showing a nanopillar, +where r and h are the radius and the height, respectively. +(b, c) Spatial maps of ENV1 derived from the simulations +using the nanopillar geometry of r, h = 100, 1500 nm and the +geometry of r, h = 70, 150 nm, respectively. The two maps +share the same color scale. ENV1 is found to be negligible +within the nanopillar but much larger inside the bulk of the +anvil. +geometries. One geometry has a radius of 100 nm and +a height of 1500 nm, as inspired by the nanopillars in +Ref.[50]. The other geometry has a radius of 70 nm and +a height of 150 nm, which is similar to the size of our +140-nm NDs. We simulate the stress distribution under +the boundary conditions stated in Section V, where the +boundary loads follow the data at the highest pressure +point p6 of DAC1. +In Fig. 7(b) and (c), we map the +spatial distribution of ENV1 in the implanted anvil for +the two geometries, revealing that the indicator of non- +hydrostaticity, E, is close to zero within the nanopillar +but it is much larger inside the bulk of the anvil. This +hints at a good hydrostatic condition in the nanopillar, +such that the mixing of NV ground-state spin sublevels +can be minimized and the INVs inside the nanopillar can +perform robust quantum sensing based on ODMR spec- +troscopy. +IX. +DISCUSSION AND SUMMARY +Integrating different NV sensor types has some use- +ful sensing applications. For instance, one may integrate +INVs and NDs to study a liquid-solid phase boundary, +where the liquid and solid properties can be sensed by +immersed NDs and shallow INVs respectively. +Under- +standing liquid-solid interfaces at a microscopic scale is a +prevailing challenge in quantum chemistry [54], and NV +sensing may provide new opportunities to the field. +In summary, this work has revealed a noticeable dif- +ference in the local stress environments encountered by +INVs and NDs in the same DAC. Note that NV is just +a platform for reconstructing the stress components, and +our results should be generalized to the stress discrep- +ancy between different parts in a high-pressure device: +more hydrostatic at the pressure medium interface and +more anisotropic inside the force-transmitting solid ele- +ments, given a hydrostatic pressure medium below Pc. +Moreover, our experiments and simulations demonstrate +the sensitivity of NV centers to different stress profiles. +Although INVs can be a versatile non-invasive tool in +diamond-based pressure devices, NDs appear to be a +better option for gauging hydrostatic pressure and have +a longer working range for pressure detection with zero +magnetic field. In fact, any type of NV sensor can be a +legitimate pressure gauge as long as it is well-calibrated, +and our work is exactly dedicated to characterizing the +behaviors of different NV sensors in a confined pressure +device. +We want to emphasize that the choice of NV +sensors heavily depends on the experimental purpose so +that their unique advantages can be fully utilized. Fur- +thermore, this work provides insights on different aspects +of the NV energy structure. +We confirm a pressure- +induced repulsion between NV electronic ground and ex- +cited states by measuring the blue shift of ZPL in the NV +PL spectrum. +We also show that the electron-phonon +coupling in the NV system would not be significantly +modified by local stress anisotropy, as seen from the mea- +sured stability of the INV T1 time. With a deeper under- +standing of the pressure-tuned NV system, more different +sensing applications of the NV center are expected in the +future. +Our +work +also +addresses +the +tolerance +to +non- +hydrostaticity when NV centers are applied as versatile +sensors in pressurized environments, which is a key ques- +tion from the NV community. Non-hydrostaticity hin- +ders NV sensing in the following ways: (i) the ground- +state spin sublevels are mixed in the energy eigenstates +under a non-hydrostatic stress field (see Eq. (1)), lower- +ing the efficiency of ODMR spectroscopy; (ii) magnetic +field sensing using NV centers would be inaccurate if the +E term in Eq. (9) is comparable to the magnetic field. +Our empirical results show that when the indicator of +non-hydrostaticity, E, reaches O(101) MHz, one of the +resonances in the ODMR spectrum of NV centers at zero +bias field is heavily suppressed, and the sensing accuracy +is thus decreased. This would impose restrictions on the +maximum working pressure of NV sensing. +There are +three solutions when we encounter this threshold of E ∼ +O(101) MHz. A straightforward solution would be to use +NDs in a pressure medium with a sufficiently high Pc to +ensure a hydrostatic environment around the NV centers. +Another solution would be to fabricate a diamond anvil +with some nanopillars on the anvil culet. The NV centers +encompassed in the nanopillars would perceive a more + +X +Y +Z12 +10 +8 +6 +4 +2112 +hydrostatic environment similar to NDs drop-casted on +the culet, as demonstrated in the finite-element simu- +lations in Supplementary Materials. Besides, the crys- +tal orientation of the nanopillers is definite in the lab- +oratory frame, which is an advantage over NDs. +The +third solution would be to apply a bias magnetic field +of γ∥B∥ ≫ E such that the non-hydrostaticity is not +the dominant term in the Hamiltonian, which has been +demonstrated in the literature [18, 19]. +ACKNOWLEDGMENTS +We +thank +P. +T. +Fong +for +the +fruitful +discus- +sion. +K.O.H acknowledges financial support from +the Hong Kong PhD Fellowship Scheme. +S.K.G. ac- +knowledges financial support from Hong Kong RGC +(GRF/14300418, GRF/14301020, and A-CUHK402/19). +S.Y. acknowledges financial support from Hong Kong +RGC (GRF/14304419). +Kin On Ho, Man Yin Leung, and Wenyan Wang con- +tributed equally to this work. +[1] A. P. Drozdov, M. I. Eremets, I. A. Troyan, V. Kseno- +fontov, and S. I. Shylin. Conventional superconductiv- +ity at 203 kelvin at high pressures in the sulfur hydride +system. Nature, 525(7567):73–76, Sep 2015. 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PMID: 22277079. + diff --git a/rtE5T4oBgHgl3EQfKA5U/content/tmp_files/load_file.txt b/rtE5T4oBgHgl3EQfKA5U/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9f194b46215be568cebe0b403ee8c7fd26caef40 --- /dev/null +++ b/rtE5T4oBgHgl3EQfKA5U/content/tmp_files/load_file.txt @@ -0,0 +1,1443 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf,len=1442 +page_content='Spectroscopy Study on NV Sensors in Diamond-based High-pressure Devices Kin On Ho,1, ∗ Man Yin Leung,2, ∗ Wenyan Wang,1, ∗ Jianyu Xie,1 King Yau Yip,1 Jiahao Wu,2 Swee K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' Goh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' 3 Andrej Denisenko,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content='4 J¨org Wrachtrup,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content='4 and Sen Yang1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' † 1Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' The Chinese University of Hong Kong,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' Shatin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' New Territories,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' Hong Kong,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' China 2Department of Physics and the IAS Centre for Quantum Technologies,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' The Hong Kong University of Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' Clear Water Bay,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' Kowloon,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' Hong Kong,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' China 3Shenzhen Research Institute,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' The Chinese University of Hong Kong,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' Shatin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' New Territories,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' Hong Kong,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' China 43rd Institute of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' University of Stuttgart and Institute for Quantum Science and Technology (IQST),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' Pfaffenwaldring 57,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' D-70569,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' Stuttgart,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' Germany (Dated: January 16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' 2023) High-pressure experiments are crucial in modern interdisciplinary research fields such as engineer- ing quantum materials,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' yet local probing techniques remain restricted due to the tight confinement of the pressure chamber in certain pressure devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' Recently, the negatively charged nitrogen-vacancy (NV) center has emerged as a robust and versatile quantum sensor in pressurized environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' There are two popular ways to implement NV sensing in a diamond anvil cell (DAC), which is a conventional workhorse in the high-pressure community: create implanted NV centers (INVs) at the diamond anvil tip or immerse NV-enriched nano-diamonds (NDs) in the pressure medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' Nonetheless, there are limited studies on comparing the local stress environments experienced by these sensor types as well as their performances as pressure gauges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' In this work, by probing the NV energy levels with the optically detected magnetic resonance (ODMR) method, we experimentally reveal a dramatic difference in the partially reconstructed stress tensors of INVs and NDs incorpo- rated in the same DAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' Our measurement results agree with computational simulations, concluding that INVs perceive a more non-hydrostatic environment dominated by a uniaxial stress along the DAC axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' This provides insights on the suitable choice of NV sensors for specific purposes and the stress distribution in a DAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' We further propose some possible methods, such as using NDs and diamond nanopillars, to extend the maximum working pressure of quantum sensing based on ODMR spectroscopy, since the maximum working pressure could be restricted by non-hydrostaticity of the pressure environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' Moreover, we explore more sensing applications of the NV center by study- ing how pressure modifies different aspects of the NV system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' We perform a photoluminescence (PL) study using both INVs and NDs to determine the pressure dependence of the zero-phonon line (ZPL), which helps developing an all-optical pressure sensing protocol with the NV center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' We also characterize the spin-lattice relaxation (T1) time of INVs under pressure to lay a foundation for robust pulsed measurements with NV centers in pressurized environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' INTRODUCTION Pressure is an important thermodynamic parameter for engineering quantum materials because it allows one to tune material properties without altering the chemical composition, and some long-searched-for quantum phases are expected to emerge under ultra-high pressure, such as room-temperature superconductivity [1–4] and metallic hydrogen [5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' High-pressure experiments are, however, non-trivial to perform since one needs robust pressure devices and compatible measurement techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' One reliable pressure device is the diamond anvil cell (DAC), which has been widely used in the high-pressure community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' The pressure is achieved by mechanically pressing two opposing diamond anvils towards a tightly confined pressure chamber in the middle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' The pressure medium filling up the chamber remains hydrostatic be- low its critical pressure Pc, and it undergoes solidification or glass transition at Pc where pressure inhomogeneity ∗ These authors contributed equally to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' † phsyang@ust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content='hk starts building up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' It is crucial to understand whether the medium is hydrostatic during the experiment, since the subsequent data processing and interpretation may be inappropriate if the artifacts from pressure inhomo- geneity are not taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' As to the suitable measurement techniques in high- pressure experiments, quantum sensing with negatively charged nitrogen-vacancy (NV−) centers has emerged as a strong candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' We always denote NV− as NV in the rest of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' The NV center is a color defect in diamond which consists of a substitutional nitrogen atom, an adjacent atomic vacancy, and an extra electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' Its ground state is an electron spin S = 1 system and the spin sublevels are responsive to temperature, stress field, magnetic field, electric field, and the surrounding spin bath, making the NV center a versatile sensor for these physical quantities [7–14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' In practice, we mea- sure the electron spin resonance (ESR) spectrum of the NV center using the optically detected magnetic reso- nance (ODMR) method, which relies on the spin-state- dependent fluorescence rate of the NV center caused by the spin-state-dependent decay route back to the ground state (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' 1(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' In ODMR spectroscopy, a green laser is used for initialization and read-out of the NV spin arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content='05462v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content='mes-hall] 13 Jan 2023 2 state while a microwave (MW) is used for spin-state ma- nipulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' The ODMR spectrum encodes information about the NV energy structure and hence the environ- ment around the NV center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' Due to the superior reso- lution and sensitivity, quantum sensing with NV centers has become a promising experimental technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' It has been demonstrated that NV sensing is highly compatible with DACs, and NV centers have outstand- ing sensing performance even under the demanding con- ditions inside DACs [15–23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' There are mainly two ways to incorporate NV sensors in DACs: (1) create a layer of implanted NV centers (INVs) at a suitable depth inside the diamond anvil tip [18, 19, 22], (2) drop-cast some NV- enriched nano-diamonds (NDs) at the pressure medium interface inside the pressure chamber [16, 17, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' In general, they are employed to study different kinds of materials under pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' INVs are commonly used to probe 2D or 3D materials with flat surfaces, while NDs are often applied to examine materials with irregular sur- faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' Moreover, INVs and NDs have their own advan- tages and drawbacks in sensing applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' INVs pro- vide an easy way to detect vector fields because of the known orientation of the bulk diamond crystal in the lab- oratory frame, yet, the spatial resolution is restricted by the optical diffraction limit and the spatial uniformity of INVs is constrained by imperfections in the implantation procedures;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' on the other hand, NDs present high spatial resolution controlled by the ND size given the NDs are sparsely distributed and NV centers in the NDs are in close proximity to the sample, yet, the crystal orienta- tions of NDs are random and require individual calibra- tion in the laboratory frame and spin decoherence times of NDs are generally shorter than INVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' Some obvious pros and cons of INVs and NDs are long known, but to the best of our knowledge, no studies have directly com- pared the pressurized environments perceived by these two types of NV sensors in a single DAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' This incom- plete understanding of the pressurized environments at different locations in a DAC may hinder the accurate choice of NV sensors for different experimental purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' Another prevailing question from the NV community is the maximum pressure that NV centers can work with as quantum sensors, especially as magnetic field sensors since the probing of local magnetic fields with high spa- tial resolution is crucial for material research and phase transition studies [17–19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' Ultra-high pressure can bring detrimental effects on quantum sensing with NV centers, including the quenching of ODMR contrasts due to the spin-sublevel mixing in a non-hydrostatic environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' To realize magnetic field sensing in pressurized systems, some previous studies have demonstrated the use of a bias magnetic field to overcome the effects of uniaxial stresses [17–19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' Nonetheless, a strong bias field is re- quired for large uniaxial stresses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' This may impose tech- nical difficulties on the experimental setup, and a strong bias field may undesirably change the properties of the material under investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' Thus, it is of interest to explore other complementary solutions for extending the working pressure of NV sensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' In this work, we first incorporate both INVs and NDs in the same DAC and analyze the difference in effec- tive pressure transmissions from the hydrostatic pressure medium to these two types of sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' We partially re- construct the local stress tensors perceived by INVs and NDs using information from the respective ODMR spec- tra, and we also perform finite-element simulations to cross-check our experimental findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' These analyses serve to calibrate the local pressurized environments of the two sensor types, to compare their performances as hydrostatic pressure gauges, and to determine their op- timal working ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' By comparing the pressure condi- tions of the two sensor types, we demonstrate how non- hydrostaticity restricts the maximum working pressure of NV sensing, and we further propose some possible so- lutions to conquer the non-hydrostaticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' Besides, thor- oughly characterizing the stress responses of NV sensors may pave the way for simultaneous detection of multiple physical parameters via ODMR spectroscopy, like pres- sure and temperature or pressure and magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' Next, we employ our ODMR-calibrated NV sensors to investigate from different perspectives the pressure-tuned energy structure of the NV center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' We measure the pho- toluminescence (PL) spectra of both INVs and NDs to study the pressure dependence of the zero-phonon line (ZPL), which represents the energy spacing between elec- tronic orbitals of the NV center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' We also measure the spin-lattice relaxation (T1) time of INVs against pres- sure to probe the electron-phonon coupling in the solid- state defect system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' Combining various spectroscopic techniques ranging from continuous-wave (cw) to pulsed measurements and from ESR to PL measurements, we hereby provide a multi-dimensional understanding of the NV quantum system under high pressure, which helps fostering more accurate and distinct applications of NV sensing in extreme conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' Such applications include an all-optical pressure sensing protocol based on PL spec- troscopy and robust implementation of pulse sequences at high pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' THEORETICAL BACKGROUND In a single crystalline diamond with an ensemble of NV centers, there are four possible spatial orientations for the NV centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' We thus have five relevant reference frames: the crystal frame (X, Y, Z) and the principal axis frames for the four NV orientations (x, y, z)k, k ∈ {NV1, NV2, NV3, NV4}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' The four NV frames can be related by simple rotation transformations due to the symmetry of the diamond crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' In this work, we follow Barfuss et al.’s conventions of the five frames and the coordinate transformations between them [24], and we always take compressive stresses to be positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' The NV center is a robust stress sensor due to the spin-stress coupling effect [11, 14, 19–21, 24–37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' Under a stress field affecting the spin-spin interaction,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' the ground- 3 state Hamiltonian for each NV orientation in its principal axis frame can be written as [24–26] Hk = (D0 + M k z )S2 z + M k x(S2 y − S2 x) + M k y {Sx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' Sy} + N k x{Sx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' Sz} + N k y {Sy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' Sz},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' (1) where S is the spin-1 operator,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' D0 = 2870 MHz in am- bient conditions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' and in the hybrid representation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' the NV-frame quantities M k x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content='y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content='z and N k x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content='y can be expressed in terms of the components σIJ of the crystal-frame stress tensor σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' For NV1 along [111] crystal direction, M NV1 z = a1(σXX + σY Y + σZZ) + 2a2(σY Z + σXZ + σXY ), (2) M NV1 x = b(2σZZ − σXX − σY Y ) + c(2σXY − σY Z − σXZ), (3) M NV1 y = √ 3b(σXX − σY Y ) + √ 3c(σY Z − σXZ), (4) N NV1 x = d(2σZZ − σXX − σY Y ) + e(2σXY − σY Z − σXZ), (5) N NV1 y = √ 3d(σXX − σY Y ) + √ 3e(σY Z − σXZ), (6) where a1, a2, b, c, d, and e are the spin-stress coupling constants in the hybrid representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' To obtain the above expressions for the other three NV orientations, we need to replace σIJ by (Kl·σ·(Kl)T)IJ in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' (2) to (6), where Kl are the coordinate transformations from NV1 to l ∈ {NV2, NV3, NV4} as defined in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' The re- sulting expressions for NV2-4 are different from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' (2) to (6) only by sign flips in some of the off-diagonal tensor components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' See Supplementary Materials for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' Experiments have found that a1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content='486±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content='0002, a2 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content='37 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content='002, b = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content='147 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content='0002, c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content='342 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content='0007 MHz/kbar [19, 25], agreeing well with the theoretical values from a density functional theory (DFT) study [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' This DFT study also reports d = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content='012(1) and e = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content='066(1) MHz/kbar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' Since d and e are an order of magnitude smaller than the rest of the coupling con- stants, we can neglect the N k x and N k y terms in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' (1) for our first-order discussion here, and the three eigenvalues of the Hamiltonian Hk can thus be analytically solved as follows, f k 0 = 0, f k ± = D0 + M k z ± � (M kx)2 + (M ky )2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' (7) Hence, f k ± are the two resonance frequencies detectable by ODMR spectroscopy, with their center and splitting being Dk and 2Ek respectively (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' 1(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' In the regime of small shear stresses, the four NV ori- entations have close f+’s and close f−’s, leading to two overall resonances in the ODMR spectrum of the whole NV ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' We further assume equal population for the four NV orientations in the diamond crystal, such that the two overall ODMR resonances should be aver- ages of f k + and f k − over k ∈ {NV1, NV2, NV3, NV4}, with their center D and splitting 2E written respectively k k FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' (a) A simplified energy level diagram of the NV cen- ter showing the spin-state-dependent transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' (b) An il- lustration of the DAC configuration in our experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' The MW antenna is fabricated on the implanted diamond anvil culet, while some 140-nm NDs are drop-casted on the other un-implanted anvil culet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' (c, d) ODMR responses of a 140-nm ND (labeled as ND1) and a patch of INVs (labeled as INV1) to the change in pressure of DAC1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' The decrease in their ODMR contrasts is due to the stress-induced mixing of NV spin states and the degradation of the MW structure, where the latter factor takes a heavier toll on ND1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' as D = D0 + 1 4 � k M k z , (8) E = 1 4 � k � (M kx)2 + (M ky )2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' (9) These expressions reveal that D scales with pressure, while E results from the imbalance between uniaxial stresses along the three orthogonal directions and the presence of shear stresses, or in other words E is an indi- cator of hydrostaticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' When we compress the diamond crystal, both D and E will increase in general, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' the ODMR resonances will shift to the right and split further apart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' With Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' (8) and (9) in hand, we can employ ODMR spectroscopy to partially reconstruct the crystal-frame stress tensor σ perceived by the NV ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' This the- ory section is applicable for both INVs and NDs, and to have meaningful interpretations of the reconstructed crystal-frame stress tensors, we must also understand how the INV and ND crystal frames are related to the laboratory frame, which we will discuss in Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' (d)(arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' units) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content='0 DAC1 ND1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content='9 Fluorescence d 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content='8 2850 2925 3000 Frequency (MHz)DAC1INV1 (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content='9 Fluorescence 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content='8 2850 2925 3000 Frequency (MHz)(a) (b) Excited states 0 Metastable states D Ground (c) states (d)(c)4 III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' EXPERIMENTAL SETUP Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' 1(b) illustrates our customized DAC design where both INVs and NDs are incorporated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' We utilize (100)- oriented diamond anvils, and the layer of INVs is located at the culet of one of the anvils.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' This implanted anvil is prepared by 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content='8 keV 15N ion implantation at a dose of ∼1012 N/cm2 and subsequent annealing at 950oC in a high vacuum (P < 10−6 mbar) for 2 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' The re- sulting implantation area has a diameter of 200 µm and is at a depth of ∼10 nm below the culet surface that has a surface roughness of ∼1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content='5 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' On the other hand, some 140-nm NDs with nitrogen concentration of 3 ppm are sparsely drop-casted on the culet surface of the other un-implanted diamond anvil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' To perform ODMR spec- troscopy with these two types of NV sensors, a 150- µm-radius Omega-shaped gold microstructure is fabri- cated on the implanted anvil for MW transmission [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' As to the pressure chamber in our design, a 300-µm- diameter hole is drilled in the middle of a beryllium- copper gasket and the hole is completely filled with a 4:1 methanol:ethanol mixture as the pressure medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' At room temperature, this particular medium remains hy- drostatic up to ∼100 kbar [39–43] which fully covers our experimental pressure range, enabling us to compare the local pressurized environments of INVs and NDs given the medium is in an excellent hydrostatic condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' An- other reason for choosing this medium is that most of the common phase transitions tuned by pressure in con- densed matter physics lie within 100 kbar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' Therefore, it is of technical significance to study the stress distribution in a DAC, which is a popular pressure device in material research, using a medium with the hydrostatic limit up to 100 kbar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' We have prepared two DACs based on the above- described design, where all the cell configurations are the same except for the thickness of the pre-indented gasket (150 µm in one DAC and 200 µm in the other).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' We will denote these two DACs as DAC1 and DAC2 respectively hereafter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' In our experiments with the DACs, a home- built confocal microscope with a 520-nm laser diode and a long-working-distance objective (50X Mitutoyo Plan Apo SL) is used to optically address the NV sensors, and the local pressure is calibrated by ∂D/∂P = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content='49 MHz/kbar [20] and the D value at ambient pressure measured by the corresponding NV sensor (the ambient D values have only tiny deviations from D0 = 2870 MHz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' Since the implanted anvil is (100)-oriented, it is nat- ural to define the INV crystal frame (X, Y, Z) with the X axis along the DAC axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' On the other hand, it is not that trivial to determine how the crystal frames of individual NDs are oriented with respect to the labora- tory frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' We need to first apply a static magnetic field along the DAC axis and measure the ODMR spectrum of the target ND.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' Then by studying the Zeeman splittings in the spectrum, we can obtain the projection angles of the DAC axis onto the four NV orientations [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' The unit direction of the DAC axis in the ND crystal frame can thus be computed by solving an effective problem of the intersection of three cones (see Supplementary Mate- rials for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' The subsequent stress analysis should not depend on exactly how we assign the four angles to the four NV orientations (NV1-4) under our assumption of the equal population for the four orientations, and we will explicitly check that this is the case in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' For PL measurements, we use the 520-nm laser diode to excite NV electrons from the electronic ground state to the phonon band above the electronic excited state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' The NV electrons would decay to the zero-phonon mode of the excited state via emitting infrared (IR) radiation, then to the phonon band of the ground state via emit- ting red PL, and finally to the zero-phonon mode of the ground state via emitting IR radiation [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' The ZPL in the resulting PL spectrum is produced by NV elec- trons that decay from the zero-phonon mode of the ex- cited state directly back to the zero-phonon mode of the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' The PL spectra of INVs and NDs are col- lected using a commercial spectrometer (Princeton In- strument IsoPlane SCT-320) with a 550-nm long-pass fil- ter in front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' To obtain a PL spectrum solely originating from the NV centers in a targeted sensor, we subtract the PL spectrum measured under an applied MW field at one of the ODMR resonance frequencies from the spec- trum without any exerted MW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' This method makes use of the spin-state dependence of the NV fluorescence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' To enhance the data quality, we choose to drive whichever one of the two ODMR resonances with higher contrast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' COMPARISONS OF LOCAL PRESSURIZED ENVIRONMENTS DAC1 (DAC2) is pressurized in an ascending pressure sequence from the ambient pressure p0 up to p6 (p5), ex- cept that p4 (p5) is a reduced pressure point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' Through- out the experiment with DAC1 (DAC2), we have tracked three (five) 140-nm NDs and six (six) 500-nm patches of INVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' Note that our confocal microscope has a lateral resolution of ∼500 nm, and we will number the tracked sensors in DAC1 and DAC2 with Arabic numerals and in alphabetical order respectively, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' ND1, INV1, NDa, INVa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' In general, the difference between the local pres- surized environments of NDs and INVs becomes more significant as we increase the DAC pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' Using data from DAC1 as examples, we present in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' 1(c) and (d) how the raw ODMR spectra of ND1 and INV1 change with the DAC1 pressure respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' Their spectral changes can be compared in terms of the center D and splitting 2E of the ODMR resonances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' At p0, ND1 and INV1 agree well on D but ND1 has a larger E than INV1, indicating a larger intrinsic lattice distor- tion in ND1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' When DAC1 is pressurized to p2, ND1 shows a greater rightward shift in D while INV1 exhibits a more noticeable increase in E, and such differences in their spectral responses become more pronounced at p6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' These reveal that when we press the diamond anvils to- 5 (a) (b) (c) (d) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' (a, b) Plots of SD of pressure and average Enet against average pressure for three (five) NDs and six (six) patches of INVs in DAC1 (DAC2), where Enet is the net change in E with respect to the ambient value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' The two DACs reveal very similar data trends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' The NDs only show a tiny increase in SD while the INVs have no observable SD at all, signifying the pressure homogeneity at the pressure medium interface and ∼10 nm deep in the culet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' Moreover, the INVs reveal a much greater increase in average Enet, implying a more hydrostatic environment around the NDs and a more anisotropic environ- ment around the INVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' Here, the markers are joined in a way to indicate the pressure sequences in the experiments, while the purple arrows drawn are to emphasize the significantly different behaviors of NDs and INVs at the highest pressure point achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' (c, d) The differences in average D, aver- age P, and average Enet between the three NDs and the six patches of INVs in DAC1 along the pressure sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' In gen- eral, the NDs experience much stronger hydrostatic pressure compared with the INVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' For all subfigures, one of the three NDs in DAC1 is replaced by another ND for the statistics at p4 and p6, due to the occasionally weakened fluorescence of those NDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' wards each other, ND1 experiences stronger local pres- sure from a more hydrostatic environment at the pressure medium interface, while INV1 is subjected to weaker lo- cal pressure from a more directional stress environment inside the anvil culet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' The stress anisotropy around INV1 may have produced a symmetry breaking between the two ground-state sublevel transitions, as seen from the increasingly unequal contrasts of the two ODMR reso- nances at p2 and p6 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' 1(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' On the other hand, both ND1 and INV1 show decreases in D and E at the reduced pressure point p4, reflecting the expected stress relaxation when we loosen the diamond anvils.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' Note that the decline in ODMR contrasts for ND1 and INV1 is due to the stress-induced mixing of NV spin states and the degradation of MW structure, where the latter factor takes a heavier toll on ND1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' Apart from the artifact of the deteriorated MW structure, all the mentioned main features in the ODMR responses of the two NV sensor types can be reproduced in the independent experiment with DAC2 (see Supplementary Materials).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' To go beyond describing the raw spectra, we perform statistical comparisons of the local environments per- ceived by NDs and INVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' 2(a) and (b), we plot the standard deviation (SD) of pressure and average Enet against average pressure for the tracked NDs and the tracked patches of INVs in DAC1 and DAC2, where Enet is the measured E offset by the ambient value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' It is evi- dent that the two DACs give rise to very similar results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' First, the NDs only show a tiny increase in the SD of pres- sure while the patches of INVs have no observable SD at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' This suggests we have highly homogeneous pressure at both the medium interface and ∼10 nm deep in the culet, and the small SD from the NDs also hints at an ex- cellent hydrostatic condition of the pressure medium (4:1 methanol:ethanol mixture) within the pressure range un- der investigation [20, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' Second, the average pressure detected by the NDs becomes increasingly greater than that detected by the patches of INVs, and the patches of INVs have a much more remarkable increase in the aver- age Enet compared with the NDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' These statistics verify our previous inference that a more hydrostatic environ- ment exists at the medium interface to produce stronger local pressure, while a more anisotropic environment ex- ists inside the anvil culet to give weaker local pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' Third, at the reduced pressure points, the NDs and the patches of INVs show much smaller differences in the av- erage pressure, SD of pressure, and average Enet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' This implies relaxation of the DAC may tend to “unify” the pressurized environments at the medium interface and inside the culet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' Note that for DAC2, the data of NDs at p1 may be affected by the instability of the pressure medium due to insufficient buffer time between the pres- surization of the DAC and measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' As a more direct comparison to supplement the above discussions, we plot in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' 2(c) and (d) the quantitative differences in average D, average pressure P, and average Enet be- tween the NDs and the patches of INVs in DAC1 along the pressure sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' QUANTITATIVE STRESS TENSOR ANALYSIS In this section, we will consider net effects of the DAC pressure on the stress tensors of the two NV sensor types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' First, we assume the stress tensors induced by the DAC pressure to be quasi-hydrostatic, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' a hydrostatic pres- sure P plus a first-order correction from a uniaxial stress of magnitude λP along the DAC axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' This assumption is intuitive since any non-hydrostaticity in the DAC is likely to arise from the symmetry breaking due to the external force applied along the DAC axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' Under this assumption, the crystal-frame stress tensor of an ND can be written as σND = � � P 0 0 0 P 0 0 0 P � � + U T n � � λP 0 0 0 0 0 0 0 0 � � Un, (10) DAC1NDs 4 DAC1NVs (kbar) DAC2 NDs DAC2 NVs 2 D S 0 12 (ZHW) 8 4 V 0 20 40 60 80 0 Average pressure (kbar)30 20_ 一DAC1 台 M 20 V 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE5T4oBgHgl3EQfKA5U/content/2301.05462v1.pdf'} +page_content=' P 10 D VV >i V (kbar) ND 0